modelling and analysing moving objects and travelling subjects
TRANSCRIPT
Modelling and Analysing
Moving Objects and Travelling Subjects
Bridging theory and practice
Matthias Delafontaine
To the memory of my grandfather Paul
Copyright © Matthias Delafontaine, Department of Geography, Faculty of Sciences, Ghent
University, 2011. All rights reserved. No part of this publication may be reproduced, stored
in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical,
photocopying, recording, or otherwise, without permission in writing from the proprietor(s).
ISBN: 978-94-90695620
Legal deposit: D/2011/12.134/12
NUR: 755/905/983/984
The research reported in this dissertation was conducted at the CartoGIS research unit,
Department of Geography, Faculty of Sciences, Ghent University, and funded by the
Research Foundation – Flanders.
Faculty of Sciences
Department of Geography
Modelling and Analysing
Moving Objects and Travelling Subjects
Bridging theory and practice
Dissertation submitted in accordance with the requirements for the degree of
Doctor of Sciences : Geography
Modelleren en Analyseren
van Bewegende Objecten en Personen die Zich Verplaatsen
Een brug tussen theorie en praktijk
Proefschrift aangeboden tot het behalen van de graad van
Doctor in de Wetenschappen : Geografie
by / door
Matthias Delafontaine
Supervisors
Prof. Dr. Nico Van de Weghe
Ghent University
Dr. Tijs Neutens
Ghent University
Members of the Reading Committee
Prof. Dr. Robert Weibel
University of Zurich
Prof. Dr. Roland Billen
University of Liege
Prof. Dr. Frank Witlox
Ghent University
Remaining members of the Examination Committee
Prof. Dr. Anthony G. Cohn
University of Leeds
Prof. Dr. Christophe Claramunt
Naval Academy Research Institute
Prof. Dr. Ben Derudder
Ghent University
Prof. Dr. Philippe De Maeyer
Ghent University
“And crawling on this planet's face, some insects called the human race.
Lost in time. And lost in space.” (The Rocky Horror Picture Show)
Table of contents I
Table of contents
Preface ................................................................................................................. VII
List of figures ......................................................................................................... IX
List of tables ......................................................................................................... XIII
List of algorithms .................................................................................................. XV
1 Introduction ..................................................................................................... 1
1.1 Background and motivation ....................................................................................... 1
1.2 Rationale and synopsis ............................................................................................... 4
References ............................................................................................................................ 10
Part I – Moving Objects ................................................................................. 19
2 A Qualitative Trajectory Calculus to reason about moving point objects ......... 21
2.1 Introduction .............................................................................................................. 21
2.2 Background ............................................................................................................... 22
2.3 The Qualitative Trajectory Calculus ......................................................................... 22
2.3.1 Simplifications ...................................................................................................... 22
2.3.2 Continuity, conceptual neighbours, and transitions ............................................ 23
2.3.3 Types of QTC ......................................................................................................... 24
2.4 QTC – Basic (QTCB) ................................................................................................... 25
2.5 QTC – Double Cross (QTCC) ...................................................................................... 27
2.6 Representing and reasoning with QTC ..................................................................... 31
2.6.1 Conceptual neighbourhood diagrams .................................................................. 31
2.6.2 Composition tables ............................................................................................... 33
2.6.3 Incomplete knowledge ......................................................................................... 36
2.7 Extending QTC .......................................................................................................... 37
2.7.1 Multiple MPOs ...................................................................................................... 38
2.7.2 Multiple time points and intervals ....................................................................... 38
2.7.3 Multiple topological relations .............................................................................. 38
2.8 Example case ............................................................................................................ 39
2.9 Future research directions ....................................................................................... 41
2.10 Conclusion ................................................................................................................ 41
References ............................................................................................................................ 42
3 Inferring additional knowledge from QTCN relations ....................................... 45
3.1 Introduction .............................................................................................................. 45
3.2 Qualitative versus quantitative questions ............................................................... 46
3.3 The Qualitative Trajectory Calculus ......................................................................... 47
II
3.4 The Qualitative Trajectory Calculus on Networks .................................................... 49
3.4.1 Definitions and restrictions concerning networks and moving objects ............... 49
3.4.2 Definition of QTCN relations ................................................................................. 53
3.5 Composition ............................................................................................................. 56
3.5.1 Composition of QTCN relations ............................................................................. 57
3.5.2 Temporal Constraints ........................................................................................... 57
3.5.3 Spatial Constraints ............................................................................................... 58
3.6 Transforming QTCN into the Relative Trajectory Calculus on Networks .................. 60
3.7 Discussion ................................................................................................................. 65
3.7.1 A Police/Gangster Example .................................................................................. 65
3.7.2 A Collision Avoidance Application ........................................................................ 67
3.8 Conclusions and future work ................................................................................... 68
References ............................................................................................................................ 69
4 Qualitative relations between moving objects in a network changing its
topological relations ............................................................................................. 73
4.1 Introduction .............................................................................................................. 73
4.2 The Qualitative Trajectory Calculus ......................................................................... 74
4.3 The Qualitative Trajectory Calculus on Networks .................................................... 74
4.3.1 Definition .............................................................................................................. 74
4.3.2 Relations in QTCN .................................................................................................. 76
4.3.3 Transitions in QTCN ............................................................................................... 77
4.3.4 Theory of Dominance ........................................................................................... 79
4.4 Topological changes in networks: QTCDN’ ................................................................ 80
4.4.1 Topological Change and Dynamic Networks ........................................................ 80
4.4.2 Relations in QTCDN’ ............................................................................................... 81
4.4.3 Transitions in QTCDN’ ............................................................................................. 81
4.5 Conclusions and Future work ................................................................................... 82
References ............................................................................................................................ 85
5 Implementing a qualitative calculus to analyse moving point objects ............. 87
5.1 Introduction .............................................................................................................. 87
5.2 The Qualitative Trajectory Calculus (QTC) ............................................................... 89
5.2.1 Types of QTC ......................................................................................................... 89
5.2.2 Unconstrained movement .................................................................................... 90
5.3 A QTC-based information system ............................................................................ 92
5.3.1 Trajectory representations ................................................................................... 92
5.3.2 Conceptual model ................................................................................................. 92
5.3.3 Implementation prototype ................................................................................... 94
5.4 Case studies .............................................................................................................. 97
Table of contents III
5.4.1 Cars on a street .................................................................................................... 97
5.4.2 Squash rally ........................................................................................................ 100
5.5 Discussion ............................................................................................................... 101
5.6 Conclusions and outlook ........................................................................................ 104
References .......................................................................................................................... 105
Appendix A ......................................................................................................................... 109
Appendix B ......................................................................................................................... 110
6 Modelling moving objects in geospatial sketch maps ................................... 113
6.1 Introduction ............................................................................................................ 113
6.2 Extended Sketch Maps ........................................................................................... 114
6.3 Moving objects in geospatial sketch maps ............................................................ 116
6.3.1 Moving point objects and geospatial lifelines .................................................... 116
6.3.2 Lifeline glyphs ..................................................................................................... 118
6.3.3 Typology of lifeline representations ................................................................... 121
6.3.4 Multiple lifelines ................................................................................................. 121
6.4 Conclusions and outlooks ....................................................................................... 122
References .......................................................................................................................... 123
Part II – Travelling subjects .......................................................................... 125
7 Analysing spatiotemporal sequences in Bluetooth tracking data .................. 127
7.1 Introduction ............................................................................................................ 127
7.2 Sequence Alignment Methods ............................................................................... 129
7.2.1 Background......................................................................................................... 129
7.2.2 Methodology ...................................................................................................... 129
7.3 Case study .............................................................................................................. 131
7.3.1 Data collection .................................................................................................... 131
7.3.2 Data preparation ................................................................................................ 133
7.3.3 Sequence alignment ........................................................................................... 135
7.3.4 Results ................................................................................................................ 136
7.4 Conclusions ............................................................................................................. 140
References .......................................................................................................................... 141
8 Modelling potential movement in constrained travel environments using rough
space–time prisms .............................................................................................. 145
8.1 Introduction ............................................................................................................ 145
8.2 Background ............................................................................................................. 147
8.3 A space-time prism in an unconstrained travel environment ............................... 149
8.4 A rough space-time prism in an unconstrained travel environment ..................... 151
8.5 A space-time prism in an obstacle-constrained travel environment ..................... 153
IV
8.6 A rough space-time prism in an obstacle-constrained travel environment .......... 159
8.6.1 Combination of approaches ............................................................................... 159
8.6.2 Algorithm ............................................................................................................ 161
8.6.3 Example .............................................................................................................. 163
8.7 Conclusions ............................................................................................................. 166
References .......................................................................................................................... 168
9 Reconciling place-based and person-based accessibility: a GIS toolkit .......... 171
9.1 Introduction ............................................................................................................ 171
9.2 Related tools........................................................................................................... 173
9.3 PrismMapper .......................................................................................................... 175
9.3.1 Accessibility measures ........................................................................................ 175
9.3.2 System ................................................................................................................ 179
9.4 Example case .......................................................................................................... 183
9.5 Conclusion .............................................................................................................. 190
References .......................................................................................................................... 191
10 The relationship between opening hours and accessibility of public service
delivery ............................................................................................................... 195
10.1 Introduction ............................................................................................................ 195
10.2 Space-time demands, opening hours and accessibility ......................................... 197
10.3 Method ................................................................................................................... 199
10.3.1 Measuring accessibility .................................................................................. 199
10.3.2 Optimising opening hours in terms of accessibility ........................................ 202
10.4 Case study .............................................................................................................. 206
10.4.1 Data ................................................................................................................ 206
10.4.2 Data preparation ............................................................................................ 211
10.4.3 Results ............................................................................................................ 212
10.5 Conclusion and avenues for future work ............................................................... 217
References .......................................................................................................................... 219
11 The impact of opening hours on the equity of individual space-time
accessibility ......................................................................................................... 225
11.1 Introduction ............................................................................................................ 225
11.2 Method ................................................................................................................... 227
11.2.1 Scheduling procedure ..................................................................................... 227
11.2.2 Equity approaches .......................................................................................... 228
11.3 Case study .............................................................................................................. 229
11.3.1 Input data ....................................................................................................... 230
11.3.2 Evaluation functions and computation .......................................................... 236
Table of contents V
11.3.3 Results ............................................................................................................ 238
11.4 Conclusion .............................................................................................................. 249
References .......................................................................................................................... 250
12 Discussion and conclusions ....................................................................... 253
References .......................................................................................................................... 263
Samenvatting (Dutch summary) .......................................................................... 268
References .......................................................................................................................... 271
Biographical sketch ............................................................................................. 273
Preface VII
Preface
Writing a doctoral dissertation has been a long journey, and it is true: you can never finish a
PhD, you just quit working on it. Nevertheless, the compilation of four years of intense
endeavour and engagement somehow releases me and offers me the opportunity to look
back. Rather than as a static collection of scientific achievements, my dissertation appears to
me as a chronology reflecting in many respects the evolution that my research and the way I
dealt with it have gone through. In that sense, I am elated by the impression of having
acquired and developed certain skills that are much more universal than the sheer context
of research.
Beyond my personal movement, much has happened and changed in my environment over
the course of this study. Of these events, probably the most drastic one has been the loss of
my dear grandfather Paul Hubert, to whom I owe great respect. To heartily acknowledge
him, I have dedicated this work to his fond memory.
Pursuing a PhD is often regarded as a lonesome and isolating experience. I do recognize this
perception and I must admit that numerous solitary activities may have enabled me to find
the necessary reflection. Yet, I could by no means have carried out this research without the
support of many others. Whereas it would lead me too far to mention each name
exhaustively, I would like to express my sincere appreciation to all the people that have
thereby assisted me in any respect. Let me explicitly address some of them.
To begin with, I wish to thank my supervisors Nico Van de Weghe and Tijs Neutens for their
admirable academic guidance and for their willingness to be always available for me. Nico
Van de Weghe was the person who picked me up after graduating in geography and
convinced me to apply for a doctoral grant. I have come to know him as a reliable, pure and
devoted scientist. I am grateful to him for giving me the opportunity to conduct doctoral
research and to offer me the necessary academic freedom. I greatly appreciate his openness
to discuss and explore fresh ideas and his courage to face novel research challenges.
I also owe much gratitude to my other supervisor, Tijs Neutens. He has been the person with
the unparalleled ability to constantly motivate and inspire me, even in the more awkward
circumstances. I highly commend his great enthusiasm, solidity and diligence, but also his
fine research skills and abundant sense of humour which have made working with him a
great pleasure.
Furthermore, I am indebted to Anthony Cohn for his genuine interest in and contribution to
this research. Our fruitful collaboration has benefited from his kind and honest attitude, his
constructive and discerning comments, and his excellent academic advice.
VIII
In addition, I would like to thank my former academic guide and colleague Peter Bogaert for
his obligingness, open-hearted support and contribution to this work. My grateful
acknowledgment goes to five other contributors as well: thank you Tim Schwanen, Philippe
De Maeyer, Frank Witlox, Mathias Versichele, and Hossein Chavoshi.
Besides the already named, my deep appreciation further extends to other colleagues at the
Department of Geography at Ghent University – especially to our matchless secretary Helga
Vermeulen and to my colleague assistants. Moreover, I gratefully recognize the Research
Foundation – Flanders for funding this research.
Beyond the academic context, I have been strongly supported by my friends and family.
Special thanks to my friends for the many gorgeous and memorable moments of distraction
that gave me precious time to breath and kept my mind and – to a lesser extent also my
body – in good shape.
The most unconditional and all-embracing help and assistance come from my family. Thanks
ever so much to my grandmother Germaine, my parents Ann and Joris, Martine and Hendrik,
my sisters and brothers Ruth and Bram, Pieter and Kimberly.
Last but not least, I must acknowledge my fiancée and best friend. Lien, thanks awfully for
taking care of me and offering me your unqualified love, encouragement and understanding.
I’ll cherish our wonderful years in Ghent as a dear memory and I look forward to marrying
you and continuing our life in Torhout.
Ghent, March 15, 2011.
List of figures IX
List of figures
Figure 2.1 – Simplification in QTC of a real-life situation (a) by taking cumulatively account of
the relational simplification (b), the object simplification (c), and the temporal
simplification (d) (simple arrows for trajectories, double arrows for instantaneous
velocity vectors). .......................................................................................................... 23
Figure 2.2 – Two MPOs represented in a typical two-dimensional QTCB (a), QTCC (b), and
QTCN (c) setting. The frame of spatial reference is represented by the dashed line... 25
Figure 2.3 – QTCB1 (a), and QTCB2 (b) relation icons. ............................................................... 27
Figure 2.4 – Different use of the double cross in the Double Cross Calculus (Galton 2001) (a),
and the QTCC calculus (b). ............................................................................................ 28
Figure 2.5 – QTCC1 relation icons. ............................................................................................. 29
Figure 2.6 – QTCC2 relations and the minimal number of spatial dimensions supporting them:
respectively the dotted, dashed, and straight boundaries for one, two, and three
dimensions.................................................................................................................... 30
Figure 2.7 – CNDs for QTCB1 in n-dimensional space (a), for QTCB2 in a one-dimensional space
(b), and for QTCB2 in a two- or higher-dimensional space (c). The straight, dashed and
dotted lines respectively represent the conceptual distances one, two and three. ... 31
Figure 2.8 – CND for QTCC1 in a two-or higher-dimensional space. Links have been gray-
shaded according to the conceptual distance between the adjacent relations. ......... 32
Figure 2.9 – Configuration of two cars k and l at sample time stamps during an overtake
event. ............................................................................................................................ 40
Figure 3.1 – Bifurcating (a) and non-bifurcating (b) shortest paths. ....................................... 52
Figure 3.2 – A shortest path omitting node pass event. ......................................................... 52
Figure 3.3 – 57 Canonical cases for QTCN at level 2. ................................................................ 55
Figure 3.4 – Animations for the composition of (+ −) and (− 0); a movement arrow next to
an object indicates that the object is passing a node. ................................................. 57
Figure 3.5 – Possible relative movement configurations in QTCN for R1(k, l) R2(l, m) where
m lies on the simple shortest path between k and l and none of the objects is located
at a node. ...................................................................................................................... 59
Figure 3.6 – Possible relative movement configurations in QTCN for R1(k, l) R2(l, m) where k
lies on the simple shortest path between m and l and none of the objects is located at
a node. .......................................................................................................................... 60
Figure 3.7 – A transition in QTCN from (− 0 +) via (0 0 +) to (+ 0 +). ............................ 64
Figure 3.8 – Examples of transformations from QTCB to RTC. ................................................. 65
Figure 3.9 – Simplified animation of three policemen chasing a gangster. ............................. 66
Figure 3.10 – Two scenes without collision danger for two moving objects. .......................... 67
Figure 4.1 – Speed Change Event. ............................................................................................ 78
Figure 4.2 – Node Pass Event. .................................................................................................. 78
X
Figure 4.3 – Continuous Shortest Path Change Event. ............................................................ 78
Figure 4.4 – The CND of QTCN. ................................................................................................. 79
Figure 4.5 – Discontinuous Shortest Path Change Event. ........................................................ 82
Figure 4.6 – Transition in QTCDN’ due to a combination of a Discontinuous Shortest Path
Change Event with a Node Pass Event. ........................................................................ 82
Figure 4.7 – Possible QTCDN’ transitions due to a Discontinuous Shortest Path Change Event
(combined and/or otherwise) (a) and due to the combination of a Discontinuous
Shortest Path Change Event with a Speed Change Event (b)....................................... 83
Figure 4.8 – Possible QTCDN’ transitions due to a combination of a Discontinuous Shortest
Path Change Event with a Node Pass Event or (exclusive) with a Continuous Shortest
Path Change Event........................................................................................................ 83
Figure 4.9 – Possible QTCDN’ transitions due to the combination of a Speed Change Event
with a combined Discontinuous Shortest Path Change Event (a) and due to the
combination of a Speed Change Event with the combination of a Discontinuous
Shortest Path Change Event and a Node Pass Event or (exclusive) a Continuous
Shortest Path Change Event (b). .................................................................................. 84
Figure 5.1 – Two MPOs represented in a typical QTCB (a), QTCC (b), and QTCN (c) setting. The
frame of spatial reference is represented by the dashed line. .................................... 89
Figure 5.2 – Properties of two MPOs k and l at a time instant t. ............................................. 90
Figure 5.3 – UML class diagram for a QTC-based information system .................................... 93
Figure 5.4 – A continuous MPO trajectory (a) and a representation of it according to
Assumption 5.1 with fixes (crosses) per second (b). .................................................... 95
Figure 5.5 – Schematic sketch of the study area. .................................................................... 97
Figure 5.6 – Duration (gray bars) and frequency (black bars) for 24 fourth order simple
permutable patterns in QTC-C22. .............................................................................. 101
Figure 5.7 – Trajectories of two objects k and l during a time interval for two situations (A
and B) according to two representations: realistic representation (a), (c);
representation satisfying Assumptions 5.1-5.3 (b), (d). Crosses represent fixes. ..... 103
Figure 6.1 – ER diagram of the extended sketch map ontology. ........................................... 116
Figure 6.3 – Map of a geospatial lifeline of a butterfly moving from A to B, passing flowers on
its way (own illustration after (Laube 2005)). ............................................................ 117
Figure 6.2 – Single-stroke glyph drawn in CogSketch (Forbus et al. 2008). ........................... 117
Figure 6.4 – A Shrewd Sketch Interpretation and Simulation Tool (ASSIST) (Davis 2002). ... 118
Figure 6.5 – Sketch map representations of the butterfly lifeline in Figure 6.3: explicit single-
stroke lifeline glyph (a), explicit multi-stroke lifeline glyph (b), explicit multi-stroke
lifeline glyph (c), implicit representation by means of six flower glyphs and four arrow
glyphs. ......................................................................................................................... 119
Figure 6.6 – Typology of lifeline representations in geospatial sketch maps. ....................... 121
Figure 7.1 – Pairwise alignment. ............................................................................................ 130
List of figures XI
Figure 7.2 – Schematic map of Flanders Expo with indication of entrances and exits for
visitors (arrows), exhibition halls (H1-H8, black rectangles), and Bluetooth nodes (A-T,
x-marks) with 20m radio range (black circles). .......................................................... 132
Figure 7.3 – Distribution of Bluetooth device classes across observed devices .................... 133
Figure 7.4 – Histogram of observed days per device ............................................................. 133
Figure 7.5 – Histogram of device-day duration. ..................................................................... 133
Figure 7.6 – Extract of transcoded Bluetooth sequences. ..................................................... 135
Figure 7.7 – Sequence alignment scoring matrix. .................................................................. 136
Figure 7.8 – Multiple alignment guide tree with clusters and subclusters labeled at their root
node. ........................................................................................................................... 137
Figure 7.9 – Extract of the sorted and colour coded multiple alignment. ............................. 138
Figure 8.1 – Space-time prism obtained from the intersection of a forward cone and a
backward cone. .......................................................................................................... 150
Figure 8.2 – An uncertain space-time prism modelled by its lower (grey), and upper (black
outlines) approximation. ............................................................................................ 153
Figure 8.3 – Travel environment constrained by university buildings A, B, and C. ............... 154
Figure 8.4 – Shortest paths (black lines) from the origin (big dot) to all obstacle vertices
(small dots). ................................................................................................................ 156
Figure 8.5 – Shortest paths (black lines) from the destination (big dot) to all obstacle vertices
(small dots). ................................................................................................................ 156
Figure 8.6 – Parent forward reachability body (grey) with indication of the parent vertex
(black dot) and the spatial extrusion zones (black outlines). ..................................... 158
Figure 8.7 – Parent backward reachability body (grey) with indication of the parent vertex
(black dot) and the spatial extrusion zones (black outlines). ..................................... 159
Figure 8.8 – Obstacle-constrained space-time prism (grey) with indication of obstacles
(black). ........................................................................................................................ 160
Figure 8.9 – Obstacle-constrained lower space-time prism (grey) with indication of obstacles
(black). ........................................................................................................................ 165
Figure 8.10 – Obstacle-constrained upper space-time prism (grey) with indication of
obstacles (black). ........................................................................................................ 165
Figure 8.11 – Cross section through time of lower (dark grey) and upper (light grey) prisms
along the axis origin (o) – destination (d), with indication of vertical obstacle
extrusions (white rectangles). .................................................................................... 166
Figure 9.1 – Space-time prism and related concepts. ............................................................ 176
Figure 9.2 – Cross section through space-time of a STP (left) and a RSTP (right). ................ 177
Figure 9.3 – PrismMapper system architecture. .................................................................... 179
Figure 9.4 – PrismMapper workflow. ..................................................................................... 182
Figure 9.5 – PrismMapper main application window. ........................................................... 183
Figure 9.6 – Public libraries in Ghent (Belgium). .................................................................... 184
Figure 9.7 – Map of ACCESS on Monday. ............................................................................... 185
XII
Figure 9.8 – Map of ACCESS on Tuesday. ............................................................................... 185
Figure 9.9 – Map of CUMF on Monday. ................................................................................. 186
Figure 9.10 – Map of CUMF on Tuesday. ............................................................................... 186
Figure 9.11 – Map of MINT on Monday. ................................................................................ 187
Figure 9.12 – Map of MINT on Tuesday. ................................................................................ 187
Figure 9.13 – Map of MINTF on Monday. .............................................................................. 188
Figure 9.14 – Map of MINTF on Tuesday. .............................................................................. 188
Figure 9.15 – Map of MAXD on Monday. ............................................................................... 189
Figure 9.16 – Map of MAXD on Tuesday. ............................................................................... 189
Figure 10.1 – Cross section through space (horizontal axis) and time (vertical axis) of the
space-time prism (grey) between fixed activities xj and xj+1 of an individual i, with the
indication of the PAW with respect to the opening hour interval hk of service facility f.
.................................................................................................................................... 200
Figure 10.2 – Study area and sampled households. .............................................................. 207
Figure 10.3 – Spatial distribution of government offices. ..................................................... 207
Figure 10.4 – Estimation of distance decay parameters. ....................................................... 211
Figure 10.5 – Total accessibility for all 900 optimal regimes with indication of the current
regime. ........................................................................................................................ 213
Figure 11.1 – Public libraries in Ghent. .................................................................................. 230
Figure 11.2 – Sampled households and population density in Ghent. .................................. 234
Figure 11.3 – Composition of the lower and upper halves in terms of employment status. 238
Figure 11.4 – Box-and-Whisker diagrams of the accessibility level per regime. ................... 247
Figure 11.5 – Theil index of the accessibility level per regime. ............................................. 248
Figure 11.6 – Box-and-Whisker diagrams per regime for the lower (left) and upper (right)
halves of the population. ............................................................................................ 248
List of tables XIII
List of tables
Table 1.1 – Manuscripts included in the dissertation. ............................................................... 5
Table 2.1 – The number of base relations, transitions, theoretical combinations of base
relations, and the ratio transitions / theoretical combinations for the Basic and
Double Cross QTC calculi. ............................................................................................. 33
Table 2.2 – CT for QTCB1 in a one-dimensional space. ............................................................. 34
Table 2.3 – CT for the speed constraint. .................................................................................. 35
Table 2.4 – CRT for QTCC1 in a two-dimensional space. ........................................................... 36
Table 2.5 – Intersection of coarse solutions to obtain fine knowledge, with U0 = U \{0}. ...... 37
Table 2.6 - QTCB1 matrix for four MPOs k, l, m, and n at time t. .............................................. 38
Table 2.7 – Trajectory sample points of two cars k and l during an overtake event. .............. 39
Table 3.1 – Composition table for QTCN at level 1 restricted to relations lasting over time
intervals; A0 and B0 stand for the set {−, +}. ................................................................. 58
Table 3.2 – Composition table for relative movement in QTCN, for R1(k, l) R2(l, m) where m
lies on the simple shortest path between k and l and none of the objects is located at
a node. .......................................................................................................................... 59
Table 3.3 – Composition table for relative movement in QTCN, for R1(k, l) R2(l, m) where k
lies on the simple shortest path between m and l and none of the objects is located at
a node. .......................................................................................................................... 60
Table 3.4 – Transformations from all QTCN canonical cases to RTCN relations. ...................... 64
Table 3.5 – Transformations from QTCN into RTCN relations. .................................................. 65
Table 3.6 – Composition results inferred over [t1, t3] due to spatial and temporal constraints.
...................................................................................................................................... 66
Table 5.1 – Relation syntax for QTCB and QTCC subtypes. ....................................................... 92
Table 5.2 – Transition table for QTC relation symbols at transition instant t ......................... 96
Table 5.3 – Summary of QTC-B21 relations with their cumulative instant, interval, and total
frequencies, and duration for 503 car pairs. ................................................................ 98
Table 5.4 – Summary of QTC-B22 relations with their cumulative instant, interval, and total
frequencies, and duration for 503 car pairs. ................................................................ 99
Table 5.5 – Complete sequence, transition time and duration of QTC-C21 relations between
two squash opponents during a rally lasting 37 s. ..................................................... 112
Table 6.1. – Export of spatial and temporal ink of the stroke in Fig. 2 as a set of timestamped
polyline vertices. ......................................................................................................... 117
Table 7.1 – Number of members and common patterns per cluster. Pattern episodes are
colour coded to hall location and annotated with hall numbers or node characters.
Hollow episode symbols represent episodes at one of the eight exhibition halls. ... 139
Table 7.2 – Median (top) and average (bottom) sequence per cluster. ................................ 140
Table 8.1 – Space-time prism volumes in m².s according to four different scenarios. ......... 164
XIV
Table 10.1 – Locational benefit calculation example ............................................................. 202
Table 10.2 – Current regime of opening hours for the government offices in Ghent (1-15). 208
Table 10.3 – Congestion factor according to day type, day time and road class. ................. 210
Table 10.4 – Optimal 405-hour regime. ................................................................................. 214
Table 10.5 – Contiguous sub-optimal 405-hour regime. ....................................................... 216
Table 11.1 – Library collection size (2010) and attractiveness estimate. .............................. 231
Table 11.2 – Opening hours of public libraries in Ghent ....................................................... 231
Table 11.3 – Utilitarian regime of 209 opening hours, with indication of the allocation order
of each hour in the scheduling procedure. Allocated hours are gray-scaled according
to an equal interval classification into five classes of the allocation order. .............. 240
Table 11.4 – Egalitarian regime of 209 opening hours, with indication of the allocation order
of each hour in the scheduling procedure. Allocated hours are gray-scaled according
to an equal interval classification into five classes of the allocation order. .............. 242
Table 11.5 – Distributive regime of 209 opening hours, with indication of the allocation order
of each hour in the scheduling procedure. Allocated hours are gray-scaled according
to an equal interval classification into five classes of the allocation order. .............. 244
Table 12.1 – Main and application-oriented contributions. .................................................. 254
List of algorithms XV
List of algorithms
Algorithm 8.1 – Main algorithm for computation of rough obstacle-constrained space-time
prisms. ........................................................................................................................ 162
Algorithm 10.1 – Computational procedure to determine the optimal n-MOI regime. ....... 203
Algorithm 10.2 – Computational procedure to determine the (sub)optimal connected n-MOI
regime. ........................................................................................................................ 205
Algorithm 11.1 – Iterative scheduling procedure. ................................................................. 227
Introduction 1
1 Introduction
1.1 Background and motivation
The act of moving through geographical space takes an essential and inherent part in the
daily life of human beings, animals, goods, and data. The rationale that motion is a
fundamental and omnipresent phenomenon which by definition relates space to time –
which in turn equals capital – feeds a general and everlasting scientific interest in modelling
and analysing moving entities. Moving objects and travelling subjects therefore constitute a
principal unit of analysis in many major domains of both theoretical and applied scientific
research, including artificial intelligence, behavioural sciences and ethology, geographical
information science (GIScience), knowledge representation, robotics, sports science, and
transportation and operations research. This scientific versatility has produced a broad
variety of theories, methodologies, applications, and technologies to collect, explore,
represent, reason about, analyse and extract information from data about moving objects.
This dissertation intends to contribute to these scientific developments – those in GIScience
in particular – with the implicit aim to close the gap between theory and practice by
implementing fundamental theoretical knowledge into practicable applications.
Over the past decades, technological evolutions have importantly re-established the topic of
moving objects as an active and even cutting-edge research issue. First of all, the
development of increasingly advanced means of transport has triggered the movement of
increasing volumes at increasing speeds over increasing distances. Second, major advances
have been made in technologies for positioning and tracking moving objects. The
establishment of satellite navigation systems in the 1970s, or rather the public disclosure of
the use of the Global Positioning System (GPS) in 1983, have been milestones in the
evolution of tracking systems enabling the collection of detailed trajectory data. They
heralded a worldwide production and distribution of mobile position-aware devices able to
determine and store their trajectories over time. GPS units are nowadays commonly
integrated in navigation systems of motorised vehicles as well as in portable appliances such
as smart phones. Apart from satellite navigation systems, other tracking technologies have
recently arisen, some of which complement the capabilities of GPS tracking. These include,
among others, video surveillance systems, mobile positioning systems, wireless tracking
systems, and radio-frequency identification (RFID). Third, along with the progress in tracking
and positioning systems, rapid advances have been made in information and communication
technology (ICT) in general, which support information systems to store, manipulate,
process, query and communicate increasingly larger data volumes. In particular, the
emergence and democratisation of geographical information systems (GIS) capable of
handling spatiotemporal information has been significant in this respect.
2 Chapter 1
This dissertation fits in the broad scope of GIScience. As for geography, geographical
information and GIS1, GIScience has been defined in many different ways. According to
Goodchild (2010), who initiated the term (Goodchild 1992), perhaps the most
comprehensive and meanwhile fairly succinct definition was published by Mark (2003, p. 1-
2) and adopted by the University Consortium for Geographic Information Science (UCGIS):
“The development and use of theories, methods, technology, and data for understanding
geographic processes, relationships, and patterns.” GIScience is generally recognised as the
science behind GISs, which can be described as information systems that integrate
hardware, software, and data for capturing, managing, analysing and displaying geographical
information (i.e. information with a geographical reference). Research in GIScience
concerning moving objects has, given the spatiotemporal nature of motion, followed a more
general and long-lasting research trend addressing the incorporation and exploration of the
temporal component of geospatial information, in order to support time-integrative or
genuine spatio-temporal GISs (Langran 1993, Peuquet 1994, Raper 2000, Ott & Swiaczny
2001). This evolution has been characterised by a major shift in the perception of
geoinformation as dynamic information – sometimes referred to as geo-temporal
information (O'Connor, Zerger & Itami 2005) – rather than a collection of static facts with a
fixed geospatial extent as represented through traditional maps. It has been generally
acknowledged that the modelling of time-dependent geographical information is unique to
GIS and therefore takes an integral part in GIScience which cuts across most of its other
topics (Goodchild 1992, Goodchild 2004, Mark 2003). In spite of the community’s consensus
on the important role of time for geographic information handling, the development of true
spatiotemporal GISs has progressed slowly (Laube 2005) and even continues today.
Together with the growing capabilities to collect data about moving objects, the integration
of time has initiated another vital shift in GIScience from a place-based to a person-based
perspective (Miller 2003, Miller 2007). According to the place-based perspective, attributes
are in the first place related to locations, which are by consequence the principal unit of
analysis to start from. The place-based perspective is predominant in geography and
cartography, and in turn in traditional GIScience and GISs. While definitely viable and
valuable, a growing need existed for a complementary perspective which places individuals,
rather than places, at the centrepoint. Especially the critique that many place-based
approaches tend to reduce the individual and his/her space-time behaviour to a set of
attributes associated to a static location (e.g. the individual’s residence), has fostered the
need for a person-based perspective. According to Miller (2007, p. 527), the person-based
perspective “focuses on individuals in space and time and their allocation of activities in the
physical and virtual worlds”. The growing attention to person-based approaches has
1 Already in the early years of GIS, there has been considerable ambiguity about its definition (e.g. Maguire
1991, Raper & Livingstone 1995, Dangermond 1988, Chan & Williamson 1997). A further discussion thereof is considered out of this dissertation’s scope.
Introduction 3
provoked a renewed interest into Hägerstrand’s famous work What about people in
Regional Science? (Hägerstrand 1970) and a significant revival of the classical time
geography building on his milestone oeuvre (Timmermans, Arentze & Joh 2002, Kwan 2002,
Kwan 2004, Levinson & Krizek 2005, Miller 2007, Couclelis 2009, Neutens, Schwanen &
Witlox 2011).
Both the above mentioned trends have, until present, importantly coloured GIScience in
general and the research on moving objects in particular. More than that, they are especially
reflected within this dissertation and intertwine its contributions in various ways (see section
1.2). Above all, they propelled the modelling and analysis of moving objects and travelling
subjects to the forefront of the GIScience research agenda. Perhaps this development is best
illustrated by the multitude of international research initiatives and publications dedicated
to the topic. First of all, numerous recent international meetings have brought together
leading world-class scientists in the field to exchange their knowledge and ideas. These
include seminars such as those at Dagstuhl (Bitterlich et al. 2008, Sack et al. 2010),
workshops (e.g. Van de Weghe et al. 2008, Gottfried et al. 2009, Billen et al. 2010), and
summer schools (e.g. http://mss2010.modap.org/). Second, a number of major international
projects, such as MOVE (http://www.move-cost.info), GeoPKDD (http://www.geopkdd.eu),
and MODAP (http://www.modap.org), support a more systematic research of the topic.
Third, an unlimited body of publications has explicitly addressed the topic of moving objects.
Without aiming for an exhaustive overview, some important themes within this very wealth
of contributions may be reported here. A first frequently researched issue concerns the
tracking of and collection of data about moving objects (e.g. Cohen & Medioni 1999, Aslam
et al. 2003, Tseng et al. 2003, Civilis, Jensen & Pakalnis 2005, O'Connor, Zerger & Itami 2005,
Yilmaz, Javed & Shah 2006, Shoval & Isaacson 2007b, Wang et al. 2007, Renso et al. 2008,
Shoval 2008, Ahas 2010). A second important theme is the modelling, storing, and querying
of moving objects in databases (i.e. moving objects databases or briefly MODs) (e.g. Hornsby
& Egenhofer 2002, Brakatsoulas, Pfoser & Tryfona 2004, Jensen, Lin & Ooi 2004, Wolfson &
Mena 2005, Rodríguez 2005, Güting & Schneider 2005, Güting, de Almeida & Ding 2006,
Saltenis et al. 2000, de Almeida & Güting 2005, Revesz 2010). Building on the development
of MODs, many research efforts have addressed knowledge discovery from and data mining
of moving objects data, including contributions on the extraction of clusters (e.g. Li, Han &
Yang 2004, Zhang & Lin 2004, Buzan, Sclaroff & Kollios 2004, Nanni & Pedreschi 2006,
Jensen, Lin & Ooi 2007, Rinzivillo et al. 2008), patterns (e.g. Du Mouza & Rigaux 2004, Laube,
Imfeld & Weibel 2005, Gudmundsson, van Kreveld & Speckmann 2007, Laube, Duckham &
Wolle 2008, Wilson 2008, Dodge, Weibel & Lautenschutz 2008, Demšar & Virrantaus 2010),
and similarity within moving objects data (e.g. Van Kreveld & Luo 2007, Pelekis et al. 2007,
Lin & Su 2008, Dodge, Weibel & Forootan 2009). In addition, visualisation techniques
supporting the analysis of moving objects have been extensively studied (e.g. Rinzivillo et al.
2008, Andrienko & Andrienko 2008, Andrienko et al. 2009, Willems et al. 2010, Andrienko et
4 Chapter 1
al. 2010). Finally, a significant share of publications is dedicated to the prediction and
simulation of moving objects and/or travelling agents (e.g. Elnagar & Gupta 1998, Bors &
Pitas 2000, Wahle & Schreckenberg 2001, Brinkhoff 2002, Ray & Claramunt 2003, Mostafavi
& Gold 2004, Chih-Yu & Yu-Chee 2004, Chen, Jin & Yue 2007, Shoshany, Even-Paz & Bekhor
2007, El-Geneidy, Krizek & Iacono 2007, Hoyoung et al. 2008).
1.2 Rationale and synopsis
This dissertation consists of a compilation of ten international peer-reviewed manuscripts
(Chapters 2-11) 2, each of which intends to fulfil two general objectives. The first objective
reads as follows:
Objective 1 To present an original contribution to GIScience and its potential for
modelling and analysing moving objects and travelling subjects in particular.
The first objective confirms the focus on GIScience and its research about modelling and
analysing moving objects. Furthermore, the objective implies a claim on the originality and
scientific contribution of each of the included manuscripts that has also been required by
their editors and/or publishers. One of the critiques on contributions in GIScience is that
they often negate the scientific responsibility of GIScience to underpin GISs and somehow let
GIScience exist as a science separated from its technologies, tools, practices, and users
(Pickles 1997, Aitken & Michel 1995, Elwood 2006, Leszczynski 2009). This discordance
between GIScience and GISs is, among others, reflected in the results of citation analyses
(e.g. Nelhans 2007) and the incompatibility of the vocabularies used in both domains
(Schuurman 2000). Therefore, on top of the first objective, a second objective is pursued:
Objective 2 To add to or enhance the practical usefulness of existing theoretical
contributions and thus facilitate closing gaps between science and
technology, and between theory and applications.
The second objective is particularly reflected through manuscripts that address the
implementation and/or empirical application of fundamental formalisms that have hitherto
remained largely theoretical.
An overview of the academic manuscripts included in this dissertation is provided in Table
1.1. The majority of these have been published or are forthcoming in an international peer
reviewed book or journal, while the others have been submitted for possible publication and
are under review at the time of writing. The content of the dissertation has been divided
into the chapters that deal with moving objects in general (Part I, Chapters 2-6), and the
2 The major part in the development of each manuscript is to be attributed to the first author, although for
chapter 10, it should be noted that the second author has addressed the entire implementation and experimental design.
Introduction 5
Cap Title Authors Outlet Status P
ar
t I
–
Mo
vin
g o
bj
ec
ts
2 A Qualitative Trajectory Calculus to reason about moving point objects
Delafontaine M. Chavoshi S. H. Cohn A. G. Van de Weghe N.
Qualitative Spatio-Temporal Representation and Reasoning: Trends and Future Directions (book chapter)
published 2011
3 Inferring additional knowledge from QTCN relations
Delafontaine M. Bogaert P. Cohn A. G. Witlox F. De Maeyer P. Van de Weghe N.
Information Sciences (journal article)
published 2011
4
Qualitative relations between moving objects in a network changing its topological relations
Delafontaine M. Van de Weghe N. Bogaert P. De Maeyer P.
Information Sciences (journal article)
published 2008
5
Implementing a qualitative calculus to analyse moving point objects
Delafontaine M. Cohn A. G. Van de Weghe N.
Expert Systems With Applications (journal article)
published 2011
6 Modelling moving objects in geospatial sketch maps
Delafontaine M. Van de Weghe N.
AGILE Workshop on Adaptation in Spatial Communication (conference paper)
published 2009
Pa
rt
II
– T
ra
ve
llin
g s
ub
je
ct
s 7
Analysing spatiotemporal sequences in Bluetooth tracking data
Delafontaine M. Versichele M. Neutens T. Van de Weghe N.
Environment and Planning B (journal article)
submitted
8
Modelling potential movement in constrained travel environments using rough space-time prisms
Delafontaine M. Neutens T. Van de Weghe N.
International Journal of Geographical Information Science (journal article)
in press
9 Reconciling place-based and person-based accessibility: a GIS toolkit
Delafontaine M. Neutens T. Van de Weghe N.
International Journal of Geographical Information Science (journal article)
submitted
10
The relationship between opening hours and accessibility of public service delivery
Neutens T. Delafontaine M. Schwanen T. Van de Weghe N.
Journal of Transport Geography (journal article)
in press
11
The impact of opening hours on the equity of individual space-time accessibility
Delafontaine M. Neutens T. Schwanen T. Van de Weghe N.
Computers, Environment and Urban Systems (journal article)
in press
Table 1.1 – Manuscripts included in the dissertation.
chapters that address travelling subjects, i.e. the travel behaviour of individuals (part II,
chapters 7-11).
6 Chapter 1
Part I – Moving objects
The first part of the dissertation considers moving objects in their most general form of
entities whose position or geometric attributes change over time. In this respect, moving
objects have often been modelled as moving points which are referred to as moving point
objects (MPOs). This representation is predominant in GIScience (Laube, Imfeld & Weibel
2005) and also underlies the research presented in Part I. MPOs are a purified
conceptualisation of moving objects that is appealing for two reasons. On the one hand, it
allows analysts to focus strictly on the movement of the entity at hand and abstract away
from other, often irrelevant, geometric properties that may otherwise obscure the analysis.
On the other hand, compared to any other geometry of a higher dimension, MPOs are much
more elegant and efficient to handle from a computational point of view.
The contributions in Part I are situated in qualitative reasoning (QR), a research field that has
remained principally theoretical, despite its considerable potential of applications. QR is a
major field of artificial intelligence, which has been adopted in GIScience, especially its
subfield of qualitative spatial reasoning (QSR) (Freksa 1992, Egenhofer & Mark 1995, Cohn &
Renz 2007). QR seeks to develop techniques to enable information systems to reason about
the behaviour of physical systems, without the kind of precise quantitative information
needed by conventional analysis techniques such as numerical simulators (Weld & de Kleer
1989, Iwasaki 1997). Key to QR are qualitative representations: symbolic representations of
discrete quantity spaces, such that “the distinctions made in these discretisations are
relevant to the behaviour being modelled” (Cohn 1996, p. 124). One common algebraic
framework in QR for representing and reasoning is a qualitative calculus3. In brief, and
although there is no precise definition, a qualitative calculus arises from a set of jointly
exhaustive and pairwise disjoint relations, together with a set of operations to reason about
these relations (Ligozat & Renz 2004).
The following interrelated research questions are addressed in Part I:
Question 1 How to formally describe the relations between MPOs adequately in a
qualitative manner such that a calculus is obtained to represent and reason
about these relations?
Question 2 How to implement this calculus in an information system?
Chapters 2-4 formulate an answer to the first question, whereas the second question is
tackled in Chapter 5. To this end, a dedicated qualitative calculus for handling qualitative
relations among MPOs, namely the Qualitative Trajectory Calculus (QTC), is considered. QTC
was introduced in 2004 in the doctoral dissertation of Van de Weghe (2004). His
3 Plural: calculi.
Introduction 7
fundamental work has been complemented by a number of later contributions. In this light,
Chapter 2 presents a theoretical overview of the fundamental types of QTC and
demonstrates how the calculus implements major reasoning concepts, how it can be
extended, and how it can be employed in order to represent raw moving object data. This
chapter has been published in Qualitative Spatio-Temporal Representation and Reasoning:
Trends and Future Directions (Delafontaine et al. 2011b). Chapter 3 and 4 add to Chapter 2 in
that they further elaborate a specific type of QTC: the Qualitative Trajectory Calculus on
Networks (QTCN). QTCN considers objects that are constrained in their movement by
networks, as is the case for most transportation means. Chapter 3 – published in Information
Sciences (Delafontaine et al. 2011a) – introduces a formal axiomatisation of QTCN, explores
its reasoning power and its ability to infer additional knowledge. In QTCN, the networks that
delineate the movements are assumed to be static, i.e. they remain topologically unaltered
over time. This assumption, however, does not always mirror the reality. In multi-modal
transportation networks, for instance, temporal connections or disconnections such as the
opening of a bridge over a waterway might be common. Therefore, Chapter 4, published in
Information Sciences (Delafontaine et al. 2008), extends QTCN to networks that may change
topologically by introducing the Qualitative Trajectory Calculus on Changing Networks
(QTCDN’).
Although in Chapters 2-4 applications of QTC are touched, the main focus is on the
theoretical framework. Chapter 5, published in Expert Systems With Applications
(Delafontaine, Cohn & Van de Weghe 2011), aims to provide a more systematic basis for
QTC-based applications by addressing the implementation of QTC in an information system.
A prototype QTC-based information system, named QTCAnalyst, is developed and
illustrated. Starting from raw trajectory data, QTCAnalyst is able to automatically generate
and export QTC representations that model relations among moving objects. Typically, the
input data will be obtained from tracking systems such as GPS devices, although alternative
data sources and input modalities are worthwhile exploring. Given that reasoning with
qualitative information particularly reflects human cognition, communication and decision
making, the incorporation of human-originated data merits particular attention, especially in
view of the growing capabilities for human-computer interaction. As Forbus et al. (2003)
argue that “qualitative spatial reasoning is essential for working with sketch maps” (Forbus,
Usher & Chapman 2003, p. 61), a promising extension of QTCAnalyst may be to support
input from freehand sketching. To this end – as a branch-line in this dissertation – Chapter 6
addresses the modelling of moving objects in geospatial sketch maps and how MPO
trajectories can be obtained from such sketch maps. The chapter has been presented at the
AGILE Workshop on Adaptation in Spatial Communication (Delafontaine & Van de Weghe
2009).
8 Chapter 1
Part II – Travelling subjects
In line with the trend towards a person-based GIScience (see section 1.1), the second part of
this dissertation addresses the movement behaviour of individuals, in lieu of unspecified
entities. Unlike other entities, individuals most often control to a large extent the actions
they undertake in space over time. This reasoning supports two research perspectives on
analysing travel behaviour. On the one hand, the examination of revealed behaviour may
offer insights on individuals’ decisions and rationale underlying their activities. Knowledge
extracted from the monitoring and analysis of revealed travel behaviour constitutes the
basis for location-based services and may importantly feed decisions about transport (e.g.
navigation, logistics, traffic management) and security issues (e.g. evacuation and rescue
policies, crowd control).
On the other hand, individuals are subject to a set of constraints on their space-time
behaviour which provides a foundation for analysing potential behaviour. Research on
potential behaviour may improve our assessment and understanding of the effects of
different space-time constraints on individual movement and on the feasibility for
individuals to participate in activities and to travel in between them. Important applications
related to potential behaviour lie in planning, development and prospection, especially
within the context of regional science and transport studies (e.g. urban planning,
transportation planning, traffic forecasting, agent-based modelling, accessibility assessment,
wayfinding).
Part II focuses on the following fundamental research question related to the revealed and
potential space-time behaviour of individuals:
Question 3 How to model and analyse revealed and potential behaviour of individuals
starting from raw tracking data?
Partial answers to this question are given in Chapters 7 and 8. Chapter 7 (Delafontaine et al.
2011d, submitted) scrutinises a key aspect of revealed behaviour, i.e. the chronological
sequence of observed activities. In particular, Chapter 7 examines behavioural sequences
within tracking data of visitors walking around at a big trade fair. The chapter is innovative in
using tracking data obtained from Bluetooth sensing, i.e. a burgeoning, yet largely
unexplored tracking approach, and in applying sequence alignment methods, which have a
long tradition in bioinformatics, but have only recently been adopted as a data mining
technique in GIScience (Shoval & Isaacson 2007a).
Chapter 8, forthcoming in the International Journal of Geographical Information Science
(Delafontaine, Neutens & Van de Weghe 2011a), concentrates on the modelling of potential
behaviour. Potential behaviour plays a central role in time geography (Hägerstrand 1970)
and is especially captured in its key concept of a space-time prism (STP). A STP embodies the
set of space-time points that an individual may reach given a set of spatial and temporal
Introduction 9
constraints. This delineated volume and its spatial projection are usually referred to as
potential path space and potential path area respectively. Given the renewed interest in
time geography (see section 1.1), many recent implementations of STPs have considered
network-constrained travel environments (cf. Chapters 3-4, e.g. Neutens et al. 2008, Miller &
Bridwell 2009, Kuijpers & Othman 2009, Kuijpers et al. 2010). Chapter 8, extends these
implementations in at least two ways by introducing rough obstacle-constrained STPs. First,
building on the tenets of rough set theory (Pawlak 1982), these space-time prisms account
for the uncertainty of space-time constraints, especially the uncertainty related to space-
time constraints stemming from tracking data (cf. question 3). Second, they represent
potential path spaces within travel environments that can be modelled as open spaces
constrained by discrete obstacles (i.e. obstacle-constrained environments), rather than
linear networks.
The remaining contributions of Part II also rely on the modelling of potential behaviour and
the related time-geographical framework. In line with the second objective, however, their
main focus is on the more application-centric issue of accessibility, which can be defined as
an individual’s ability to travel and participate in activities given the available transport and
land use system (Pirie 1979, Pooler 1987). Therefore, two additional research questions have
been investigated:
Question 4 How to measure the accessibility of opportunities to individuals through the
consideration of space-time constraints?
Question 5 How do time constraints of opportunities affect their accessibility and how
can these be manipulated in order to control individual accessibility?
The fourth question is addressed starting from the general concern that, until present,
despite the important efforts that have stressed the advantages and need for a person-
based assessment (Kwan 2009, Miller 2007, Neutens et al. 2010), there remains a strong
tradition to evaluate accessibility from a place-based perspective in empirical research. In
this dissertation it is argued that the place-based and people-based perspectives
complement each other in many respects. Therefore, Chapter 9 (Delafontaine, Neutens &
Van de Weghe 2011b, submitted) aims to combine both perspectives into a new kind of STP
denoted as a reverse STP. Reverse STPs implement some important space-time constraints
of conventional person-based STPs, while supporting the advantages of place-based
approaches such as the ability to generate area-covering maps. More than that, based on
reverse STPs, Chapter 9 introduces and illustrates a novel GIS toolkit, named PrismMapper,
for measuring and mapping an individual’s accessibility to services.
Beyond combining place-based and person-based approaches, the PrismMapper toolkit
contributes through the explicit consideration of service facility opening hours in its
assessment of accessibility. Opening hours have often been overlooked as temporal
10 Chapter 1
constraints that delimit the accessibility of services to individuals which should be taken into
account when measuring person-based accessibility. This observation has lead to the fifth
research question. Chapters 10 and 11 seek for an answer to this question through
examining how opening hours of services affect individual accessibility and how this
accessibility can be amended by adopting different opening hour schedules. Chapter 10,
forthcoming in Journal of Transport Geography (Neutens et al. 2011) presents a procedure
to determine a regime of opening hours of services which optimises the absolute level of
person-based accessibility to these services. The procedure is implemented in a case study
focussing on the accessibility of government offices to citizens in the city of Ghent (Belgium).
Finally, in Chapter 11, forthcoming in Computers, Environment and Urban Systems
(Delafontaine et al. 2011c), the approach of Chapter 10 is generalised, such that a regime
can be derived which maximises any arbitrary function of individual accessibility, rather than
only its absolute sum across the population. This generalised algorithm is then applied
according to evaluation functions that draw on different equity principles in order to assess
the effects of (re)scheduling opening hours on the equity of person-based accessibility
among individuals. This is illustrated in a case study considering the accessibility of public
libraries in Ghent.
To conclude this dissertation, Chapter 12 summarises its main achievements and results, and
evaluates the postulated objectives and research questions within the wider research
context.
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Moving Objects
Part I
“Among entities there must be some cause which moves and combines
things. There must be a principle of such a kind that its substance is
activity.” (Aristotle)
A Qualitative Trajectory Calculus to reason about moving point objects 21
2 A Qualitative Trajectory Calculus to reason about moving
point objects
Delafontaine M., Chavoshi S. H., Cohn A. G., Van de Weghe N.
in Hazarika S. M. (Ed.): Qualitative Spatio-Temporal Representation and Reasoning:
Trends and Future Directions (2011)
Copyright © IGI Global
Abstract. A number of qualitative calculi have been developed in order to reason
about space and time. A recent trend has been the emergence of integrated
spatiotemporal calculi in order to deal with dynamic phenomena such as motion. In
2004, Van de Weghe introduced the Qualitative Trajectory Calculus (QTC) as a
qualitative calculus to represent and reason about moving objects. This chapter
presents a general overview of the principal theoretical aspects of QTC, focusing on
the two most fundamental types of QTC. It is shown how QTC deals with important
reasoning concepts, and how the calculus can be employed in order to represent raw
moving object data.
Kewords. Qualitative calculus – Spatio-temporal reasoning – Moving point objects
2.1 Introduction
Reasoning about spatial and temporal information takes a central place in human daily life. A
number of qualitative calculi have been developed to represent and reason about spatial or
temporal configurations. Most of them focus on one of the two domains, whereas a few are
true spatiotemporal calculi that deal with spatiotemporal phenomena. One such is the
Qualitative Trajectory Calculus, which in the remainder of this chapter will be referred to as
QTC. QTC is a qualitative calculus to reason about a specific spatiotemporal phenomenon:
moving objects.
The remainder of this chapter is structured as follows. First, relevant background issues are
discussed. Second, some general characteristics of QTC are explained and a brief overview of
all QTC calculi that have been elaborated so far is given. The two most fundamental QTC
calculi, QTCB and QTCC, are then presented in detail. The following sections discuss
representing and reasoning with QTC, as well as how QTC can be extended. An application
section follows in order to highlight the potential of implementing QTC in information
systems. The final sections mention opportunities for further research and conclusions.
22 Chapter 2
2.2 Background
In Artificial Intelligence, several qualitative calculi exist to reason about either spatial or
temporal information, the most well-known being Allen’s Interval Calculus (Allen 1983),
which has about 1200 citations in the ISI Web of Science by the time of writing. According to
Wolter & Zakharyaschev (2000), an apparent and natural step is to combine both spatial and
temporal formalisms in order to reason about spatiotemporal phenomena. A crucial and
fundamental phenomenon at this cross-pollination of space and time is motion. Note that
motion is an inherently spatiotemporal phenomenon (Peuquet 2001). Dealing with motion is
essential to spatial and geographical information systems, where an evolution from static to
dynamic formalisms and representations has been made. A specific type of motion is
associated with moving objects, i.e. objects whose position moves through space in time.
In the past decade, the modelling of moving objects has been a hot topic in fields such as
GIScience, Artificial Intelligence and Information Systems (Bitterlich et al. 2008). In
qualitative reasoning, however, considerable work has focused on the formalisation of
motion, or moving objects in particular. Some examples are Muller (2002), Ibrahim (2007),
Hallot & Billen (2008), and Kurata & Egenhofer (2009). These approaches have in common
that they rely on topological models such as the Region Connection Calculus (Randell, Cui &
Cohn 1992) or the 9-Intersection model (Egenhofer & Franzosa 1991). However, a general
shortfall of topological models is their inability to further differentiate between disjoint
relations. This makes their applicability to represent and reason about continuously moving
objects questionable, as in many cases moving objects remain disjoint for most of the time.
For instance, cars in a traffic situation are usually disjoint, apart from the exceptional case of
an accident.
In order to overcome this inability, the Qualitative Trajectory Calculus (QTC), was proposed
by Van de Weghe (2004). QTC provides a qualitative framework to represent and reason
about moving objects which enables the differentiation of groups of disconnected objects.
The development of QTC has been inspired by some major qualitative calculi: the Region
Connection Calculus (Randell, Cui & Cohn 1992), the temporal Semi-Interval Calculus (Freksa
1992a), and the spatial Double Cross Calculus (Freksa 1992b, Zimmermann & Freksa 1996).
2.3 The Qualitative Trajectory Calculus
2.3.1 Simplifications
Information systems usually represent knowledge according to an underlying model of the
real world, making simplifications in order to abstract away from the mass of details that
would otherwise obscure essential aspects. To this end, QTC makes four simplifications
(Figure 2.1). First and foremost, QTC considers the relation between two objects, i.e. binary
relations (relational simplification, Figure 2.1b), as is common in spatial and temporal
reasoning (Cohn & Renz 2007). Second, moving objects are spatially simplified into moving
A Qualitative Trajectory Calculus to reason about moving point objects 23
point objects or MPOs (object simplification, Figure 2.1c), as is common in GIScience and
geoinformatics (Laube 2005, Gudmundsson, van Kreveld & Speckmann 2004, Güting et al.
2000, Noyon, Claramunt & Devogele 2007). There are only two topological relations (disjoint
and equal) between two MPOs. Since the relation between two equal MPOs is trivial, the
third simplification in QTC is the restriction to disjoint MPOs (topological simplification).
Finally, in order to understand the temporal dimension in depth, it is important to find out
what happens at one time point. Hence, QTC relations are relations that hold at a particular
time point (temporal simplification, Figure 2.1d).
Figure 2.1 – Simplification in QTC of a real-life situation (a) by taking cumulatively account of the
relational simplification (b), the object simplification (c), and the temporal simplification (d)
(simple arrows for trajectories, double arrows for instantaneous velocity vectors).
2.3.2 Continuity, conceptual neighbours, and transitions
QTC assumes space and time, and thus the motion of objects, to be continuous. As a
consequence, QTC relations change in time according to the laws of continuity. Along with
continuity comes the important concept of conceptual neighbourhood as introduced by
Freksa (1992b). Two QTC relations between the same pair of MPOs are conceptual
neighbours if and only if these relations can directly follow each other through continuous
motion of the MPOs, without the necessity for a third relation to hold at an intermediate
point in time. A transition then denotes the continuous change of one relation into a
conceptual neighbouring relation. Each transition thereby happens at a certain instant or
point in time, which we will term a transition instant. A conceptual neighbourhood can be
represented by a conceptual neighbourhood diagram (CND), i.e. a visualisation of a graph
which nodes represent relations, and where two nodes are connected if they are conceptual
neighbours of each other.
All QTC calculi are associated with a set of jointly exhaustive and pairwise disjoint (JEPD)
base relations. Consequently, there is one and only one relation for each pair of coexisting
MPOs at each time instant. In addition, due to continuity, the concurrent movement of two
MPOs over a given time interval is uniquely mapped to a sequence of conceptually
neighbouring base relations.
24 Chapter 2
All QTC relations are formed by a tuple of labels (representing different primitive qualitative
relations) that all have the same three-valued qualitative domain , which we will
denote as in the remainder of this chapter. A ‘0’ symbol corresponds to a landmark value,
and as Galton (2001) points out, this value always dominates both ‘’ and ‘+’ values. Hence:
A ‘0’ must always last over a closed time interval (of which a time instant is a special
case);
A ‘’ / ‘+’ must always last over an open time interval;
Only transitions to or from ‘0’ are possible (transitions from ‘’ / ‘+’ to ‘+’ / ‘’ are
impossible) and transition instants always correspond with a ‘0’ value.
Based on the notion of topological distance introduced by Egenhofer & Al-Taha (1992), the
conceptual distance can be defined as a measure for the closeness of QTC relations (Van de
Weghe & De Maeyer 2005). We take the conceptual distance between ‘0’ and another
symbol to be one. This is the smallest conceptual distance, apart from zero (i.e. the distance
between a symbol and itself). Since a direct transition is impossible, the conceptual distance
between ‘’ and ‘+’ is equal to two (one for ‘’ to ‘0’ and one for ‘0’ to ‘+’). The overall
conceptual distance between two QTC relations can then be calculated by summing the
conceptual distance over all relation symbols. For instance, for two QTC relations consisting
of four symbols, the conceptual distance ranges from zero to eight.
2.3.3 Types of QTC
Due to the consideration of different spaces and frames of reference, the following types of
QTC have been elaborated:
Basic type – QTCB (Van de Weghe et al. 2006), Figure 2.2a
Double Cross type – QTCC (Van de Weghe et al. 2005a), Figure 2.2b
Network type – QTCN ) (Bogaert et al. 2007), Figure 2.2c
Shape type – QTCS (Van de Weghe et al. 2005b)
The Basic (QTCB) and the Double Cross (QTCC) types both deal with MPOs that have a free
trajectory in an n-dimensional space. QTCB relations are determined by referring to the
Euclidian distance between two MPOs (Figure 2.2a). QTCC relations on the other hand rely
on the double cross, a concept introduced by Zimmerman and Freksa (1996), as a spatial
reference frame (Figure 2.2b). QTCB and QTCC will be discussed in detail in the next two
sections.
QTCN (Network) focuses on the special case of MPOs which trajectories are constrained by a
network, such as cars in a city. Since both the Euclidean distance and the double cross
concepts ignore the spatial configuration of a potential underlying network, they are not
well suited for QTCN. Therefore, QTCN relations rely on the shortest paths in the network
between the considered MPOs (Figure 2.2c). In essence, QTCN employs the philosophy of
A Qualitative Trajectory Calculus to reason about moving point objects 25
QTCB in the context of a space constrained by a network. QTCN will not be considered further
in this chapter.
Finally, QTCS (Shape) employs the double cross concept in order to describe trajectory
shapes or even arbitrary undirected polylines in a qualitative way. Thus, QTCS deals with the
relative configuration of a trajectory, rather than with the relation between MPOs. Due to
this different focus, it is out of the scope of this chapter.
Figure 2.2 – Two MPOs represented in a typical two-dimensional QTCB (a), QTCC (b), and QTCN (c)
setting. The frame of spatial reference is represented by the dashed line.
2.4 QTC – Basic (QTCB)
An MPO is always characterised by an origin and a destination, whether explicit or implicit.
Hence, a basic dichotomy concerning MPOs, perhaps the most fundamental one, is the
distinction between towards and away from relations. This very generic idea underlies QTCB
where this binary relation is evaluated on the basis of Euclidean distance in an
unconstrained n-dimensional space. In addition, also the relative speed between both
objects can be taken into account. As mentioned earlier, QTC relations consist of qualitative
symbols that share the threefold domain . QTCB relations are constructed from
the following relationships:
Assume: MPOs and and time point
denotes the position of an MPO at
denotes the Euclidean distance between two positions and
denotes the velocity vector of at
denotes that is temporally before
A. Movement of with respect to at (distance constraint): −: is moving towards :
(2.1)
+: is moving away from :
(2.2)
26 Chapter 2
0: is stable with respect to (all other cases)
B. Movement of with respect to at (distance constraint), can be described as in A with and interchanged, and hence:
−: is moving towards (2.3)
+: is moving away from (2.4)
0: is stable with respect to (all other cases)
C. Relative speed of with respect to at (speed constraint): −: is moving slower than :
(2.5)
+: is moving faster than :
(2.6)
0: and are moving equally fast:
(2.7)
Two levels of QTCB relations have been proposed: a first level QTCB1 that only considers the
distance constraints (relationships A and B), and a second level QTCB2 taking account of the
speed constraint (relationship C) as well. The resulting relation syntaxes are respectively the
tuples (A B)B1
and (A B C)B2
. Note that relationship C dually represents the relative speed
of l with respect to k, and hence trivialises a fourth relationship.
Relation icons for QTCB are shown in Figure 2.3, where is always on the left side, and on
the right side. The line segments and crescents represent potential motion areas. Note that
their boundaries are open, and, for the crescents, the straight boundaries correspond to
elements of another relation. A filled dot indicates that an MPO might be stationary,
whereas an open dot means that it must be moving. Dashed lines represent uncertain
boundaries that follow from the ignorance of relative speed. The Roman numerals below the
icons specify the minimum number of spatial dimensions required for a relation to be
feasible.
There are 9 (3²) base relations in QTCB1 (Figure 2.3a). All these relations are possible in a
one- or higher-dimensional space. QTCB2 on the other hand has 27 (3³) base relations (Figure
2.3b), which are all possible in two- or higher-dimensional spaces. However, in a one-
dimensional space, only 17 (63.0%) QTCB2 relations can occur. This reduction follows from a
dependency between the distance constraints and the speed constraint in the case of a 1D
space. In a 1D space, the direction of movement is always collinear with the direction of
Euclidean distance, and hence a ‘0’ in the distance constraints always corresponds to a
stationary MPO. As a consequence, it is impossible for an MPO to be stationary and to have
a higher speed than another MPO. In a two- or higher-dimensional space on the other hand,
A Qualitative Trajectory Calculus to reason about moving point objects 27
a ‘0’ distance constraint does not necessarily indicate a stationary object, e.g. in the case of
‘tangential motion’ such as when one MPO is circling around the other MPO.
Figure 2.3 – QTCB1 (a), and QTCB2 (b) relation icons.
2.5 QTC – Double Cross (QTCC)
In addition to the towards / away from dichotomy of QTCB, QTCC employs another
fundamental distinction in navigation, i.e. the left / right dichotomy. Hence, an intrinsic ‡-
shaped frame of reference is obtained, called the Double Cross, after a concept introduced
by Freksa (1992b) (Figure 2.4). The reference line associated with the left / right distinction is
the straight connection line between both MPOs. Besides the left / right dichotomy, QTCC
also takes into account the relative difference in relative motion angle with respect to this
reference line.
28 Chapter 2
Figure 2.4 – Different use of the double cross in the Double Cross Calculus (Galton 2001) (a), and
the QTCC calculus (b).
Assume: MPOs and and time point
denotes the reference line through and
denotes the minimum absolute angle between
and
D. Movement of with respect to at (side constraint): −: is moving to the left side of :
(2.8)
+: is moving to the right side of :
(2.9)
0: is moving along (all other cases)
E. Movement of with respect to at (side constraint), can be described as in D with and interchanged, and hence:
−: is moving to the left side of (2.10)
+: is moving to the right side of (2.11)
0: is moving along (all other cases)
F. Angle constraint:
−:
(2.12)
+:
(2.13)
0: all other cases (2.14)
As for QTCB, two levels of QTCC have been defined: a first level QTCC1 which simply considers
the towards / away from and left / right distinctions and a second level QTCC2 considering
the speeds and angle constraints as well. The relational syntaxes are respectively
(A B D E)C1
and (A B D E C F)C2
. Let us consider relationship F if one of the objects is not
moving at . The object can move in every direction at and at . Assume that F is ‘’ at
A Qualitative Trajectory Calculus to reason about moving point objects 29
and ‘+’ at . Since we assume continuous motion, F has to be ‘0’ at . Thus, if at least one
MPO is stationary, F will be ‘0’.
QTCC1 and QTCC2 respectively have 81 (34) and 729 (36) theoretical JEPD base relations
(Figure 2.5, Figure 2.6). In a one-dimensional space, left and right of the reference line
through and cannot be distinguished, and hence the side and angle constraints will
always be ‘0’. Thus, in essence, QTCC1 reduces to QTCB1 and QTCC2 reduces to QTCB2 for the
one-dimensional case, with respectively 9 (11.1% of the theoretical number) and 17 (2.3%)
base relations.
Figure 2.5 – QTCC1 relation icons.
30 Chapter 2
Figure 2.6 – QTCC2 relations and the minimal number of spatial dimensions supporting them:
respectively the dotted, dashed, and straight boundaries for one, two, and three dimensions.
In two dimensions, all QTCC1 base relations exist (as for all higher dimensions), whereas only
305 (42.4%) QTCC2 relations are possible, due to the interdependence of relational symbols.
Since in 2D space, objects must be stationary whenever their distance and side constraints
are ‘0’, the speed constraint is restricted and the angle constraint must be ‘0’ in that case.
Also, an object with a ‘0’ distance constraint and non-‘0’ side constraint must be moving with
a bigger or equal angle with respect to , and hence the restriction on the angle
constraint. Analogously, objects with a non-‘0’ distance constraint and ‘0’ side constraint
move with smaller or equal angles with respect to . Moreover, when the latter two rules
A Qualitative Trajectory Calculus to reason about moving point objects 31
are combined, the inequality turns into either a strict equality, or a strict inequality, and
thereby restricts the angle constraint to a singleton.
In three-dimensional space, the distance and side constraints are insufficient to deduce the
stationarity of the objects. However, as in 2D, they determine whether the direction of
movement must be along or perpendicular to . From this observation restrictions follow
on the angle constraint. The special case where both the distance and side constraints are ‘0’
may indicate either a stationary object, or an object moving perpendicular to and
perpendicular to both the left / right and towards / away from directions. We obtain 591
(81.1%) feasible QTCC2 relations in 3D.
2.6 Representing and reasoning with QTC
QTC has been confronted with key concepts in qualitative reasoning. In this section, we will
discuss three of these issues, respectively conceptual neighbourhood diagrams (CNDs),
composition tables (CTs), and incomplete knowledge.
2.6.1 Conceptual neighbourhood diagrams
As mentioned earlier, the construction of CNDs for QTC is based on the concepts of
dominance (Galton 2001) and conceptual distance. For an in depth description, we refer to
Van de Weghe and De Maeyer (2005). CNDs for the Basic and Double Cross QTC calculi in 2D
space are respectively shown in Figure 2.7 and Figure 2.8. For each link between conceptual
neighbours the conceptual distance between the adjacent relations has been indicated. The
CND for QTCC1 in 1D has been omitted, since it would be the same as the CND for QTCB1
except for two additional ‘0’ values in each relation. The CND for QTCC2 is also not shown, as
it is too complex to visualise on a two-dimensional medium.
Figure 2.7 – CNDs for QTCB1 in n-dimensional space (a), for QTCB2 in a one-dimensional space (b),
and for QTCB2 in a two- or higher-dimensional space (c). The straight, dashed and dotted lines
respectively represent the conceptual distances one, two and three.
32 Chapter 2
Figure 2.8 – CND for QTCC1 in a two-or higher-dimensional space. Links have been gray-shaded
according to the conceptual distance between the adjacent relations.
A Qualitative Trajectory Calculus to reason about moving point objects 33
From the CNDs, we learn that, due to the laws of continuity, the conceptual neighbours of
each particular relation constitute only a subset of base relations. This set comprises the
candidate relations that may directly precede or follow the relation at hand in time, i.e. the
set of possible transitions from/to this relation. This set of candidates is thereby highly
limited when compared to the set of theoretical possibilities, as can be seen from Table 2.1.
Note that each pair of conceptual neighbours and is associated with two transitions,
i.e. a transition from to , and its converse from to . Similarly to a CND, a transition
graph can be constructed with directed links to represent existing transitions. However, for
QTCB and QTCC, all converse transitions do exist and thus one conceptual neighbour relation
can be counted for two transitions (Table 2.1). Note that this may not be the case for other
types of QTC, e.g. for QTCDN’ (Delafontaine et al. 2008).
Another notable finding is that all CNDs are completely symmetric with respect to the
relation consisting solely of ‘0’ values. We call this symmetric and reflexive relation the zero-
relation. Symmetry with respect to the zero-relation is due to the central position of ‘0’ in
the qualitative set , as well as to the symmetry of conceptual neighbourhood
for converse QTC relations.
QTC calculus # spatial dimensions # base relations # transitions # combinations ratio
B1 1+ 9 32 72 44.4%
B2 1 17 64 272 23.5%
B2 2+ 27 196 702 27.9%
C1 2+ 81 1 088 6 480 16.8%
Table 2.1 – The number of base relations, transitions, theoretical combinations of base relations,
and the ratio transitions / theoretical combinations for the Basic and Double Cross QTC calculi.
Furthermore, every relation is a conceptual neighbour of the zero-relation (and vice versa),
as is consistent with our intuition. For instance, it is highly reasonable that, whatever the
relation between two MPOs at a certain moment, they may always become stationary the
next moment, in which case their relation turns into the zero-relation.
2.6.2 Composition tables
Another important reasoning tool, apart from the CND, is the composition table (CT). The
idea behind a CT is to compose existing relations to obtain new relations, i.e. if two existing
relations and share a common element they can be composed into a new
relation . The composition operation is represented here by the symbol, e.g.
. CTs are cross tables that usually contain the composition
results of all combinations of base relations for a certain calculus, with in the left column,
in the top row, and in each of the entries. CTs are very useful from a
computational point of view, since a simple table look-up can be far more efficient than
complex theoretical deduction (Bennett 1997, Skiadopoulos & Koubarakis 2004, Vieu 1997).
34 Chapter 2
In addition, CTs play an important role when working with incomplete information and
larger inference mechanisms.
The CT for one-dimensional QTCB1 is shown in Table 2.2. Of its 81 (9²) compositions, 20 are
unique, 20 of them are twofold, 4 are threefold, and the remaining 36 yield no solution
(empty set). The empty solutions come along with the inconsistent cases where the common
MPO must be moving in the first relation and must be stationary in the second relation, and
vice versa. Hence, in order to avoid them, one might use two separate CTs: one for the case
of a moving common MPO, the other one for the stationary case.
(--) (0-) (+-) (-0) (00) (+0) (-+) (0+) (++)
(--)
(-+)˅
(+-)
(--)
(0+)˅
(0-)
(0-)
(--)˅
(++)
(+-)
(0-)
(-0)˅
(+0)
(-0) (00) (00)
(-0)˅
(+0)
(+0)
(+-)
(--)˅
(++)
(-+)
(0+)˅
(0-)
(0+)
(-+)˅
(+-)
(++)
(-0)
(-+)˅
(--)˅
(+-)
(0+)˅
(0-)
(--)˅
(+-)˅
(++)
(00) (0+)˅
(0-)
(00) (0+)˅
(0-)
(+0)
(--)˅
(-+)˅
(++)
(0+)˅
(0-)
(-+)˅
(++)˅
(+-)
(-+) (--) (-+)˅
(+-)
(0-) (0+)˅
(0-)
(+-) (--)˅
(++)
(0+) (-0) (-0)˅
(+0)
(00) (00) (+0) (-0)˅
(+0)
(++) (-+) (--)˅
(++)
(++) (0+)˅
(0-)
(++) (-+)˅
(+-)
Table 2.2 – CT for QTCB1 in a one-dimensional space.
Complete CTs can be constructed for QTCB2. However, this would generally be a bad idea,
since the speed constraint (relationship C) is independent from the distance constraints
(relationships A & B). Hence, a more efficient solution is to use separate CTs for the distance
and speed constraints, and then recombine the results afterwards. Table 2.3 presents the CT
for the speed constraint, with seven unique results and two universe results. Note that these
universe sets are further reduced whenever at least one of the objects is stationary, as can
be deduced from the distance constraints in one-dimensional QTCB.
A Qualitative Trajectory Calculus to reason about moving point objects 35
− 0 +
− − − U
0 − 0 +
+ U + +
Table 2.3 – CT for the speed constraint.
The CTs for two-dimensional QTCC1 and QTCC2 would respectively contain 6561 (81²) and
93025 (305²) entries, each of these entries containing a set of up to 81 and 305 elements.
Since CTs soon become very large, a so-called composition-rule table (CRT) was introduced
by Van de Weghe et al. (2005c). A CRT differs from a traditional CT as it does not contain all
individual composition results. Nevertheless, a CRT does provide all the information offered
by a traditional CT. Instead of the full CT, a CRT uses a set of composition rules to generate
the composition results for the relations at hand. These rules can be implemented in
information systems in order to automatically generate compositions, which might be
preferable to CTs due to their extent.
CRTs for QTC can be obtained by using diagrammatic reasoning on the basis of relation icons
(e.g. Figure 2.3 and Figure 2.5). As and are given, their corresponding relation icons
can be translated so that the position of in the icon of matches the position of in the
icon of . Then, in order to find the composition result, two central issues have to be
considered. First, which rotation do we need, such that the velocity vector of in matches
the one of in ? Second, how is moving with respect to in , and how is moving
with respect to in ?
Let us consider the case of two-dimensional QTCC1. For the first issue, nine rotational
possibilities have to be taken into account1: the crisp rotations 0°, 90°, 180°, 270°, the range
rotations ]0°, 90°[, ]90°, 180°[, ]180°, 270°[, ]270°, 360°[, and the option of no possible match
by rotation. The latter case occurs due to the impossibility of inference between a moving
and a stationary MPO. For the second issue, only the first and the third relational symbols of
(relationships A & C) and the second and fourth symbols of (relationships B & D) have
to be considered to determine the composition result.
Table 2.4 presents the CRT for QTCC1 in 2D. It has 324 entries, which is a compression to less
than 5% compared to the original CT with 6561 entries. The CRT consists of two halves: the
upper half to look up the first and third symbol (relationships A & C) of , and the lower
half to determine the second and fourth symbol (relationships B & D).
1 According to trigonometry, we take anti-clockwise angles as being positive.
36 Chapter 2
We briefly explain the CRT with an example. The velocity vector of in the icon of
(+ + 0 0)C1
needs to be rotated over 270° in order to match the vector of in
(− 0 − −)C1
. Hence, to find , we use the column of 270° of Table 2.4.
Then, the relationships A & C of are in the row corresponding to A & C of R1, i.e.
the row of ‘’,’’, and the column of 270°: we obtain for A and ‘’ for C. Analogously, B &
D of depend on B & D of : we get ‘+’ for B, ‘’ for D. Thus, we find
(− + − −)C1, (0 + − −)
C1, (+ + − −)
C1 . Note that results for A & C in one column
always have to be combined with results for B & C in the same column, even when multiple
columns correspond to the same rotation angle, e.g. for 180°. Thus we should not take the
cross product.
0° 90° 180° 270°
0°-
90° 90°- 180° 180°- 270°
270°-
360° X
A C A C A C A C A C A C A C A C A C A C A C A C A C A C A C A C A C A C
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
– 0 – 0 – + – 0 + 0 – – – + – + – + – + 0 + + + + – 0 – – – – – – – – –
– – – – – U – – + + U – – U – U – U – U – + U + + U + – U - U – U – U –
0 – 0 – – – 0 – 0 + + – – – – – – – – – – 0 – + + + + 0 + - + – + – + –
+ – + – U – + – – + + U U – U – U – U – – – – U U + + + + U + U + U + U
+ 0 + 0 + – + 0 – 0 + + + – + – + – + – 0 – – – – + 0 + + + + + + + + +
+ + + + + U + + – – U + + U + U + U + U + – U – – U - + U + U + U + U +
0 + 0 + + + 0 + 0 – – + + + + + + + + + + 0 + – – – – 0 – + – + – + – +
– + – + U + – + + – – U U + U + U + U + + + + U U – – – – U – U – U – U
B D B D B D B D B D B D B D B D B D B D B D B D B D B D B D B D B D B D
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+ 0 + 0 + + – 0 + 0 + – + + – + 0 + + + + + + + + – + – + – 0 – – – + –
+ + + + U + – – + + + U U + – U – + U + U + U + + U + U + U + – U – + U
0 + 0 + – + 0 – 0 + + + – + – – – 0 – + – + – + + + + + + + + 0 + – + +
– + – + – U + – – + U + – U U – – – – U – U – U U + U + U + + + + U U +
– 0 – 0 – – + 0 – 0 – + – – + – 0 – – – – – – – – + – + – + 0 + + + – +
– – – – U – + + – – – U U – + U + – U – U – U – – U – U – U – + U + – U
0 – 0 – + – 0 + 0 – – – + – + + + 0 + – + – + – – – – – – – – 0 – + – –
+ – + – + U – + + – U – + U U + + + + U + U + U U – U – U – – – – U U –
Table 2.4 – CRT for QTCC1 in a two-dimensional space.
2.6.3 Incomplete knowledge
Not always everything has to be known about a situation to make inferences which are
important for the issue at hand (Frank 1996). Obviously, in these situations information may
lack for offering complete answers to queries. However, ‘a partial answer may be better
than no answer at all.’ as Freksa (1992a) argues. By abstracting away from the mass of
metrical details, qualitative representations are much more appropriate for handling such
A Qualitative Trajectory Calculus to reason about moving point objects 37
incomplete knowledge, rather than quantitative approaches (Cristani, Cohn & Bennett
2000).
As mentioned before, the development of the QTC has been inspired by some major QR
calculi, especially the temporal Semi-Interval Calculus (Freksa 1992a) and the spatial Double-
Cross Calculus (Zimmermann & Freksa 1996, Freksa 1992b). Central in these theories is the
specific attention to incomplete knowledge, and hence, one might expect QTC to be able to
handle incomplete knowledge as well.
One kind of incomplete knowledge results from natural language expressions. Consider the
expression “ is moving towards , which is not slower than ”. This expression can be
represented in QTC, for instance by (− U U+)B2
with . Hence, we obtain a union
of six solutions. Interestingly, these solutions constitute a conceptual neighbourhood, i.e.
they are mutually path-connected through conceptual neighbour relations when isolated
from the complete CND of base relations (see Figure 2.7). According to Freksa (1992a), we
achieve coarse knowledge, i.e. a kind of incomplete knowledge that allows to be represented
by a conceptual neighbourhood of relations at a certain level of granularity. When relations
between MPOs are perceived or described incompletely through natural language, the
resulting knowledge will typically be coarse.
Whenever one expression may lead us to incomplete knowledge, multiple expressions can
be combined in order to deduce finer knowledge. Table 2.5 gives an example of four
expressions, each of which has a coarse result, for which the intersection results in complete
knowledge. In addition, composition offers an appropriate inference mechanism to integrate
expressions about three or more objects.
Natural language expression QTCB2 solution Integrated solution
“ k is moving towards l ” (− U U)B2
(− U U)B2 (U0 U0 U)B2 (U + U)B2 (U U 0)B2 = ( + 0)B2
“ k and l are moving along
the same straight line ” (U0 U0 U)B2
“ l is moving away from k ” (U + U)B2
“ l is moving equally fast as k ” (U U 0)B2
Table 2.5 – Intersection of coarse solutions to obtain fine knowledge, with U0 = U \{0}.
2.7 Extending QTC
Complex real-life motions go far beyond the earlier described simplifications applied in QTC.
Can we relax these constraints? Obviously, not all simplifications can be ignored. Therefore,
we now focus on how QTC can be extended, whilst still accepting the object simplification,
i.e. the abstraction of moving objects to MPOs. In the remainder of this section we will
38 Chapter 2
discuss the respective and cumulative releases of the relational, temporal, and topological
simplifications.
2.7.1 Multiple MPOs
The relations between multiple MPOs can be represented by means of a QTC cross table or
matrix (Table 2.6). An element in this matrix represents the QTC relation between
MPOs and . A QTC matrix can be computed at each time point. The following compression
rules and techniques can be used in order to reduce its size:
The diagonal of the matrix can be excluded, as it is empty due to the topological
constraint.
Only the upper right (or lower left half) of the matrix has to be considered, as is gray
shaded in bold in Table 2.6. The lower part of the matrix holds the converse relations of
the upper part and vice versa and is therefore redundant.
t k l m n
k
--
-+ -0
l -- ++ +0
m +- ++
-0
n 0- 0+
0-
Table 2.6 - QTCB1 matrix for four MPOs k, l, m, and n at time t.
Hence, for objects, the number of elements can be reduced from to - . Note
that research has been done in order to further simplify topological relations (Rodríguez,
Egenhofer & Blaser 2003) and simplifying temporal relations (Rodriguez, Van de Weghe & De
Maeyer 2004) over multiple elements. It could be interesting to combine both in order to
simplify spatiotemporal relations, such as QTC relations. Thus, the number of elements in a
QTC matrix could be further reduced so that it only contains relevant information, i.e. no
redundancies.
2.7.2 Multiple time points and intervals
What if we consider QTC matrices at different time moments? According to the philosophy
of qualitative reasoning, new relations only need to be calculated whenever transitions
occur. As a consequence, it will be the most efficient to compute one initial matrix and to
store only relations which have transitioned in all subsequent matrices.
2.7.3 Multiple topological relations
QTC does not distinguish topological relations, and might hence be complemented by
topological calculi. As mentioned earlier, point objects only have two topological relations:
A Qualitative Trajectory Calculus to reason about moving point objects 39
disjoint and equal. Though QTC is developed to reason about disjoint objects, this constraint
might be relaxed. Note that in case of equal MPOs, we will always obtain zero-relations.
2.8 Example case
This section discusses an example application of QTC in one of the major domains of applied
science that in essence deals with objects moving in a geographical space, namely
transportation research. Ever since their invention, cars have been a focus of research for
numerous traffic engineers that have tried to represent and understand their complex
physics. A typical example is the case of an overtake event (André, Herzog & Rist 1989,
Fernyhough, Cohn & Hogg 2000). In this section, we will analyse this case in QTC starting
from raw trajectory sample points as received from position aware devices. As the left / right
distinction is crucial in overtake events, we will utilise QTCC.
Let us consider two cars and . Table 2.7 gives their two-dimensional sample coordinates
during an overtake event at regular time steps of one second. As QTC assumes continuity,
such a discrete set of sample points has to be interpolated in order to obtain continuous
trajectories. Although several approaches are possible, we will, for this example case, rely on
simple linear interpolation in space and time. This is also shown in Table 2.7.
sample
point
car k
car l
x (m) y (m) t (s) x (m) y (m) t (s)
1 15 0 0 15 10 0
2 15 5 1 15 13 1
3 10 13 2 15 17 2
4 10 23 3 15 23 3
5 10 33 4 15 28 4
6 15 41 5 15 33 5
7 15 46 6 15 38 6
Table 2.7 – Trajectory sample points of two cars k and l during an overtake event.
Figure 2.9 gives an overview of the spatial configuration of the objects and their
instantaneous velocity vector for all sample times during the overtake event. Also, the QTCC1
relations are given at and in between these sample instants. We find the following relation
40 Chapter 2
pattern: (− + 0 0)C1 (− + − +)
C1 (0 0 − +)
C1 (+ − − +)
C1 (+ − 0 0)
C1.
Since this is a pattern of subsequent conceptual neighbours, we call it a conceptual
animation (Van de Weghe et al. 2005a). It consists of five relations, four of which hold over a
time interval, whereas (0 0 − +)C1
occurs instantaneously at 3 s. Although all others last
over intervals, continuity theory induces some subtle differences between them. As pointed
out earlier, a ‘0’ value must always last over a closed time interval (of which a time instant is
a special case), whereas ‘’ and ‘+’ must always hold over an open time interval. Therefore,
it follows that (− + − +)C1
and (+ − − +)C1
persist over open time intervals, whereas
(0 0 − +)C1
occurs at an instantaneous closed time interval. Note that, as Table 2.7 does
not provide a preceding and following sample point for respectively the first and the seventh
sample point, the change in movement direction is unknown at these instants.
Consequently, the beginning of (− + 0 0)C1
and the end of (+ − 0 0)C1
are unknown, and
hence their corresponding time intervals are half-closed (Figure 2.9). With this knowledge, a
more complete description of the complete conceptual animation would be:
]0, 1]:(− + 0 0)C1
]1, 3[:(− + +)C1
[3]:(0 0 − +)C1
]3, 5[:(+ − − +)C1
[5, 6[:(+ − 0 0)C1
.
time 0 s 1 s 2 s 3 s 4 s 5 s 6 s
instantaneous
QTCC1 relation unkown (-+00)
C1 (-+-+)
C1 (00-+)
C1 (+--+)
C1 (+-00)
C1 unkown
interval QTCC1
relation (-+00)
C1 (-+-+)
C1 (-+-+)
C1 (+--+)
C1 (+--+)
C1 (+-00)
C1
Figure 2.9 – Configuration of two cars k and l at sample time stamps during an overtake event.
The overtake event illustrated here would be a legal manoeuvre if we consider right-hand
driving in Continental Europe. What about the case of left-hand driving? In that case, the
A Qualitative Trajectory Calculus to reason about moving point objects 41
trajectories of and are mirrored along the main road axis, and hence left interchanges
with right. As can be expected for QTCC1, we obtain a symmetrical animation with the last
two characters inverted, while the first two remain: (− + 0 0)C1 (− + + −)
C1
(0 0 + −)C1 (+ − + − )
C1 (+ − 0 0)
C1.
Similarly to the overtake event, numerous other traffic situations can be modelled by means
of conceptual animations. A qualitative framework can then be composed of such QTC
patterns in order to reason about, recognise or simulate traffic events.
2.9 Future research directions
A major direction for further research is the extension of QTC theory. New types of QTC,
perhaps application-specific types, can be elaborated. To this end, opportunities lie in the
relaxation of one or more of the simplifications that were made, e.g. to allow moving line,
region and body objects, next to the conventional MPOs. Also, more realistic scenarios,
spatial and temporal constraints, or frames of reference can be taken into account.
Delafontaine et al. (2008) already made an attempt in that direction by studying the
implications for QTCN when considering dynamic instead of static networks.
Other directions lie in the implementation, application and evaluation of QTC. Efforts in that
direction have already been undertaken by Delafontaine et al. (Delafontaine & Van de
Weghe 2008, Delafontaine 2008). We plan to evaluate the usefulness of QTC-based
information systems through extensive case studies.
Finally, future research may consider how QTC relates to other domains, such as
psychological, cognitive and behavioural sciences, linguistics, information visualisation, and
human-computer interaction. It is our aim to study how QTC relates to cognition and natural
language, e.g. in simple prepositions such as ‘towards’ and ‘away from’ (Bogaert et al. 2008).
2.10 Conclusion
This chapter has presented the Qualitative Trajectory Calculus as a qualitative
spatiotemporal calculus to handle the relations between moving objects adequately. The
development of QTC and which spatial and temporal calculi inspired QTC has been
discussed. The chapter has focused on the two most general and fundamental QTC calculi,
i.e. the Basic and Double Cross types, as they constitute the basis of all other types. The
principal reasoning mechanisms such as conceptual neighbourhoodness and composition
have been considered in some detail, as well as the ability for QTC to deal with incomplete
knowledge. The usefulness and applicability of QTC has been illustrated in a simple case
where, starting from raw trajectory data, a conceptual QTC animation is obtained. Finally,
although QTC is not yet fully theoretically well-documented, we presented three useful
directions for future research.
42 Chapter 2
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Inferring additional knowledge from QTCN-relations 45
3 Inferring additional knowledge from QTCN relations
Delafontaine M., Bogaert P., Cohn A. G., Witlox F., De Maeyer P., Van de Weghe N.
in Information Sciences (2011), Volume 181, Issue 9
Copyright © Elsevier Science
Abstract. It is widely held that people tend to use qualitative rather than quantitative
phrases when raising or answering questions about moving objects. Queries about
whether an object is moving towards or away from another object or whether
objects are getting closer to each other or further away from each other, require
qualitative responses. This characteristic should be reflected in a calculus to be used
to describe and reason about continuously moving objects. In this chapter, we
present a qualitative trajectory calculus of relations between two disjoint moving
objects, whose movement is constrained by a network. The proposed calculus (QTCN)
is formally introduced and illustrated. Particular attention is placed on how to infer
additional knowledge from QTCN relations by means of composition tables and the
transformation of QTCN relations into relations defined by the Relative Trajectory
Calculus on Networks (RTCN).
Keywords. Moving Objects – Qualitative information – Networks – Spatio-temporal
Reasoning
3.1 Introduction
Continuously moving objects are prevalent in many domains such as human movement
analysis (such as traffic planning or sports scene analysis) and animal behaviour science
(Laube, Imfeld & Weibel 2005). Most applications focus on the positional movement of the
object, abstracted to a single point1. Recent advances in various positioning technologies
(e.g. GPS, LBA, wireless communication) (Zeimpekis, Giaglis & Lekakos 2002) allow the
capture and storage of large quantities of such moving point data. Research has addressed
the generation (Brinkhoff 2002, Pfoser & Theodoridis 2003), indexing (Saltenis et al. 2000,
Agarwal, Arge & Erickson 2003, Lee et al. 2007, Francis, Madria & Sabharwal 2008),
modelling (Hornsby & Egenhofer 2002, Güting, de Almeida & Ding 2006, Hornsby & King
2008) and querying (Sistla et al. 1997, Erwig et al. 1999, Laube et al. 2007, Gao et al. 2010) of
moving objects in spatiotemporal databases. However, only recently has work been
conducted in reasoning about the relations between moving point objects and the
1 In the rest of this chapter, when we refer to “moving point objects”, we mean such a moving object whose
spatial extent has been abstracted to a single point, for example its centroid.
46 Chapter 3
transitions between these relations, especially in a qualitative framework (Clementini, Di
Felice & Hernandez 1997, Van de Weghe 2004). A specific proposal for qualitative relations
between disjoint moving point objects is the Qualitative Trajectory Calculus (QTC), which
formally defines qualitative relations between disjoint moving point objects (Van de Weghe
2004).
In this chapter, building on Van de Weghe (2004), QTC is adapted to objects moving in
networks, resulting in QTCN, and its power for representing and reasoning with qualitative
information for objects moving in networks is shown. The chapter is structured as follows.
Section 3.2 describes the difference between qualitative and quantitative information and
explains why qualitative information can be useful. Section 3.3 briefly introduces the
Qualitative Trajectory Calculus (QTC), which is the basis for the Qualitative Trajectory
Calculus for Networks (QTCN) and which is formally outlined in section 3.4. The next two
sections focus on reasoning with QTCN relations. Section 3.5 presents the composition of
QTCN relations, while section 3.6 shows how QTCN relations can be transformed into
relations defined by the Relative Trajectory Calculus on Networks (RTCN). Section 3.7
discusses the usefulness of QTCN in possible applications, leading to conclusions and
directions for further research in section 3.8.
3.2 Qualitative versus quantitative questions
When raising or answering questions about moving objects, both qualitative and
quantitative responses are possible. Typically, when responding to a question in a
quantitative sense, a predefined unit of a quantity on a continuous measuring scale is used
(Goyal 2000). For example, when asked for the speed of a car, the most likely quantitative
answer to that question would be that the car drives at, say, 30 km/h. As Galton (2000) says,
quantitative information is ‘measured by quantity’. In the qualitative approach, the expected
answer will be ‘the car is driving slowly’. Qualitative information is concerned with
information which ‘depends on a quality’ (Galton 2000). A key aspect of qualitative
information, is to find ways to represent continuous aspects of the world (space, time,
quantity, etc.) by a small set of symbols (Forbus 1997, Clementini, Di Felice & Hernandez
1997). In the qualitative approach, continuous information is qualitatively discretised by
landmarks separating neighbouring open intervals, resulting in discrete quantity spaces
(Weld & de Kleer 1989). For instance, one might say that a car driving more than 30 km/h is
driving fast, and a car driving less than 30 km/h is driving slowly.
When describing the movement of objects, a qualitative description can sometimes give a
more satisfactory answer than a quantitative one. For example, if one does not know the
exact speed of a car and a bicycle, but one knows that the speed of the car is higher than the
speed of the bicycle, one can say that the car is moving faster than the bicycle, labelling this
with the qualitative value ‘+’. One could also say that the bicycle is moving slower than the
car, by assigning the qualitative value ‘−’ to this relation. Finally, both objects can also move
Inferring additional knowledge from QTCN-relations 47
at the same speed, resulting in a qualitative value ‘0’. Note that a distinction is only
introduced if it is relevant to the current context (Byrne & Johnson-Laird 1989, Clementini, Di
Felice & Hernandez 1997).
Of particular interest in describing qualitative information, are representations that form a
finite set of jointly exhaustive and pairwise disjoint (JEPD) relations (Renz & Nebel 2007). In a
set of JEPD relations, any two entities are related by exactly one of these relations, and they
can be used to represent definite knowledge with respect to the given level of granularity.
Incomplete or partial knowledge can be specified by coarse relations representing unions
(i.e. disjunctions) of possible JEPD relations.
There are a variety of other grounds why reasoning with qualitative information can be
considered complementary to reasoning in a quantitative way, in areas such as Artificial
Intelligence and Geographic Information Science. A key motive is the fact that human beings
are more likely to prefer to communicate in qualitative categories, supporting their intuition,
rather than using quantitative measures (Freksa 1992b). Representing and reasoning with
qualitative information can overcome information overload. Information overload occurs
whenever more information has to be handled than can be processed (O'Reilly 1980). For
example, it is easier to communicate a certain slope characteristic of a region (e.g. flat,
steep, hilly) than to provide over a thousand height points (Donlon & Forbus 1999). Also,
spatial expressions in natural language are rarely precise (e.g. the library is located in the
centre of the town; he is moving towards the cinema) (Guesgen & Albrecht 2000); in other
words, they usually do not provide enough information to identify the exact geographical
location of an object or event (Kalashnikov et al. 2006). Abstract, non-coordinate-based
methods are necessary to deal with these uncertainties (Frank 1996). Although reasoning
with qualitative information may lead only to a partial answer, such an answer is often
better than having no answer at all (Freksa 1992a). In addition, since the information is more
granular, qualitative reasoning can be computationally easier than its quantitative
counterpart (Freksa 1992b). Finally, qualitative data often provides an ideal way to deliver
insights into a particular problem rapidly, in order to identify potential issues that warrant a
more detailed quantitative analysis (Iwasaki 1997).
3.3 The Qualitative Trajectory Calculus
Mereotopology is the most developed area of qualitative spatial reasoning (Bennett 1997,
Cohn & Renz 2007). However, when it comes to moving point objects, topological models
such as the 9-intersection model merely distinguish two trivial topological relations between
two point objects: equal and disjoint (Egenhofer & Herring 1991). Since in the real world the
mereotopological relationship between most moving objects is that of being disjoint, and
topological models cannot further differentiate between disjoint objects, nor indeed can any
purely topological representation, important questions remain unanswered. An obvious
example is the case of two airplanes, where it is imperative to know whether both airplanes
48 Chapter 3
are likely to stay in a disjoint relation; if not, the consequences are catastrophic. In order to
represent and reason about moving objects the Qualitative Trajectory Calculus (QTC) was
introduced (Van de Weghe 2004). This calculus deals with qualitative relations between two
disjoint moving point objects. QTC can distinguish a number of basic binary relationships
between two moving objects. An object can be moving towards another object; it can be
moving away from another object; or it can be stable with respect to the other object. In
(Van de Weghe 2004), two QTC calculi are defined. The Qualitative Trajectory Calculus –
Double Cross (QTCC) (Van de Weghe 2004, Van de Weghe et al. 2005a) examines relations
between moving point objects based on three reference lines forming a so-called double
cross. The Qualitative Trajectory Calculus – Basic (QTCB) (Van de Weghe & De Maeyer 2005,
Van de Weghe et al. 2006) defines these relations by comparing differences in distance over
time. In order to elaborate a QTC calculus for network-based moving objects, we will build
on QTCB since QTCC is not suitable to use in a network environment, as it utilises a direction-
based spatial reference for defining relations. In the remainder of this section we will briefly
introduce QTCB as defined in (Van de Weghe 2004).
In QTC, time is assumed to be continuous and linear. This time line can be represented by
the set of real numbers ( ) and it has a total order associated with it. This implies that one
cannot identify two time points next to each other. The density of allows no notion
“nextness” (Mortensen 1999). In order to formally define the qualitative relations available
in QTCB, we introduce the following notations and definitions:
denotes the position of an object at time point .
denotes the speed of at time point .
denotes the distance between two positions and .
Definition 3.1 A relation in QTCB at level 1 between a first object and a second object
at a time point is defined by a two character label. This label represents
the following two relationships:
1. Movement of with respect to at :
−: is moving towards :
(3.1)
+: is moving away from :
(3.2)
0: is stable with respect to : all other cases
2. Movement of with respect to at :
Can be described as in 1, with and interchanged, hence:
−: is moving towards (3.3)
Inferring additional knowledge from QTCN-relations 49
+: is moving away from (3.4)
0: is stable with respect to
Definition 3.2 A relation in QTCB at level 2 between a first object and a second object
at a time point is defined by a three character label. The first two
characters are defined as in Definition 3.1. The third character represents
the relative speed and is defined as follows:
3. Relative speed of with respect to at :
−: is moving slower than :
(3.5)
+: is moving faster than :
(3.6)
0: and are moving equally fast:
(3.7)
3.4 The Qualitative Trajectory Calculus on Networks
Having introduced QTC, we will now elaborate the definition of QTC on networks. Moreira et
al. (1999) differentiate between two kinds of moving objects: objects that have a completely
free trajectory, only constrained by the dynamics of the object itself (e.g. a bird flying
through the sky) and objects that have a constrained trajectory (e.g. a train on a railway
track). Many trajectories involving humans are bounded to a network. Hence, there is a need
to develop a calculus that defines qualitative relations between two disjoint moving objects
on trajectories constrained by a network. An informal description and definition of QTCN was
presented in (Van de Weghe 2004, Van de Weghe et al. 2004, Bogaert et al. 2007), while a
conceptual neighbourhood diagram for QTCN was presented in (Bogaert et al. 2007). In this
chapter, QTCN is defined formally. Also, we explore the power of this calculus to infer
additional information from the basic QTCN-relations.
3.4.1 Definitions and restrictions concerning networks and moving objects
A network, such as a road, rail or river network, is usually described as a set of
interconnected linear spatial features; each such linear feature can be regarded as a curve,
describing a linear path through the space it is embedded in. Thus, in essence, a network is a
co-dimensional structure. The concept of co-dimensionality can be used to express the
difference in dimension between spatial entities (point: zero-dimensional; line: one-
dimensional, region: two-dimensional, etc.) and the space they are embedded in (Galton
2000). In the case of a network, one-dimensional structures (a set of interconnected lines)
are embedded in a two dimensional (co-dimension one) or three dimensional space (co-
dimension two). Therefore, we assume an underlying spatial framework S for specifying
locations. Typically this would be , but could be any set with a metric distance function
50 Chapter 3
obeying the triangle inequality, and a notion of curve defined, such that
denotes the set of simple non-closed curves in .
In order to formally define QTCN relations for two moving point objects, using the network in
which they are embedded as a reference frame, three functions are defined on curves. For
any curve :
denotes the length of ;
is true if is an endpoint of ;
if and are two points incident in , then denotes the subcurve of
between and including and .
The network in which objects move in QTCN is characterised by a graph, whose edges
represent a set of linear features and the nodes of the graph represent the endpoints which
bound these linear features (Definition 3.3). A function embeds these nodes and
edges in the spatial framework (Definition 3.4 and 3.5). As stated above, the edges should
represent simple non-closed curves. To formally define this property, we do not allow two
nodes to lie at the same location (Restriction 3.1), the edges should be bounded by two
different nodes (Definition 3.5) and two different edges can only intersect at their respective
endpoints (Restriction 3.2). The number of edges representing curves which intersect at a
node denotes the degree of that node (Definition 3.6).
Definition 3.3 If is a network then is its set of nodes and is its set
of edges.
Definition 3.4 If is a node then is the spatial location of in .
Restriction 3.1 If is a network then .
Definition 3.5 If is a network and then is the curve
denoted by in , and
.
Restriction 3.2 If is a network then
nodes( ).
Definition 3.6 If is a network then the degree of a node ,
.
The movement of objects in QTCN is restricted by the network, which implies that the
location of an object should at all times be situated on an edge (Definition 3.7). As stated in
Inferring additional knowledge from QTCN-relations 51
section 3.3, QTC only considers relations between disjoint objects, thus, two different
objects cannot be at the same place at the same time (Restriction 3.3).
Definition 3.7 An object at a time point is located in a network iff
.
Restriction 3.3 All two non-identical objects and are not instantaneously coincident at
a time point : .
To relate two objects in QTCN, there needs to be at least one path between both objects (see
section 3.4.2). A path is composed of a connected sequence of edges. Since the objects do
not necessarily lie at the endpoint of an edge, a notion of edge segments is required
(Definition 3.8). The notation denotes an edge segment which represents (i.e.
whose location is) that part of an edge between a point and an endpoint of the edge
(including and ). If is the other endpoint of , then the edge segment equals the edge
(as a special case). Thus, a path between two objects is composed of a sequence such that
the first and last elements are edge segments on which the two objects are located (possibly
the same segment), and any intermediate edges form a connected path, such that no edge
occurs more than once (Definition 3.9). The length of a path is defined as the sum of the
length of its edges and edge segments (Definition 3.10). A shortest path is defined as a path
such that there is no path having a shorter length between the same two nodes (Definition
3.11). There can be more than one shortest path between two objects at the same time. If,
in this special case, the first edge segment is different for all of these shortest paths, we
refer to these shortest paths as bifurcating shortest paths (Definition 3.12 and Figure 3.1).
Definition 3.8 If is an edge then is an edge segment of iff
.
Definition 3.9 A path between two different objects and in a network at time
point is a sequence such that
=seg( , | , )
loc( ).
Definition 3.10 is the length of a path .
Definition 3.11 A shortest path in a network from an object to an object at a
time point is a path such that there is no path from to of length less
than . We may write when is such a shortest path.
52 Chapter 3
Definition 3.12 If there are at least two different shortest paths from an object
to an object at a time point , then there is a bifurcating shortest path
from to at iff
.
Figure 3.1 – Bifurcating (a) and non-bifurcating (b) shortest paths.
It is obvious that objects moving on a network do not always move along the same edge
simultaneously. Objects can move from one edge to another. When doing so, they pass a
node (Definition 3.13). If passes a node lying at the intersection of two edges and at
time point , and neither of these edges is along a shortest path from to at , this event is
referred to as a shortest path omitting node pass event (Definition 3.14 and Figure 3.2).
Definition 3.13 An object on a network is in a node pass event along edges at a
time point , iff
.
Definition 3.14 An object on a network is in a shortest path omitting node pass event
with respect to another object at a time point iff
.
Figure 3.2 – A shortest path omitting node pass event.
Inferring additional knowledge from QTCN-relations 53
3.4.2 Definition of QTCN relations
The reference used to qualitatively assess the relation between two objects is the distance
measured along the shortest path. If there is no path between two objects, then there is no
QTCN relation between these objects. Put differently, these objects are either not moving
along a network, or they occupy disjoint parts of a disconnected network and will hence
remain disjoint. The shortest path is chosen because it seems to encode what it means for
one object to approach or recede from another object in a network. (In Euclidean space, one
might naturally define approaching in terms of an angular measure, but this is not applicable
in networks, and shortest path is the appropriate equivalent notion.) In a network, an object
can only approach another object if and only if it moves along a shortest path between these
two objects (Bogaert et al. 2007). Using this property, we can state that an object can only
approach another object at a time point in a network if it does not lie on
immediately before and if it lies on immediately after . moves away from at if
it is on immediately before and if it does not lie on
immediately after . If
lies on only at , but not immediately before and immediately after , or if is on
immediately before and immediately after , then is stable with respect to
(although this relation may only last for an instantaneous moment in time). This property
allows reformulating conditions (3.1, 3.2) of Definition 3.1 for the construction of the first
level relation of QTCB to a QTCN setting.
Definition 3.15 A relation in QTCN at level 1 between a first object and a second object
on a network at a time point is defined by a two character label. This
label represents the following two relationships:
1. Movement of with respect to at :
−: is moving towards :
(3.8)
+: is moving away from :
(3.9)
0: is stable with respect to (all other cases):
(3.10)
(3.11)
2. Movement of with respect to at :
Can be described as in 1 with and interchanged, hence:
54 Chapter 3
−: is moving towards (3.12)
+: is moving away from (3.13)
0: is stable with respect to (3.14, 3.15)
The second level relation of QTCN is defined identically to the definition in QTCB – cf.
Definition 3.2 (Definition 3.16).
Definition 3.16 A relation in QTCN at level 2 between a first object and a second object
in a network at a time point is defined by a three character label. The
first two characters are defined as in Definition 3.15. The third character
represents the relative speed and is defined as follows:
3. Relative speed of with respect at :
−: is moving slower than :
(3.16)
+: is moving faster than :
(3.17)
0: and are moving equally fast:
(3.18)
Based on Definition 3.16, we can construct all canonical cases for QTCN relations at level 2.
Let us analyse all possible movements of a first object with respect to a second object in
a QTCN relation at time point . can be stationary, i.e. not moving with respect to the
network, or not. If is stationary at , it will be located on a shortest path to at (and
immediately before and immediately after ), and therefore the definition yields ‘0’ for the
first character in the label (i). If is moving at , then by definition there are four possibilities
(ii – v). can be on a shortest path to immediately before and not immediately after ,
which returns ‘+’ for the first character in the label (ii). can be on a shortest path to
immediately after but not immediately before , which returns ‘−’ for the first character in
the label (iii). When is in a shortest path omitting node pass event with respect to , it will
not be on a shortest path to just before and after , resulting in a ‘0’ for the first character
in the label (iv). If there is a bifurcating shortest path from to , then will be on a shortest
path to just before and after , which also yields ‘0’ for the first character in the label (v).
The same five cases exist for the movement of the second object in the relation. Hence,
there exist 25 (5²) canonical cases looking at the first level of QTCN. When considering the
second level, the additional three possibilities for the third label character might be
expected to yield 75 (25*3) canonical cases. However, due to the impossibility for a
stationary object to move faster than or equally as fast as a non-stationary object, 18 of
these relations cannot physically occur. The remaining 57 canonical cases are presented in
Figure 3.3. The first column in the figure presents the QTCN relation. In the other columns, an
icon is sketched for all canonical cases. A ‘0n’ denotes a ‘0’ due to a shortest path omitting
node pass event. A ‘0b’ denotes a ‘0’ due to the existence of a bifurcating shortest path
Inferring additional knowledge from QTCN-relations 55
Figure 3.3 – 57 Canonical cases for QTCN at level 2.
56 Chapter 3
Figure 3.3 (continued)
between the objects. The left and right dots represent the positions of (the first object)
and (the second object), respectively. A dot is filled if the object can be stationary. The
arrow symbols represent the potential movement directions of the objects. The arrows can
have different lengths indicating the difference in relative speed.
3.5 Composition
People often make inferences of and from qualitative relations in daily life (Byrne & Johnson-
Laird 1989). For example, if we know that Nico is taller than Philippe and Frank is taller than
Nico, we infer that Frank is taller than Philippe. A specific type of inference mechanism,
which is a fundamental part of a relational calculus, is the composition of its relations (Tarski
1941). The idea behind composition is to compose a finite set of new facts and rules from
existing ones, i.e. if two existing relations and share a common object ( ),
they can be composed into a new relation , denoted by:
– note that may be a disjunction of base relations. If, for a set of relations, the
compositions of all combinations of base relations can be computed, they are usually stored
in a composition table. Composition tables make sense from a computational point of view,
since a compositional inference can simply be looked up, instead of needing complex
computations (Bennett 1997, Vieu 1997). Ever since their introduction, composition tables
have been precomputed for many different temporal (e.g. the interval calculus (Allen 1983)
and the semi interval calculus (Freksa 1992a)), spatial (e.g. topological calculi (Randell, Cui &
Cohn 1992, Egenhofer 1994), directional calculi (Frank 1991, Freksa 1992b), distance calculi
Inferring additional knowledge from QTCN-relations 57
(Hernández, Clementini & Di Felice 1995)), and spatiotemporal calculi (e.g. QTC (Van de
Weghe et al. 2005b)).
3.5.1 Composition of QTCN relations
Since the composition of relative speed (represented by the third character of a level 2 QTCN
relation) is straightforward, this section will focus on the composition of QTCN at level 1.
Nine (3²) QTCN base relations can be distinguished at level 1. As a consequence, the
composition table at level 1 has 81 (9²) entries, each of which potentially contains a subset
of these nine relations. Thus, 729 (93) possible combinations of three relations need to be
examined for their existence or non-existence. For each possibility that actually exists, a
simple ‘animation’ can be drawn to demonstrate its existence. Examples of such animations
for the composition of (+ −) and (− 0) are shown in Figure 3.4.
Figure 3.4 – Animations for the composition of (+ −) and (− 0); a movement arrow next to an
object indicates that the object is passing a node.
Since each composition yields the entire set of base relations, the construction of a
composition table is trivial. This triviality results from the fact that QTCN relations do not
provide sufficient information about the spatiotemporal configuration of the network.
Therefore, in order to obtain sparser composition tables, additional knowledge of the
relation between the network and the moving objects is required. This can be acquired by
imposing temporal as well as spatial constraints.
3.5.2 Temporal Constraints
As a first approach to achieve sparser composition tables, temporal constraints can be
considered. One valuable temporal constraint, perhaps the only general one, is to consider
which relations last over a time interval (rather than those holding only instantaneously). A
‘0’ in a level 1 QTCN label can only hold over a time interval when an object is stationary with
58 Chapter 3
respect to the network, as can be proven using the constraints of continuity (Bogaert et al.
2007). As a consequence, an object which is stationary with respect to one object will also be
stationary with respect to any other object. The composition table according to this
restriction is provided in Table 3.1. The composition table consists of five fine results (i.e.
singleton base relations), all being (0 0), 18 disjunctions of two relations, 22 disjunctions of
four relations and 36 inconsistent compositions (denoted by the empty set). Thus, the total
number of possibilities is reduced from 729 to 129.
Table 3.1 – Composition table for QTCN at level 1 restricted to relations lasting over time intervals;
A0 and B0 stand for the set {−, +}.
3.5.3 Spatial Constraints
While the composition results in Table 3.1 are already much sparser than those obtained
without constraints, they merely provide five fine results. Therefore, as a second approach,
spatial constraints can be imposed on top of the temporal restriction. As shown in section
3.4, the determination of a level 1 QTCN relation merely involves knowledge about the
relative movement with respect to the shortest path(s) between the objects concerned. In
composition, this relative movement is known for the first two object pairs, while nothing is
known about the shortest path(s) of the third pair, leaving all relations possible to occur. For
three objects , and on a network , assume that the relations and
are given and is unknown, implying that and
are known and that
is unknown. If it is known that is on
or that is on , a simple non-
closed curve can be drawn containing the positions of all three objects at . On this curve,
each object has three movement possibilities: it can be stable or move in one of two
opposite directions. Hence, there are 27 (33) movement configurations of these three
objects. An illustration of each specific configuration is shown in Figures 5 and 6, respectively
illustrating the cases of lying on and lying on
. The associated
composition tables are presented in Table 3.2 and Table 3.3. This kind of composition is very
useful, since it always leads to exact knowledge: both tables contain 27 fine composition
results, whereas 54 compositions are inconsistent.
− − − 0 − + 0 − 0 0 0 + + − + 0 + +
− − A0 B0 A
0 0 A
0 B0 A
0 B0 A
0 0 A
0 B0
− 0 A0 B0 A
0 0 A
0 B0
− + A0 B0 A
0 0 A
0 B0 A
0 B0 A
0 0 A
0 B0
0 − 0 B0 0 0 0 B
0 0 B
0 0 0 0 B
0
0 0 0 B0 0 0 0 B
0
0 + 0 B0 0 0 0 B
0 0 B
0 0 0 0 B
0
+ − A0 B0 A
0 0 A
0 B0 A
0 B0 A
0 0 A
0 B0
+ 0 A0 B0 A
0 0 A
0 B0
+ + A0 B0 A
0 0 A
0 B0 A
0 B0 A
0 0 A
0 B0
Inferring additional knowledge from QTCN-relations 59
Figure 3.5 – Possible relative movement configurations in QTCN for R1(k, l) R2(l, m) where m lies
on the simple shortest path between k and l and none of the objects is located at a node.
− − − 0 − + 0 − 0 0 0 + + − + 0 + +
− − − + − 0 − −
− 0 − + − 0 − −
− + − + − 0 − −
0 − 0 + 0 0 0 −
0 0 0 + 0 0 0 −
0 + 0 + 0 0 0 −
+ − + + + 0 + −
+ 0 + + + 0 + −
+ + + + + 0 + −
Table 3.2 – Composition table for relative movement in QTCN, for R1(k, l) R2(l, m) where m lies on
the simple shortest path between k and l and none of the objects is located at a node.
60 Chapter 3
Figure 3.6 – Possible relative movement configurations in QTCN for R1(k, l) R2(l, m) where k lies
on the simple shortest path between m and l and none of the objects is located at a node.
− − − 0 − + 0 − 0 0 0 + + − + 0 + +
− − + − + 0 + +
− 0 + − + 0 + +
− + + − + 0 + +
0 − 0 − 0 0 0 +
0 0 0 − 0 0 0 +
0 + 0 − 0 0 0 +
+ − − − − 0 − +
+ 0 − − - 0 − +
+ + − − − 0 − +
Table 3.3 – Composition table for relative movement in QTCN, for R1(k, l) R2(l, m) where k lies on
the simple shortest path between m and l and none of the objects is located at a node.
3.6 Transforming QTCN into the Relative Trajectory Calculus on Networks
Having defined the QTCN relations between a pair of moving objects, a set of trivial
qualitative questions can be answered. For example, by looking at the third character of the
label, one can identify which object is moving the fastest. Looking at the first two characters
Inferring additional knowledge from QTCN-relations 61
of the QTCN label, queries such as whether an object is moving towards or away from
another object can be resolved. In addition to these trivial questions, QTCN at level 2 has the
power to answer additional questions using the information contained by all three
characters in the label. This information can be obtained by transforming QTC relations into
relations defined by the Relative Trajectory Calculus (RTC) (Van de Weghe 2004).
In contrast to QTC, where distances between objects at different times are compared (e.g. in
Definitions 3.1 and 3.15), RTC defines relations based on the relative motion of an object
against another object at the same moment in time (Van de Weghe 2004) (Definition 3.17).
Definition 3.17 A relation in RTC between a first object and a second object at a time
point is defined by a single character label. This label represents the
comparison of the distance between and immediately before with the
distance between and immediately after . This results in three
possibilities:
−: the distance between and decreases:
(3.19)
0: the distance between and remains the same:
(3.20)
+: the distance between and increases:
(3.21)
RTCN describes the RTC relations on networks. In what follows, we will show that every QTCN
relation can be mapped onto a single RTCN relation. This allows QTCN at level 2 to answer
questions such as whether two objects are getting closer to each other or whether they are
getting further away from each other. To this end, we will first consider the cases where the
union of all shortest paths over the entire time span can be described as a simple curve
without junctions. Note that this excludes, among others, the case of bifurcating shortest
paths (Figure 3.1) and shortest path omitting node pass events (Figure 3.2). Hence, the
following equalities apply for the QTCN relation between the objects and at time point :
A ‘−’ in the first character of the relation label implies:
(3.22)
A ‘+’ in the first character of the relation label implies:
(3.23)
62 Chapter 3
Analogous reasoning applies for ‘−’ and ‘+’ in the second label character, yielding 3.24 and
3.25. Regardless of the QTCN relation it follows from (3.22-3.25) that:
(3.26)
Theorem 3.1
A QTCN relation (− − −) between the objects and at a time point can be transformed
into an RTCN relation (−), such that the RTCN relation is true whenever the QTCN relation is
true.
Proof:
By definition, the first two characters of (− − −) imply:
(3.27)
(3.28)
From (3.27) and (3.28) it follows that:
(3.29)
(3.30)
(3.31)
Which is by definition equal to the RTCN relation (−).
Analogously, it can be proven that the QTCN relations {(− − 0), (− − +), (− 0 +),
(0 − −)} can be converted into the RTCN relation (−).
Theorem 3.2
A QTCN relation (+ + +) between the objects and at time point can be transformed
into an RTCN relation (+), such that the RTCN relation is true whenever the QTCN relation is
true.
Proof:
By definition, the first two characters of (+ + +) imply:
(3.32)
(3.33)
From (3.32) and (3.33) it follows that:
(3.34)
(3.35)
(3.36)
Which is by definition equal to the RTCN relation (+).
Analogously, it can be proven that the QTCN relations {(+ + 0), (+ + −), (+ 0 +),
(0 + −)} can be converted into the RTCN relation (+).
Inferring additional knowledge from QTCN-relations 63
Theorem 3.3
A QTCN relation (− + −) between the objects and at time point can be transformed
into an RTCN relation (+), such that the RTCN relation is true whenever the QTCN relation is
true.
Proof:
By definition, the third character of (− + −) implies:
(3.37)
(3.38)
(3.39)
(3.40)
(3.41)
(3.42)
(3.43)
Which is by definition equal to the RTCN relation (+).
Analogously, it can be proven that the QTCN relation (− + +) can be converted into the
RTCN relation (+), that the QTCN relations {(+ − −), (+ − +)} can be converted into the
RTCN relation (−), and that the QTCN relations {(− + 0), (+ − 0), (0 0 0)} can be
converted into the RTCN relation (0).
Note that the above mentioned theorems are not valid when the union of shortest paths
does not constitute a simple curve over the considered time span, since equations (3.22-
3.25) are not valid. Based on restrictions imposed by continuity, it can be shown that, in
these cases, there is also a unique transformation from a QTCN relation into a single RTCN
relation. Consider a qualitative variable capable of taking any of the three qualitative values
‘−’, ‘0’ and ‘+’. Due to continuity, this variable cannot make a direct change from ‘−’ to ‘+’
and vice versa, since such a change must always pass the intermediate value ‘0’ (Galton
1995). Let us consider the shortest path omitting node pass event in Figure 3.7. In Figure
3.7a there is a QTCN relation (− 0 +), which can be transformed into the RTCN relation (−),
according to Theorem 3.1. Analogously, (+ 0 +) can be transformed into (+) in Figure 3.7c.
Then, due to the above restriction imposed by continuity, the QTCN relation (0 0 +) in
Figure 3.7b must be an RTCN relation (0).
Similar transformations apply for all QTCN relations occurring at shortest path omitting node
pass events or when there are bifurcating shortest paths. Table 3.4 provides an overview of
the transformations from each canonical case in QTCN to the respective RTCN relation. A ‘0n’
denotes that a ‘0’ is due to a shortest path omitting node pass event. A ‘0b’ denotes that a
‘0’ is due to the existence of a bifurcating shortest path between the objects. A ‘0s’ denotes
64 Chapter 3
a ‘0’ is due to a stationary object. The black cells indicate that no corresponding RTCN
relations physically exist.
Figure 3.7 – A transition in QTCN from (− 0 +) via (0 0 +) to (+ 0 +).
QTCN-label RTCN-label QTCN-label RTCN-label QTCN-label RTCN-label
− − − − 0s 0s 0 0 0n 0n + 0 − − 0 − 0s 0s + 0s + − + − − + − 0b 0s − 0s + 0 − 0s − 0b 0s 0 0s + + − 0s 0 0b 0s + 0 0b + − + − 0s + − 0n 0s − 0b + 0 0 − 0b − 0 0n 0s 0 0b + + 0 − 0b 0 0 0n 0s + 0 0n + − + − 0b + − 0s 0b − 0 0n + 0 0 − 0n − 0 0s 0b 0 0n + + 0 − 0n 0 0 0s 0b + + − − − − 0n + − 0b 0b − 0 + − 0 0 − + − + 0b 0b 0 0 + − + + − + 0 0 0b 0b + 0 + 0s − − + + − 0n 0b − 0 + 0s 0 0s − − − 0n 0b 0 0 + 0s + + 0s − 0 0n 0b + 0 + 0b − 0 0s − + 0s 0n − 0 + 0b 0 0 0b − − − 0s 0n 0 + 0b + + 0b − 0 0 0s 0n + + 0n − 0 0b − + 0 0b 0n − 0 + 0n 0 0 0n − − − 0b 0n 0 0 + 0n + + 0n − 0 0 0b 0n + 0 + + − + 0n − + 0 0n 0n − 0 + + 0 + 0s 0s − 0n 0n 0 0 + + + +
Table 3.4 – Transformations from all QTCN canonical cases to RTCN relations.
Table 3.4 clearly shows that the ‘0s’, ‘0n’, and ‘0b’ labels do not influence the transformation
from QTCN to RTCN. Therefore, Table 3.4 can be reduced to Table 3.5.
Thus, there is a one-to-one mapping from QTCN to RTCN relations. This is notable since for
QTC relations of objects having a free trajectory in , this is not the case (Van de Weghe
2004). The latter is illustrated in Figure 3.8. Since the dotted line has a fixed length, the
figure shows that a QTCB relation (− + 0) can be transformed into all possible RTC relations.
Inferring additional knowledge from QTCN-relations 65
QTCN-label RTCN-label QTCN-label RTCN-label QTCN-label RTCN-label
− − − − 0 − − − + − − − − − 0 − 0 − 0 0 + − 0 0 − − + − 0 − + 0 + − + + − 0 − 0 0 0 − 0 + 0 − 0 − 0 0 0 0 0 0 0 + 0 0 0 − 0 + − 0 0 + 0 + 0 + + − + − + 0 + − + + + − + − + 0 0 0 + 0 0 + + 0 + − + + − 0 + + 0 + + + +
Table 3.5 – Transformations from QTCN into RTCN relations.
Figure 3.8 – Examples of transformations from QTCB to RTC.
3.7 Discussion
On the one hand, defining and examining the properties of a distance based calculus for
moving objects constrained by networks is a worthwhile theoretical investigation into
further aspects of QTC theory. On the other hand, we argue that this calculus is also
convenient for the use in applications. In this section, we will illustrate this usefulness by
means of two examples.
3.7.1 A Police/Gangster Example
In order to show the applicability of the composition of QTCN relations at level 1 and the
usefulness of the temporal and spatial constraints stated in section 3.5, let us consider the
following example where three policemen , and are at different locations in a city
and wish to catch a gangster along a road network (Figure 3.9). It is assumed that the
policemen know their mutual positions and therefore their mutual shortest paths at any
time, but they can only see the gangster if they are in line of sight. At time , while and
are still awaiting instructions, policeman has noticed and started chasing the gangster
who started to escape, thus yielding = (− +) (Figure 3.9a). Since all shortest paths
are simple and is on both
and
, composition can be applied using Table
3.2, such that can give the right orders to and concerning the direction in which
they should move, i.e. directs and to start moving towards (since and know
where is), which causes and to move towards just after . At , is at a junction.
Hence, composition cannot be applied, since one cannot know which turn will take (Figure
3.9b). Immediately after , will have seen turning right, and so still knows at which
66 Chapter 3
edge is, thus enabling composition with respect to and . This situation lasts until
(Figure 3.9c) and will continue after , probably until the gangster gets caught. Table 3.6
lists the respective composition results inferred over . As can be noted, results are
only lacking at , whereas during the rest of the period there is complete information due
to the existing spatiotemporal constraints.
Time Known relations Results inferred from
temporal constraints
Results inferred from
spatial constraints
= (0 −),
= (0 0),
= (− 0),
= (− +)
= (0 −)
(0 +)
= (0 −)
(0 +)
= (0 −),
= (− 0)
= (− −),
= (− +),
= (− −),
= (− +)
= (− −)
(− +)(+ −)(+ +)
= (− −)
(− +)(+ −)(+ +)
= (− −),
= (− −)
= (0 −),
= (0 0),
= (− 0),
= (− +)
None possible None possible
= (0 −),
= (0 0),
= (− 0),
= (− +)
= (− −)
(− +)(+ −)(+ +)
= (− −)
(− +)(+ −)(+ +)
= (− −)
= (0 −),
= (0 0),
= (− 0),
= (− +)
= (− −)
(− +)(+ −)(+ +)
= (− −)
(− +)(+ −)(+ +)
= (− −)
Table 3.6 – Composition results inferred over [t1, t3] due to spatial and temporal constraints.
Figure 3.9 – Simplified animation of three policemen chasing a gangster.
Inferring additional knowledge from QTCN-relations 67
3.7.2 A Collision Avoidance Application
An application where QTCN at level 2 can be useful is in collision avoidance systems. If one
wants to know if two objects are going to collide, then it is useful, as a first step, to restrict
attention to the objects that might meet. In other words, only the objects which are getting
closer to each other, i.e. objects in an RTCN relation (−), are relevant, because otherwise
they cannot collide. Thus, QTCN relations at level 2 eliminate many movements from further
examination, greatly reducing calculation times. Further examination of the remaining
relations gives information on the type of collision. The relations (− + +) and (+ − −)
indicate possible rear-end collisions, whereas (− − −), (− − 0), and (− − +) indicate
possible head-on collisions. The relations (− 0 +) and (0 − −) may indicate collisions with
a stationary object. Note that these QTCN relations indicate potential collisions that do not
necessarily result in real collisions. Related work on collision avoidance has, on the one hand,
focussed on detecting possible collisions between objects which have a completely free
trajectory in a two dimensional space (Schlieder 1995, Gottfried 2005, Dylla & Wallgrun
2007). These approaches mainly focus on the direction of movement. Although they have all
shown their usefulness when the movement of objects is unconstrained, directional
methods can not directly be transformed to networks, since they do not take into account
the spatial structure of a network. The movement in Figure 3.10a, for example, would
announce a possible collision in all the above mentioned directional approaches, while from
QTCN analysis it follows that the objects move away from each other and therefore cannot
collide. Furthermore, none of the methods above incorporates the relative speed between
two moving objects. However, the notion of relative speed is crucial for collision detection in
cases where the objects move in the same direction, while in the other cases, it may offer
appealing insights into a finer subdivision of collision types. Consider the example in Figure
3.10b. When using only directional information, this movement would trigger a collision
detection, but since is moving faster than , the distance between them increases, and
hence, there is no true collision danger. For both these reasons, directional approaches over-
predict possible collisions, while QTCN does not.
Figure 3.10 – Two scenes without collision danger for two moving objects.
Other techniques for collision avoidance considering network-constrained objects mainly
focus on railway networks. Collisions in these systems are avoided by disallowing two trains
to occupy the same track segment (Hansen 1998, Haxthausen & Peleska 2000). First of all,
these methods also over-predict possible collisions, since two trains may travel on the same
track without colliding (e.g. as in Figure 3.10b with moving slower than ). Secondly, this
68 Chapter 3
sole constraint does not capture every possible collision situation. If two trains are on
different segments, they can still be close and moving towards each other. Hence, not all
collisions can be predicted in collision avoidance systems relying on this constraint
(especially for objects colliding at network junctions).
3.8 Conclusions and future work
In this chapter, we have formally presented the Qualitative Trajectory Calculus on Networks
(QTCN) as a qualitative calculus to represent and reason about moving point objects which
are constrained in their movement by networks.
We have shown two techniques to infer additional knowledge from the basic QTCN relations.
On the one hand we have presented the composition of QTCN relations (section 3.5). It was
found that, at level 1, each QTCN base relation is a possible result for each composition of
two relations. While this result, at first, can be considered of limited use, it was shown how
sparser and more powerful composition tables may be obtained by imposing realistic
additional spatial and temporal constraints. By excluding instantaneous relations, we were
able to reduce the total of 729 possibilities to 129 (18%). In addition, by restricting to the
case where the union of shortest paths involved in the composition forms a non-closed
curve, a further reduction was made to 27 (4%) fine results (i.e. singleton base relations).
These sparser composition tables are more powerful and useful with respect to potential
applications, as has been illustrated in section 3.7.1.
On the other hand, we have demonstrated that QTCN is able to answer qualitative questions
such as whether objects on a network are moving towards or away from each other. These
queries are not limited to trivial questions which merely relate to the relationship
represented by a single QTCN relation character. Hence, a QTCN relation conveys more
information than each of its individual label characters separately. As pointed out in section
3.6, each canonical relation in QTCN at level 2 can be uniquely transformed into a RTCN
relation (this is not the case for QTCB in (Van de Weghe 2004)). Therefore, QTCN is
capable of answering questions such as whether two objects are getting closer to each other
or whether they are getting further away from each other. In section 3.7.2, we have
illustrated that the definition of QTCN and the unique transformation of its relations into
single RTCN relations can be useful, for example in collision avoidance systems.
The theoretical contributions in this chapter complement the earlier contributions vis-à-vis
other calculi of the QTC family (see Delafontaine et al. 2011 for an overview) in general, and
regarding QTCN in particular. While QTCN relations have been introduced in a brief and
informal manner in earlier work (Van de Weghe 2004, Bogaert et al. 2007), this chapter
offers a formal axiomatisation of QTCN. In addition to the conceptual neighbourhood
diagrams presented in (Bogaert et al. 2007), we have presented the composition tables for
QTCN as well as the transformation of QTCN into RTCN relations. Furthermore, we have
Inferring additional knowledge from QTCN-relations 69
explored and illustrated the reasoning power of QTCN by means of its ability to answer
qualitative queries. As has been recently shown for QTCB and QTCC (Delafontaine, Cohn &
Van de Weghe 2011), these contributions will allow QTCN to be implemented in an
information system in order to represent and reason about moving objects constrained by
networks.
Among the qualitative calculi that deal with relations between moving objects, QTCN is
unique in its consideration of network-based objects. An exception is the work of Wang et al.
(2005) who extend the Directed Interval Algebra (Renz 2001) to the Road Network Directed
Interval Algebra (RNDIA). Although their algebra is also based on the notion of shortest
paths, RNDIA differs from QTCN as it defines relations among directed network tracks rather
than relations among moving point objects. RNDIA is therefore less appropriate to represent
and reason about instantaneous events occurring among objects along their trajectories.
Collisions, for example, are not unambiguously represented in RNDIA as they may occur in
the case of different RNDIA base relations (e.g. the equal, overlay, and cross relations (Wang
et al. 2005). Given that practically all traffic movements are bounded by networks, QTCN-
based applications are promising in the field of Intelligent Transportation Systems and
Geographic Information Systems for Transportation (GIS-T) (Shaw & Rodrigue 2009).
Ongoing research involving QTCN is being conducted on cognitive aspects of the calculus.
Major questions to be investigated in this respect include what specific terms such as motion
verbs and prepositions do people attach to each of the canonical cases of the calculus.
Future findings on these issues may provide insights on the power of QTCN in natural
language processing and human computer interaction.
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Qualitative relations between moving objects in a dynamic network 73
4 Qualitative relations between moving objects in a
network changing its topological relations
Delafontaine M., Van de Weghe, N., Bogaert P., De Maeyer P.
in Information Sciences (2011), Volume 178, Issue 8
Copyright © Elsevier Science
Abstract. The Qualitative Trajectory Calculus on Networks (QTCN) defines qualitative
relations between two continuously moving point objects (MPOs) moving along a
network. As prevailing in other research, this network is presumed static in QTCN.
Actually, in many cases, networks are dynamic entities. For example in a road
network, the opening of a bridge can temporarily close the connection between two
junctions; traffic jams and traffic lights increase the time needed to travel from A to
B. Therefore, it is interesting to examine what happens with qualitative relations
between two continuously moving point objects if there are changes in the network.
In this chapter, we introduce QTCDN’, being the Qualitative Trajectory Calculus on
Changing Networks able to handle topological network changes. Potential
applications of the calculus in transportation are highlighted, clearly illustrating the
usefulness of the calculus.
Keywords. Qualitative calculus – Moving point objects – Changing networks –
Topological relations
4.1 Introduction
In recent years, time and space have grown into scarce items, being now factors taking part
in widely divergent human decisions. Choices require weighing space and time against each
other. This is possible due to their relation in the notion of ‘change’. A suchlike change, that
we are familiar with, is the movement of an object in time through space. Movements can
occur in a free space or can be spatially restricted by certain factors. A network, e.g. a road
or a river network, being a set of interconnected linear features, is an example of a space in
which object movements are restricted. Mostly, these moving objects are studied in a static
network (Monferrer & Lobo 2002, Pfoser & Jensen 2005, Güting, de Almeida & Ding 2006,
Kim et al. 2006, Li & Lin 2006, Liu, Do & Hua 2006, Lee et al. 2007). Due to this,
spatiotemporal changes of our complex and dynamic reality are neglected, which makes
static models difficult to justify and hardly tenable in a changing world counting in
nanoseconds. Therefore, this chapter concentrates on setting up a calculus of relations
74 Chapter 4
between moving objects in a network changing its topological relations. Topological
relations are the geometric relations between spatial objects that are invariant under
homeomorphisms such as translation, rotation and scaling (Bennett 1997, Vieu 1997). Since
topological relations determine the entire structure of a network, and of a changing network
as well, this chapter focuses on changes of topological relations in a network. Such changes
are present in many real world networks. Consider for example railway barriers closing a
street segment in a road network. The street concerned, first being modelled as a connected
linear feature in the network, becomes interrupted due to this event. This street now has to
be represented by two different lines being no longer connected with each other.
For the setting up of a calculus of relations in this chapter, a qualitative calculus was
preferred to a quantitative one, because qualitative information is the most essential and
pithy kind of information (Clementini, Di Felice & Hernandez 1997, Cohn & Renz 2007) and it
is the most rapidly processed by humans (Monferrer & Lobo 2002). In addition, qualitative
reasoning techniques are becoming increasingly important e.g. in Geographical Information
Science (Saygin, Ulusoy & Yazici 1999, Claramunt & Theriault 2004, Duckham et al. 2006,
Nedas, Egenhofer & Wilmsen 2007) and Artificial Intelligence (Gerevini 2005, Rebolledo
2006, Badaloni & Giacomin 2006). Appeal has been made to an existing qualitative
approach, the Qualitative Trajectory Calculus (Van de Weghe 2004), explained in section 4.2.
This calculus has already been elaborated for static networks by the introduction of the
Qualitative Trajectory Calculus on Networks (Bogaert et al. 2007), briefly stated in section
4.3. In section 4.4, we present the Qualitative Trajectory Calculus on Changing Networks able
to undergo topological change. Finally, conclusions and possible future work are mentioned
in section 4.5.
4.2 The Qualitative Trajectory Calculus
The Qualitative Trajectory Calculus (Van de Weghe 2004), shortly QTC, is a qualitative
calculus of relations between two disjoint point objects in space. Every object can be given a
relation describing its motion with respect to another object. Depending on the level of
detail and the number of spatial dimensions taken into consideration, different types of QTC
have been worked out in (Van de Weghe 2004). Since this chapter deals with networks, we
will focus on the Qualitative Trajectory Calculus on Networks or briefly QTCN.
4.3 The Qualitative Trajectory Calculus on Networks
4.3.1 Definition
The Qualitative Trajectory Calculus on Networks (Bogaert et al. 2007), shortly QTCN,
considers two point objects in a static network. The objects have the ability to move
Qualitative relations between moving objects in a dynamic network 75
continuously along the network, which remains unaltered. Bogaert et al. (2007) use the
following definitions and assumptions for further formalisation of QTCN1:
A graph is a set of edges and a set of nodes .
A network is a connected graph and a finite set of objects.
Each edge connects a pair of nodes in a network.
Each node has a degree, which is the number of edges connected to it.
At any time, each object has a position in the network, which is either at a node in ,
or is along an edge in , in which case the network at is augmented with an
additional dynamic node of degree 2 cutting the edge in two, representing .
A dynamic node is a node representing the position of the object in a network
A subgraph of a network is a set of edges and a set of nodes , such that is a subset
of the set of edges of the network and is a subset of the set of nodes of the network
A path from to at is a subgraph of the network at time , such that every node in
is of degree 2 except two nodes representing the position of and , which are of
degree 1. Thus, a path is a sequence of nodes and edges from to .
Every edge has an associated length, which is a positive number.
The length of a path is the sum of the lengths of the edges in the path.
If is a path of length and there is no path with length less than between the
same two nodes, then is a shortest path2.
A cycle is a subgraph that has at least three non dynamic nodes and contains the same
number of edges and nodes, whereby each edge is of degree 2.
In accordance with the above mentioned, a QTC network is a graph, which itself is not a
spatial structure, but needs to be embedded in a space or must be ‘spatialised’ (Galton &
Worboys 2005). For that, a function can be used, which maps each node of the graph onto a
point in the defined space, and maps each edge of the graph onto a curve segment (Galton
& Worboys 2005).
In a network, a moving object can only approach another object, if and only if it moves along
a shortest path between these two objects (Bogaert et al. 2007). Therefore, the shortest
path between two QTCN objects is used to determine relative movements of the objects with
1 We introduce the following notations for QTC:
denotes the position of an object at time ;
denotes the distance between two positions and ;
denotes the speed of at time ;
denotes that is temporally before ;
denotes the time period immediately before ;
denotes the time period immediately after ;
denotes the shortest path at time between objects and .
2 Note that multiple shortest paths between two objects might be possible.
76 Chapter 4
respect to each other. When an object approaches another object , it moves along the
shortest path between and . In that case, we can state that during and
during . Object moves away from if it lies on
during and does not lie
on during . If lies on
only at but not during or , or if it lies on
throughout , then the object will be stationary with respect to .
4.3.2 Relations in QTCN
A QTCN relation, representing the qualitative relation between two moving objects in a QTC
network, is expressed in a typical three character label. This label compromises the following
three relationships (Bogaert et al. 2007):
Assume two objects and .
1. Movement of the first object , with respect to the position of the second object at
time point :
−: moves along a shortest path:
(4.1)
+: does not move along a shortest path:
(4.2)
0: is stationary with respect to :
(4.3)
(4.4)
2. Movement of the second object , with respect to the position of the first object at
time point can be described as in case 1 with and interchanged, hence:
−: moves along a shortest path
+: does not move along a shortest path
0: is stationary with respect to
3. Relative speed of the first object at time point , with respect to the second object
at time point :
−: is slower than :
(4.5)
+: is faster than :
Qualitative relations between moving objects in a dynamic network 77
(4.6)
0: and are equally fast:
(4.7)
Thus, a (+ − 0)-label means that a first object is moving away from a second object (+),
while the second object is approaching the first object (−), and that both objects have the
same speed (0). Since a label contains three characters which can all take on three values,
there are mathematically 3³ (27) different possible relationships in QTCN. In (Bogaert et al.
2007), it is stated that all of the mathematically possible relations exist in QTCN, although not
all of them can last over an interval of time. 17 relations can hold over a time interval, while
10 relations can only exist at a certain point in time.
4.3.3 Transitions in QTCN
Freksa (1992) defines two relations between objects to be conceptual neighbours if they can
be directly transformed into one another by continuously deforming the objects in a
topological sense. The transformation of one relation into a conceptual neighbour is called a
transition. A combination of a transition between a relation A and B and a transition
between B and A, is denoted a transition pair.
A transition can only occur in QTCN because of the movement of at least one of both objects.
Bogaert, Van de Weghe and De Maeyer (2004) distinguish three possible events3 that can
cause transitions in QTCN:
1. Speed Change Event: the relative speed of the objects changes. Figure 4.1 illustrates a
Speed Change Event.
2. Node Pass Event: an object passes a node with degree of 3 or higher, whereby the
movement of the object with respect to the shortest path changes. Figure 4.2 illustrates
a Node Pass Event.
3. Continuous4 Shortest Path Change Event: the shortest path between the objects
changes, due to the continuous movement of at least one of both objects, in such way
that the movement of an object with respect to the shortest path changes. Figure 4.3
illustrates a Continuous Shortest Path Change Event.
Transitions can graphically be represented by means of a Conceptual Neighbourhood
Diagram (CND). In a CND, relations are displayed as nodes, transitions as arrows and
transition pairs as undirected lines joining two conceptual neighbours. The use of CNDs is
well established in AI applications (i.e. in Qualitative Spatial Reasoning (QSR)) to represent
3 For a full understanding of the figures presenting the events, one has to be mindful of the constraints
imposed by continuity formulated further in the ‘Theory of Dominance’. 4 Although the above defined events are all marked by continuous change, in this case the continuous
movement of the objects, the adjective ‘Continuous’ has been added here to indicate the difference with the Discontinuous Shortest Path Change Event defined in section 4.
78 Chapter 4
and reason about qualitative properties (e.g. to predict possible future events) (Randell &
Mitkowski 2004, Dylla & Wallgrun 2007). Figure 4.4 shows the CND of QTCN.
Figure 4.1 – Speed Change Event.
Figure 4.2 – Node Pass Event.
Figure 4.3 – Continuous Shortest Path Change Event.
The CND is totally bidirectional5 and it clearly shows that all of the 27 (3³) mathematically
possible relations exist in QTCN. As stated before, not all of them can last over an interval.
The ten dashed nodes represent the relations that can only exist at a specific point in time.
Not all the relations are conceptual neighbours to each other: of the 702 mathematically
possible transitions (351 transition pairs), only 152 (76 transition pairs) are possible in QTCN.
Much of these limitations can be explained by means of the theory of dominance.
5 All transitions are grouped in transition pairs
Qualitative relations between moving objects in a dynamic network 79
Figure 4.4 – The CND of QTCN.
4.3.4 Theory of Dominance
The concept of ‘dominance’ was introduced by Galton (1995b), based on Forbus’ work
(1984). Galton described the temporal nature of transitions between qualitative variables,
which are the QTCN relation characters in this context. Some important restrictions
concerning the dominance between binary qualitative relations were stated. Central in his
theory of dominance are the constraints imposed by continuity, which consequently apply to
all kinds of continuous changes. Transitions in QTCN are subject to these constraints as the
objects can only move continuously and only the objects can cause transitions in QTCN.
We will illustrate the idea of dominance with the following example. Consider a qualitative
variable capable of taking on the values ‘−’, ‘0’ and ‘+’ and able to change continuously
between them. It is clear that a direct change from ‘−’ to ‘+’ and vice versa is impossible,
since such a change must always pass the qualitative value ‘0’. This value ‘0’ only needs to
hold for a certain point in time. On the other hand, a ‘+’ or ‘−’ value must always hold over
an interval (Galton 1995a). Let us briefly explain this point. Consider two different points of a
continuous trajectory. One can always find another intermediate point in between these
points. In fact, an infinite number of points lie in between two different points. Consider now
a number line with a marked ‘0’ point. Since a ‘+’ value can stand for either positive number,
there is always an infinite number of points on the number line between ‘0’ and ‘+’. It now
follows that a ‘+’ value always holds over an interval in time. Dual reasoning applies for a ‘−’
80 Chapter 4
value. Using Galton’s (1995a) terms, we say that ‘0’ dominates ‘−’ and ‘+’, and that ‘−’ and ‘+’
are dominated by ‘0’.
Applied to QTCN, we see the impossibility of a transition between a relation with ‘−’ or ‘+’ in
one character and a relation with respectively ‘+’ or ‘−’ in the corresponding character.
Bringing this into account, 506 mathematically possible transitions become left out in QTCN,
which makes only 196 theoretically possible transitions (98 transition pairs) rest in the CND.
In addition, another 44 transitions are omitted due to restrictions imposed by the network,
which is not a free, more-dimensional space, reducing the total number of possible
transitions to 152 (76 transition pairs) (Figure 4.4).
4.4 Topological changes in networks: QTCDN’
4.4.1 Topological Change and Dynamic Networks
A topological change in a network is a change in the topological relations of a network. As
stated yet, a QTC network is a connected graph. A graph fully determines the topological
relations of a network when embedded in a space. Consequently, every topological change
of a network corresponds with a change in the graph that describes it and vice versa. The
only changes a graph, which is a set, can undergo are the addition and the deletion of one or
more of its elements. As such, we can consider every topological change of a network as an
edge addition6, an edge deletion7 or a combination of both. Additions and deletions of nodes
can be disregarded as they always imply the additions or deletions of edges in a connected
graph.
An important characteristic of topological changes is that they are discontinuous and by
consequence temporally restricted to time points: the change of a topological relation
always happens at a certain point in time and cannot take place over a time interval.
Reckoning with the theory of dominance this means that topological changes either happen
at the same time or that there is a time interval separating them.
A changing network is a network capable of undergoing change on every possible point in
time t (which does not mean that the network does change on every possible point in time).
It then follows that all relations and transitions possible in static networks (QTCN) are also
possible in dynamic networks as dynamic networks can always behave as static networks.
Where QTCN applies to static networks, we define the Qualitative Trajectory Calculus on
Changing Networks able to undergo topological changes (QTCDN’8) to be the calculus of
6 Addition of an edge includes the addition of node or (explicit) node if that node doesn’t belong to
the network yet. 7 Deletion of an edge includes the deletion of node or (explicit) if that node was only connected to
the network by . 8 In QTCDN’, N’ stands for dynamic networks (versus N for static networks) and D stands for discontinuous, as
topological changes are discontinuous changes.
Qualitative relations between moving objects in a dynamic network 81
changes in qualitative relations between two point-like disjoint objects in a changing
network able to undergo topological changes.
4.4.2 Relations in QTCDN’
Since all 27 mathematically possible relations exist in QTCN, they consequently exist in
QTCDN’. As for QTCN, 10 of these relations are restricted to time points in QTCDN’ (dashed
nodes in Figure 4.4). This restriction is due to the spatial restraints of a network, which is in
essence a one-dimensional structure embedded in a more-dimensional space.
4.4.3 Transitions in QTCDN’
As stated before, in QTCN there are three possible events causing transitions: a Speed
Change Event, a Node Pass Event and a Continuous Shortest Path Change Event. To find out
which transitions can be caused by topological network changes, one has to ask the question
which of these events can be caused by topological network changes. It is self-evident that a
topological change cannot alter the speed of the objects, what makes a Speed Change Event
to be excluded. Also a Node Pass Event is impossible since an object itself determines how it
will pass a node with degree of 3 or more. Unlike the two mentioned aspects, a topological
change in a network can certainly change the shortest path between two objects.
Nevertheless, it cannot be stated that a topological change can cause a Continuous Shortest
Path Change Event. This is because of the discontinuous nature of a topological change. Let
us briefly illustrate this point. A Continuous Shortest Path Change Event is, following from
the theory of dominance, characterized by a continuous transition of a relation character
from ‘−’ via ‘0’ to ‘+’ or vice versa. That way, there is always a time point, corresponding with
the ‘0’, on which (at least) two different shortest paths exist simultaneously in the network
(Figure 4.3). Consider now a shortest path change due to a topological change in a network
as illustrated in Figure 4.5. It shows that there is no longer a continuous transition, but a
direct change between ‘−’ and ‘+’ without passing ‘0’. Also, a time point is missing on which
two shortest paths exist at the same time. Therefore, we term such an event as a
Discontinuous Shortest Path Change Event.
Note that new transitions possible in topological changing networks are not restricted to
those caused uniquely by Discontinuous Shortest Path Change Events, but also include all
transitions due to combinations of Discontinuous Shortest Path Change Events with all other
events. Fig. 6, for example, shows a combination of a Discontinuous Shortest Path Change
Event with a Node Pass Event. Fig. 7, 8 and 9 give, by means of CNDs, an overview of all
transitions possible in QTCDN’ and impossible in QTCN. Combining all these CNDs with the
CND of QTCN (Figure 4.4) would yield the CND of QTCDN’ which is not visualized here because
of its complexity. It contains 256 transitions of which the 152 transitions of QTCN and 104
new transitions (Fig. 7, 8, 9). Of these 104 new transitions, 64 transitions cannot form
82 Chapter 4
transition pairs and therefore the CND of QTCDN’ is not bidirectional, as was the case for
QTCN.
Figure 4.5 – Discontinuous Shortest Path Change Event.
Figure 4.6 – Transition in QTCDN’ due to a combination of a Discontinuous Shortest Path Change
Event with a Node Pass Event.
4.5 Conclusions and Future work
In this chapter, we defined the Qualitative Trajectory Calculus on Changing Networks able to
undergo topological changes (QTCDN’). The possible relations and transitions in topological
changing networks were derived and presented in CNDs and conceptual figures. Note that
although the same relations exist in QTCDN’ as in QTCN, 256 transitions are possible in QTCDN’;
that is 168% of the total number of possible QTCN transitions. This illustrates the complexity
of a dynamic network versus a static network, which mirrors the complexity of a dynamic
reality versus a shortcoming static model of reality.
We strongly believe that QTCDN’ is useful in representing moving objects in the framework of
a topological changing network. Focusing for instance on transportation networks, many
changes in real world transportation networks can be considered as topological network
changes. First of all, many events caused by human management of transportation networks
can be modelled as a topological change. Next to the example of railway barriers closing a
street segment in a road network (cf. section 4.1), the same splitting up or elimination of
paths in a network can be considered when traffic lights turn red, roads are temporally
fenced off for road maintenance, bridges are opened over a waterway or floodgates in a
river network are closed. Complementary to each of these ‘network breaking’ events, one
can always find a corresponding ‘network joining’ event. In the given examples, network
Qualitative relations between moving objects in a dynamic network 83
Figure 4.7 – Possible QTCDN’ transitions due to a Discontinuous Shortest Path Change Event
(combined and/or otherwise) (a) and due to the combination of a Discontinuous Shortest Path
Change Event with a Speed Change Event (b).
Figure 4.8 – Possible QTCDN’ transitions due to a combination of a Discontinuous Shortest Path
Change Event with a Node Pass Event or (exclusive) with a Continuous Shortest Path Change Event.
84 Chapter 4
Figure 4.9 – Possible QTCDN’ transitions due to the combination of a Speed Change Event with a
combined Discontinuous Shortest Path Change Event (a) and due to the combination of a Speed
Change Event with the combination of a Discontinuous Shortest Path Change Event and a Node
Pass Event or (exclusive) a Continuous Shortest Path Change Event (b).
paths will be added or become connected again when railway barriers open, traffic lights
turn green, roads become reopened after maintenance operations, bridges over a waterway
are closed or floodgates in a river network are opened. Reversing a switch in a railway
network means the deletion of a network path together with the addition of another
network path. In addition to the temporary addition or deletion of existing paths of the
network, new paths can be (permanently) added (e.g. when a new road is laid or a new canal
is dug) or deleted (e.g. when a road is dug up or a canal is filled in).
Secondly, many applications can be found in human or natural hazards happening in
transportation networks. All kinds of events where a certain obstacle suddenly obstructs the
passage can be seen as a topological network change. The most obvious example is a
crashed vehicle blocking a street in a road network, but one should also think about
collapsing bridges and buildings, bomb attacks, demonstrations, etc. Although there can be
thought of many human–caused events, also nature phenomena, such as the blocking of
network segments by forest fires, by flooding, by a volcanic eruption, by trees that were cast
down by lightning, etc. may not be forgotten here.
Yet, the above made distinction may not be mutually exclusive, it certainly gives an idea of
the huge number of possible applications for QTCDN’ in real world transportation networks.
Given that nearly all traffic movements are bounded by a network, QTCDN’’s application field
Qualitative relations between moving objects in a dynamic network 85
in Geographic Information Systems for Transportation (GIS-T) seems to offer great potential.
We plan to evaluate QTCDN’ in this domain. We have presented QTCDN’ in a relatively informal
way concentrating on presenting ideas illustrated with simple examples. In future work, we
will fully formalize QTCDN’.
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Implementing a qualitative calculus to analyse moving point objects 87
5 Implementing a qualitative calculus to analyse moving
point objects
Delafontaine M., Cohn A. G., Van de Weghe N.
in Expert Systems With Applications (2011), Volume 38, Issue 5
Copyright © Elsevier Science
Abstract. Due to recent technological advances in position-aware devices, data about
moving objects is becoming ubiquitous. Yet, it is a major challenge for spatial
information systems to offer tools for the analysis of motion data, thereby evolving
from static to dynamic frameworks. This chapter aims to contribute to this area by
introducing an implementation prototype for an information system based on the
Qualitative Trajectory Calculus, a spatiotemporal calculus to represent and reason
about moving point objects.
Keywords. Qualitative calculus – Moving point objects – Implementation
5.1 Introduction
Capabilities to track individual moving objects have recently developed, along with the
technological advances concerning position-aware devices, navigation systems, electronic
transaction networks and surveillance systems (Laube et al. 2007). Nowadays, hi-tech
devices such as mobile phones, digicams, GPS receivers and RFIDs, are omnipresent and
allow for a low cost capture of high resolution trajectories1 of moving objects, whether these
are human beings (Wang, Hu & Tan 2003, Gau et al. 2004, Nielsen & Hovgesen 2004,
Michael et al. 2006), animals (DeCesare, Squires & Kolbe 2005, Yasuda & Arai 2005, Kritzler,
Raubal & Kruger 2007, Laube et al. 2007, Gagliardo et al. 2007), vehicles (Brunk & Davis
2002, Brakatsoulas et al. 2005, Hvidberg 2006), or even projectiles (Grace 2000). As is
generally recognized, this large potential of individually-based trajectory data heralds a new
era of movement analysis (Eagle & Pentland 2006, Laube et al. 2007) in order to feed a
broad range of application fields from ethology over traffic management to sport scene
analysis and weapon guidance.
In the past decade, GIScientists from multiple disciplines have created a sound theoretical
basis regarding the modelling, representation, analysis and extraction of knowledge from
1 Although also denoted as geospatial lifelines (Mark 1998, Hornsby & Egenhofer 2002, Laube, van Kreveld &
Imfeld 2005) others refer to trajectories (Gottfried 2008, Orlando et al. 2007, Brakatsoulas, Pfoser & Tryfona 2004, Spaccapietra et al. 2008, Gudmundsson, van Kreveld & Speckmann 2007), as we will do for consistency with the QTC calculus.
88 Chapter 5
motion data (see among others (Laube, Imfeld & Weibel 2005, Güting, de Almeida & Ding
2006, Giannotti & Pedreschi 2008, Spaccapietra et al. 2008) for an overview).
Despite these considerable efforts, common analyses of trajectory data remained limited
with respect to scope and sophistication (Laube et al. 2007), and much of this theoretical
work is not well reflected in tools offered by current spatial information systems (Wentz,
Campbell & Houston 2003).
One of the research fields which until now has remained largely theoretical is the domain of
qualitative reasoning (QR). However, one of the key motivations for QR lies in its applicability
for user interactive information systems, where qualitative information tallies much more
with human intuition, communication and decision making than quantitative information
(Egenhofer & Mark 1995, Renz, Rauh & Knauff 2000, Monferrer & Lobo 2002). In the past,
several qualitative spatial and temporal calculi have been introduced, first and foremost as a
reasoning tool: the Interval Algebra (Allen 1983), the Point Algebra (Vilain & Kautz 1986), the
Cardinal Direction Calculus (Frank 1991), the Doublecross Calculus (Freksa 1992b), the
Region Connection Calculus (Randell, Cui & Cohn 1992) and the Oriented Point Reasoning
Algebra (Moratz, Dylla & Frommberger 2005) to name but a few.
Yet, the usefulness of these calculi often remains questionable and needs a thorough
evaluation in terms of suitability, relevance and scope of potential applications. Wallgrün et
al. (2007) already made a general attempt in that direction with the development of a
qualitative spatial reasoning toolbox SparQ to allow for an easy integration in applications.
Another effort comes from El-Geresy and Abdelmoty (2004) with the introduction of a
qualitative spatial reasoning engine SPARQS for the automatic derivation of composition
tables. Another relevant line of work is that of Renz and Li (2008) who have largely
automated the task of determining the maximal tractable fragments for qualitative calculi.
Whereas most of the above mentioned calculi either stick to spatial or temporal issues, just
a few of them combine both to allow for spatiotemporal reasoning. One of them of
particular interest to the domain of moving objects is the Qualitative Trajectory Calculus
(QTC) (Van de Weghe 2004), which considers disjoint moving points objects (MPOs). We
believe that QTC constitutes a basis to represent and reason about moving objects, and thus
its implementation in an information system would provide a practical tool to support the
analysis of moving objects.
This chapter introduces an implementation prototype for the Basic (QTCB) and Double-Cross
(QTCC) calculi (Van de Weghe et al. 2006). Our aim is to show how QTC can be implemented
in an information system in a generic way, to introduce a methodology for handling
continuous data sampled at discrete times suitable for QTC, and to demonstrate the
applicability of such a system. The remainder of this chapter is organised as follows. Section
5.2 sketches a brief overview of QTC and its different types, including an informal account of
Implementing a qualitative calculus to analyse moving point objects 89
QTCB and QTCC (section 5.2). Section 5.3 introduces a conceptual modal and a prototype
QTC-based information system. In section 5.4, the use of this system is illustrated in two
different case studies. Section 5.5 presents a detailed discussion, and finally, section 5.6
draws some conclusions and considers possible future work.
5.2 The Qualitative Trajectory Calculus (QTC)
5.2.1 Types of QTC
QTC was introduced by Van de Weghe (2004) as a qualitative calculus to represent and
reason about moving objects. The QTC formalism defines relations between a pair of disjoint
MPOs. These MPOs are assumed to evolve continuously in space and time. Due to the
consideration of different spaces and frames of reference, the following types of QTC have
been elaborated (Van de Weghe 2004):
Basic type – QTCB
Double-Cross type – QTCC
Network type – QTCN
Shape type – QTCS
QTCB (Basic) and QTCC (Double-Cross) both deal with MPOs having a free trajectory in an n-
dimensional space. In QTCB, relations are determined referring to the Euclidian distance
between two MPOs (Figure 5.1a) (Van de Weghe et al. 2006), while QTCC relations use the
double cross between them (Figure 5.1b) (Van de Weghe et al. 2005a), as introduced by
Zimmerman and Freksa (1996).
QTCN (Network) (Bogaert 2008, Delafontaine et al. 2008) focuses on the special case of
MPOs which trajectories are constrained by a network, such as cars in a city. Since both the
Euclidean distance and the double cross concepts ignore the spatial configuration of a
potential underlying network, they are not well suited for QTCN. Therefore, QTCN relations
rely on the shortest paths in the network between the considered MPOs (Figure 5.1c).
Figure 5.1 – Two MPOs represented in a typical QTCB (a), QTCC (b), and QTCN (c) setting. The frame
of spatial reference is represented by the dashed line.
Finally, QTCS (Shape) is a calculus to represent and compare trajectory shapes, completely
abstracting from the actual MPOs (Van de Weghe et al. 2005b).
90 Chapter 5
In QTC, space and time (and thus the motion of MPOs) are assumed to be continuous.
Therefore, QTC relations may change over time according to the laws of continuity. In what
follows, we will use the term transition to denote the continuous change of one relation into
a conceptual neighbouring relation (Freksa 1992a), thus without passing intermediate
relations. Each transition occurs at an instant, i.e. point in time, which we will term transition
instant.
All QTC calculi are associated with a set of jointly exhaustive and pairwise disjoint (JEPD)
base relations. Consequently, there is one and only one relation for each pair of coexisting
MPOs at each time instant. In addition, due to continuity, the concurrent movement of two
MPOs over a time interval is uniquely mapped to a sequence of conceptual neighbouring
base relations.
5.2.2 Unconstrained movement
Both QTCB and QTCC were developed to represent and reason about MPO movements in a
free Euclidean space. Van de Weghe et al. (2006, 2005c) introduced four types (B11, B12,
B21, B22) of QTCB, and two types (C21, C22) of QTCC, although more subtypes could be
defined on the same basis. All relations in each of these six types are composed of multiple
relation symbols, each of which has the three-valued qualitative domain . These
symbols rely on (a subset of) the following relationships:
For a pair of MPOs and , and a time instant (Figure 5.2):
denotes the point location of at
denotes the velocity vector of at
denotes the straight line between and
denotes the positive angle between and
denotes the positive angle between and
denotes the minimum absolute angle between and
denotes the minimum absolute angle between and
Figure 5.2 – Properties of two MPOs k and l at a time instant t.
Then at :
A. : is moving towards (5.1)
Implementing a qualitative calculus to analyse moving point objects 91
+: is moving away from (5.2)
0: all other cases ( is stable with respect to )
B. : is moving towards (5.3)
+: is moving away from (5.4)
0: all other cases ( is stable with respect to )
C. : is moving to the left of (5.5)
+: is moving to the right of
(5.6)
0: all other cases ( is stable with respect to )
D. : is moving to the left of (5.7)
+: is moving to the right of (5.8)
0: all other cases ( is stable with respect to )
E. : is moving faster than
(5.9)
+: is moving slower than
(5.10)
0: all other cases ( is moving equally fast as )
F. : is moving at a smaller angle with respect to than (5.11)
+: is moving at a bigger angle with respect to than (5.12)
0: all other cases ( and are moving at the same angle with respect to )
Thus, the assessment of these six relation symbols requires knowledge on the instantaneous
location, and velocity (i.e. speed and motion azimuth) of both MPOs. Table 5.1 presents the
syntax of relations for all QTCB and QTCC types, according to the above mentioned
relationships A–F. Note that B11 and B12 have the same syntax as respectively B21 and B22:
they differ in the number of spatial dimensions taken into account. According to these rules,
a configuration where at time a zebra is moving away from a lion which in turn is
approaching and catching up with the zebra can be described in B22 by ‘+’ for A ( away
from ), ‘’ for B ( towards ), and ‘’ for E ( slower than ), which we will write as
= (+ )B22
.
92 Chapter 5
From an application point of view (cf. the lion and the zebra), the order of objects in the
relation often does not matter. Hence, converse relations have to be taken into account.
Converse relations in QTCB and QTCC can be obtained by interchanging the relation symbols
A with B, C with D, and by replacing E and F with their inverse symbols. The inverse symbol
for ‘’ is ‘+’, for ‘0’ is ‘0’, and for ‘+’ is ‘’. In the above example, the converse of
would be = ( + +)B22
.
QTC type Relation syntax
B11 (A B) B12 (A B E) B21 (A B) B22 (A B E) C21 (A B C D) C22 (A B C D E F)
Table 5.1 – Relation syntax for QTCB and QTCC subtypes.
5.3 A QTC-based information system
5.3.1 Trajectory representations
As QTC assumes spatial and temporal continuity, the location, the speed, and the motion
azimuth of an MPO are assumed to be continuous functions of time. Hence, an MPO
trajectory is a continuous set of points in space and time, which corresponds to the
conventional mathematical notion of a curve at the spatial level, and to a simple closed
interval at the temporal level.
Yet, in order to implement QTC in an information system, we need to consider how
information systems, and GISs in particular, store and represent MPO trajectories. Longley et
al. (2005, p. 70) argue that any representation is discrete, stating that “the world is infinitely
complex, but computer systems are finite”. To date, by far the most common way to store a
trajectory, is as a set of spatial locations at consecutive time steps (Orlando et al. 2007,
Turchin 1998, Yu et al. 2004, Yu & Kim 2006, Gudmundsson, van Kreveld & Speckmann 2007)
which we will term fixes, according to Laube et al. (2005). Obviously, such a discrete set of
fixes conflicts with the assumption of spatial and temporal continuity underlying QTC.
Hence, fixes need to be interpolated in space and time to obtain continuous trajectories.
5.3.2 Conceptual model
In order to implement QTC in an information system, an object-oriented design is proposed,
shown in Figure 5.3. Part of this model has been based on the MPO modelling domain after
Laube et al. (2005) where trajectories are build of a set of fixes (see section 5.3.3).
The MPO class represents dimensionless moving objects, whose spatiotemporal properties
are described by one or more trajectories (instances of Trajectory). Each Trajectory
Implementing a qualitative calculus to analyse moving point objects 93
maintains a list of one or more fixes (instances of Fix), that describe locations in space
(Point) and time (Instant). Each Trajectory has a timeSpan which equals the time interval
between its first and last fix. In order to represent continuous trajectories, the Trajectory
class may have its own detached functions to interpolate in between fixes. The MPO’s
location, speed and motion azimuth at a specific time instant are respectively returned by
the getLocation, getSpeed, and getAzimuth methods (constraint: time parameter must be
within timeSpan).
getLocation(in time) : Point
getSpeed(in time)
getAzimuth(in time)
fixes[1..*] : Fix
\timeSpan[1] : Interval
Trajectory
trajectories[1..*] : Trajectory
MPO
location[1] : Point
timeStamp[1] : Instant
Fix
1
Point
getRelation(in calculus : QTCCalculus, in time : Instant) : QTCRelation
getTransitions(in calculus : QTCCalculus, in interval : Interval)
firstTrajectory[1] : Trajectory
secondTrajectory[1] : Trajectory
\timeSpan[1] : Interval
TrajectoryPair
getSymbol(in index)
QTCRelation
1..*
1
2
0..*
numberOfSymbols[1]
relations[*] : QTCRelation
QTCCalculus
1
*
B11
C22
C21
B22
B21
B12
Instant
Interval
1
1
1
1..*
1
2
Figure 5.3 – UML class diagram for a QTC-based information system
As QTC applies to relations between two MPOs, a TrajectoryPair class is considered to
embody an ordered pair of coexisting trajectories. The firstTrajectory and secondTrajectory
properties refer to the respective Trajectory instances (constraints: firstTrajectory and
secondTrajectory belong to a different MPO; firstTrajectory and secondTrajectory have an
overlapping timeSpan). Each TrajectoryPair has a timeSpan which equals the temporal
overlap between the timeSpan of firstTrajectory and seconTrajectory. Starting from the
principles of continuity and JEPD (see section 5.2), we consider two basic operations that
comprise the necessary conditions for an implementation of QTC:
At each time instant there exists one and only one QTC relation between two MPOs
(section 5.2.1). The getRelation method returns this relation for a given type of QTC at a
given input time (constraint: the input time must be within the timeSpan of the
TrajectoryPair).
Each TrajectoryPair can be associated with a chronologically ordered set of QTC
relations and corresponding transition times (i.e. the instants at which the relations
change) over a time interval during its timeSpan. This ordered mapping is returned by
94 Chapter 5
the getTransitions method for a given type of QTC (constraint: the input time interval
must be during the timeSpan of the TrajectoryPair).
Specific types of QTC are modelled as subclasses of an abstract QTCCalculus class. They
implement two properties: relations returns their set of base relations; numberOfSymbols
returns the number of relation symbols in a relation.
QTC relations are represented by QTCRelation objects, which have a getSymbol method to
return the individual relation symbol at the specified index (constraint: the index parameter
must not exceed the numberOfSymbols of the QTCCalculus at hand).
5.3.3 Implementation prototype
Building on the conceptual model of section 5.3.2, we developed QTCAnalyst, a prototype
QTC-based information system. In the remainder of this section, we give an overview of the
assumptions and restrictions that underlie this implementation.
Trajectories
Although one can apply several methodologies to interpolate trajectory fixes in space and
time, e.g. (Yu et al. 2004), QTCAnalyst relies on the following assumptions, being the most
obvious and robust:
Assumption 5.1 – Trajectory polyline: In between two fix times, an MPO moves continuously
along the straight line segment connecting both fixes.
Assumption 5.2 – Segment speed: In between two fixes, the speed of an MPO is constant.
Thus, a trajectory is represented as a polyline of which each vertex represents a fix, as shown
in Figure 5.4. Though Assumptions 5.1 and 5.2 determine the trajectory of an MPO as a
continuous function of time, they entail two discontinuities for MPOs at fixes: a discontinuity
of motion azimuth due to Assumption 5.1 and a discontinuity of speed due to Assumption
5.2. Hence, contrary to getLocation, the getSpeed and getAzimuth methods are not defined
for time instants that correspond to trajectory fixes of the MPO at hand. In order to get
round this problem to determine the QTC relation at these instants, we will make use of a
transition table, as discussed later in this section.
To enable an objective comparison of trajectories, QTCAnalyst assumes concurrency of fixes:
Assumption 5.3 – Concurrent observation: All trajectories are sampled at the same set of fix
instants.
Whenever this assumption is not satisfied, fixes can always be resampled (interpolating as
necessary) according to Assumptions 5.1 and 5.2 so that Assumption 5.3 is met.
Implementing a qualitative calculus to analyse moving point objects 95
Figure 5.4 – A continuous MPO trajectory (a) and a representation of it according to Assumption
5.1 with fixes (crosses) per second (b).
Relations
As mentioned in section 5.3.2, an implementation of QTC requires a getRelation and a
getTransition method. getRelation can be determined according to relationships A–F (section
5.2.2). However, this will be impossible for MPOs at fixes, since getSpeed and getAzimuth are
ambiguous in that case due to Assumptions 5.1 and 5.2. In order to deduce relations for
MPOs at fixes, QTCAnalyst relies on the laws of continuity, where Galton (2001) points out
that ‘’ and ‘+’ are ‘dominated by’ ‘0’, from which follow these restrictions:
Restriction 5.1a – Intermediate ‘0’: A transition from ‘’ to ‘+’ must always pass the
intermediate value ‘0’.
Restriction 5.1b – Intermediate ‘0’: A transition from ‘+’ to ‘’ must always pass the
intermediate value ‘0’.
Restriction 5.2a – Dominated ‘’: A ‘’ lasts over an open time interval.
Restriction 5.2b – Dominated ‘+’: A ‘+’ lasts over an open time interval.
Restriction 5.2c – Dominant ‘0’: A ‘0’ lasts over either a closed time interval, or a time
instant.
Restriction 5.3a – Intermediate interval: There is always a closed time interval in between
two ‘’ relations.
Restriction 5.3b – Intermediate interval: There is always a closed time interval in between
two ‘+’ relations.
There is always an adjacent open time interval before and after a time instant. For an instant
, let us denote these intervals respectively and . Then at transition instant , due
restrictions 5.1-5.3, we obtain the relation symbols presented in Table 5.2. The
implementation of getRelation at in QTCAnalyst is now as follows:
For not at a fix time: getRelation uses relationships A – F (equations 5.1-5.12).
For at a fix time: getRelation first computates the before and after relationships (A – F)
using the locations at together with the speed and motion azimuth at respectively the
96 Chapter 5
preceding and following segment. Next, the transition table is employed to return the
relation at .
Relation symbol at t Relation symbol during t+
0 +
Relation symbol
during t
0 0 0 0 0 0 + 0 0 +
Table 5.2 – Transition table for QTC relation symbols at transition instant t
Transitions
The getTransitions method returns the chronological sequence of QTC relations and the
corresponding transition instants over a given valid time interval for the TrajectoryPair at
hand. Due to dominance theory (Galton 2001), each transition instant corresponds to a ‘0’
relationship symbol. By consequence, it suffices to assess at which instants a relationship
symbol changes to or from ‘0’. Due to Table 5.2, zero to two transitions might occur for each
relationship symbol at one time instant (e.g. at fix times).
For time instants in between consecutive fixes, relationship A will be ‘0’ when:
is stationary (intra-object coinciding fixes, no transitions);
and coincide. According to Assumptions 5.1-5.3, this situation occurs either at one
intermediate time instant (collision, two transitions), or over the complete segment
(inter-object coinciding fixes, no transitions);
is perpendicular to (equations 5.1-5.2). Due to Assumptions 5.1-5.3, this
situation occurs either at one intermediate time instant (two transitions), or over the
complete segment (no transitions).
The cases with two transitions (one from ‘’ or ‘+’ to ‘0’, the other one from ‘0’ to
respectively ‘+’ or ‘’) are mutually exclusive and can be solved analytically on the basis of
equations such as in Appendix A. Analogous to relationship A, it can be shown that there will
be at most two transitions for B – D, and four transitions for F for time intervals in between
two consecutive fixes. Obviously, no transitions can occur for E due to Assumption 5.2 (only
at fix times).
Thus, a TrajectoryPair consisting of pairs of concurrent fixes, will in a worst case scenario
have – – transitions over its total timeSpan.
Prototype application
QTCAnalyst was implemented in Visual Basic 6.5 using AutoCAD for visualisation and MS
Excel for data input and output. Through a GUI, trajectories that answer Assumptions 5.1-
5.3, can be loaded from fix data, and can be visualised in a conventional two-dimensional
space (top view perspective), or in a space-time cube. TrajectoryPair instances can then be
Implementing a qualitative calculus to analyse moving point objects 97
automatically generated for each canonical pair of coexisting Trajectory instances. Finally,
the output of the getRelation and getTransitions methods can be exported, or visualised. In
addition, QTCAnalyst is able to calculate and export simple summaries (see section 5.4.1), as
well as relation patterns (see section 5.4.2), i.e. chains of subsequent relations, from the set
of relations resulting from getTransitions.
5.4 Case studies
In this section, we utilise QTCAnalyst to analyse QTC relations in two completely different
contexts. The first case focuses on the QTCB relations between moving vehicles on a four
lane one-way street. The second case deals with QTCC relations of players in a squash
contest.
5.4.1 Cars on a street
The study area is a straight section of about 130 m of a four lane one-way road in Ghent
(Belgium), as schematized in Figure 5.5. This road is the south-north directed tail end of a
highway exit for the centre of Ghent. Another single lane road converges with it immediately
south of the study area, whereas there are traffic lights at the north end.
Figure 5.5 – Schematic sketch of the study area.
During the morning rush hour, a movie of the study area has been recorded with a steady
camera from a high building in the neighbourhood for two minutes. This movie has been
georeferenced to a local two-dimensional reference system in order to assess the relative
positions of cars – treated as MPOs – on snapshots taken at a regular time step of 1 s. The x-
axis of this system is aligned with the road’s centreline, whereas the y-axis is along the width
dimension. The resolution in y has intentionally been kept coarse, in order to eliminate
insignificant shifts of cars that stay within their lane. Hence, we obtain a data set of 44 car
trajectories with different time spans but with concurrent fixes (Assumption 3).
Of the 946 canonical trajectory pairs that exist for 44 trajectories, 503 have a temporal
overlap, and hence enable the calculation of QTC relations. Table 5.3 and Table 5.4 list some
of the QTCAnalyst results about the relations between these 503 valid pairs for the B21 and
98 Chapter 5
B22 calculi. The tables summarise the number of instantaneous occurrences, the number of
occurrences over a time interval, the total number of occurrences, and the total duration for
each relation aggregated over all valid pairs.
Relation Instants Intervals Total Duration (s)
( )B21 0 1 1 0.2
( +)B21 0 104 104 328.4
( 0)B21 3 3 6 3.0
(+ )B21 0 362 362 1 581.7
(+ +)B21 0 4 4 0.7
(+ 0)B21 58 132 190 534.0
(0 )B21 146 344 490 2 089.6
(0 +)B21 49 172 221 738.4
(0 0)B21 567 204 771 3 178.0
Total 823 1 326 2 149 8 454.0
Table 5.3 – Summary of QTC-B21 relations with their cumulative instant, interval, and total
frequencies, and duration for 503 car pairs.
From Table 5.3, we may learn that all nine B21 base relations do have at least one
occurrence. However, the occurrences are not equally distributed over this universe set.
Since there is no significant natural order for cars in a street, we will focus our discussion on
groups of converse relations. A first group is represented simply by the symmetric (0 0)B21
relation. It is the most common relation in the data set, lasting for almost 40% of the
cumulative time. Perhaps this indicates the importance of collective stops (both cars
standing still). A ‘0’ relation symbol, however, does not imply object stationarity. For
example, two cars driving next to each other in different parallel lanes are also in a (0 0)B21
relation, while both are moving. The same applies for (0 )B21
, (0 +)B21
, and their converse
relations ( 0)B21
, and (+ 0)B21
, which constitute another important group (40% of the total
time). In these cases, at least one car is moving, whereas this is unknown for its associate.
The last significant group consists of the relation ( +)B21
and its converse (+ )B21
. They
represent the regular situation of cars following one another. For the remaining two
symmetric relations, the table indicates that situations where cars are converging [( )B21
]
or diverging [(+ +)B21
] are rare: they respectively occur once and four times, lasting for only
fractions of seconds.
As shown in Table 5.4, only 21 of the 27 base relations occur in QTC-B22. The relative speed
symbol of these relations can now be used in order to further refine the interpretations
made for QTC-B21. Let us reconsider the groups of converse relations. For (0 0)B21
, we can
see that all intervals must be (0 0 0)B22
since both share the same number of interval
occurrences and have equal total duration. The remaining relations in this group, (0 0 )B22
Implementing a qualitative calculus to analyse moving point objects 99
and (0 0 +)B22
, only occur instantaneously. Due to the difference in relative speed, at least
one object must be moving in this case. These are the typical transition relations of cars
passing by each other. There number of occurrences gives a first indication of the number of
overtake events in the dataset, although relation sequences would be needed for an exact
assessment.
Relation Instant Interval Total Duration
( )B22 0 1 1 0.2
( +)B22 0 0 0 0.0
( 0)B22 0 0 0 0.0
( + )B22 0 110 110 306.5
( + +)B22 0 13 13 13.0
( + 0)B22 28 9 37 9.0
( 0 )B22 1 0 1 0.0
( 0 +)B22 1 3 4 3.0
( 0 0)B22 1 0 1 0.0
(+ )B22 0 376 376 1 197.7
(+ +)B22 0 157 157 290.1
(+ 0)B22 250 79 329 94.0
(+ + )B22 0 4 4 0.7
(+ + +)B22 0 0 0 0.0
(+ + 0)B22 0 0 0 0.0
(+ 0 )B22 3 0 3 0.0
(+ 0 +)B22 45 132 177 534.0
(+ 0 0)B22 10 0 10 0.0
(0 )B22 131 344 475 2 089.6
(0 +)B22 0 0 0 0.0
(0 0)B22 15 0 15 0.0
(0 + )B22 48 172 220 738.4
(0 + +)B22 0 0 0 0.0
(0 + 0)B22 1 0 1 0.0
(0 0 )B22 179 0 179 0.0
(0 0 +)B22 3 0 3 0.0
(0 0 0)B22 385 204 589 3 178.0
Total 1 101 1604 2 705 8 454.0
Table 5.4 – Summary of QTC-B22 relations with their cumulative instant, interval, and total
frequencies, and duration for 503 car pairs.
For the next group, we find the following correspondences for interval occurrences and
durations: ( 0)B21
with ( 0 +)B22
, (+ 0)B21
with (+ 0 +)B22
, (0 )B21
with (0 )B22
, and
(0 +)B21
with (0 )B22
. Hence, the car that is certainly moving always has the highest
100 Chapter 5
speed. Presumably, this means that in most, if not all, of the cases the other car is standing
still.
The group associated with cars that follow each other, has no instantaneous occurrences in
B22 where the relative speed symbol is ‘’ or ‘+’, as could be expected from restriction 5.2
(section 5.3.3) since those relations have no ‘0’ symbols.
Finally, the two remaining relations ( )B22
and (+ + )B22
belong to the least
represented groups, associated with convergence and divergence patterns. Due to their rare
occurrence, four base relations are not represented in these groups.
5.4.2 Squash rally
In this case study, we analyse the relation of two squash opponents in a championship duel
in QTC-C22. Therefore, we employ the public standard CVBase’06 dataset (Pers, Bon &
Vuckovic 2006). In this dataset, the trajectories of two squash players were sampled from
video frames taken at regular time steps of 0.04 seconds, by automatic computer vision
based tracking under field expert supervision. The resulting trajectories were smoothed by a
Gaussian kernel and have a positional RMS error of about 0.3 m.
Since there is only one pair of players, there is no need for a cumulative summary table as in
section 5.4.1. To simplify the discussion, we will consider the QTC-C22 relations between
both players during a rally lasting 37 s. For a complete chronological sequence of QTC-C22
relations during this rally, we refer to the Appendix B.
As stated in section 5.3.3, QTCAnalyst has the ability to compute relation patterns, i.e. chains
of two or more subsequent relations. Let us now consider some simple permutable patterns.
With simple patterns, we mean patterns that do not contain a repetition of subpatterns of a
lower order2 (i.e. complex patterns). A permutable pattern represents a pattern and all
permutations of it. For example, the patterns → → , → → , or → → all represent
the same permutable pattern. Since the smallest order is two, patterns of order two and
three are by definition simple patterns.
For the opponents in the rally, a total of 196 simple permutable QTC-C22 patterns of order
four have been found, and they are distributed non uniformly both in terms of frequency
and duration. 124 of them occur just once, i.e. only 72 patterns have at least one repetition.
Figure 5.6 presents a graph of the frequency and the total duration for the 24 patterns with
the longest total duration. All remaining patterns have a duration of less than 1.5 s, and a
frequency of eight or less. The graph shows that the three most frequent patterns also have
the highest durations. Two patterns prevail: ( + )C22
→ ( 0 )C22
→ ( )C22
→
( 0 )C22
with 40 occurrences over 7.8 s, and (+ )C22
→ (0 )C22
→
2 The number of relations a pattern consists of.
Implementing a qualitative calculus to analyse moving point objects 101
( )C22
→ (0 )C22
with 20 occurrences over 8.9 s. Interestingly, they are each
other’s converse pattern, and hence we learn that a prevailing movement behaviour during
the rally, is that one player (follower) is following its opponent (leader) until the leader
temporarily changes its moving direction towards the follower. Both players thereby
continuously remain moving to the left of each other (i.e. to the left of the reference line
connecting them). An alternation of both patterns may occur whenever the opponents are
alternatively running in clockwise cycles around each other. In squash, this behaviour may
arise whenever both players are alternating a forward move to play the ball with a backward
move to let the opponent play, taking account of the interference rule.3
Note that both patterns have ( )C22
in common: the relation with the longest overall
duration and second most occurrences (see Appendix B). Taking into account that the
patterns are permutable, it follows that they might overlap with each other.
Figure 5.6 – Duration (gray bars) and frequency (black bars) for 24 fourth order simple permutable
patterns in QTC-C22.
5.5 Discussion
This chapter has addressed the implementation of a QTC-based information system: we
proposed a conceptual model, set up a prototype system, and demonstrated its applicability 3 According to the official rules of squash (World Squash Federation 2009), “to avoid interference, the
opponent must try to provide the player with unobstructed direct access to the ball, a fair view of the ball, space to complete a swing at the ball and freedom to play the ball directly to any part of the front wall”.
102 Chapter 5
by two case studies. In what follows, we will point out some strengths and weaknesses of
our approach.
(a) Although this chapter confines itself to an implementation for QTCB and QTCC, the
proposed conceptual model is based on the principles of continuity, and of jointly
exhaustiveness and pairwise disjointness (JEPD), and is therefore generic for all QTC
calculi. QTCAnalyst is not pretended to be an end-user information system. It is a use
case independent prototype that may constitute the core of such a system, but would
need enhanced processing capabilities and automatic interpretation mechanisms in
order to become a fully fledged application. To illustrate the generality of this approach,
we chose two completely different use cases in section 5.4.
(b) QTCAnalyst provides unambiguous and consistent results. Although there are no explicit
implementations of conceptual neighbourhood diagrams (CNDs), QTCAnalyst results
(e.g. Table 5.3, Table 5.4, and Figure 5.6) are consistent with conceptual
neighbourhoodness, since the data respected the assumptions of continuity which
conceptual neighbourhoodness captures, and no gaps in recording occurred. Explicit use
of conceptual neighbourhoods could be useful in the case of incomplete data. For
instance, if at some instants the position of some MPOs is not known, possible relations
may be inferable through the assumptions of continuity as encoded in the CNDs.
Alternatively, interpolation on the actual positions could of course be used.
(c) While being a key and much studied operation in qualitative reasoning generally, the
composition of relations is not supported by QTCAnalyst. The starting point for
QTCAnalyst is trajectory data, and hence, in order to determine the relations
and , the trajectory data of all objects , , and is required. Consequently,
rather than applying the composition of and , a direct computation of
is more convenient, more efficient, and often more precise. We do not ignore
the usefulness of composition, and it provides another method, in addition to the use of
conceptual neighbourhoods mentioned in (b) above to inferring missing data; however,
composition has not been the focus in this work.
(d) Although Assumptions 5.1 and 5.2 can be used to interpolate MPO trajectories, some
open problems still remain in the MPO modelling domain. According to Laube, Imfeld
and Weibel (2005) these include the presence of uncertain and/or missing fixes as well
as granularity related issues. The granularity issue is worth discussing. Consider for
instance case A in Figure 5.7, where (0 )B22
(0 0 )B22
(0 + )B22
,
and hence has a higher speed as it covers a longer distance in reality (Figure 5.7a).
However, due to a sampling satisfying Assumptions 5.1-5.3, this case may be
represented as in Figure 5.7b, where = (0 0 0)B22
. Another example is case B,
where is circling around a stationary , and thus = (0 0)B21
(Figure 5.7c).
Implementing a qualitative calculus to analyse moving point objects 103
However, such a circular trajectory can never be represented by straight segments, and
thus will never equal (0 0)B21
over a time interval in Figure 5.7d. In both
cases, wrong relations are obtained due to the sampling of fixes and the application of
Assumptions 5.1-5.3. Moreover, they illustrate how relative speed as well as relative
direction may be influenced by a sampled representation. Note that both cases deal
with pertaining stable relations, i.e. relations with a ‘0’ symbol holding over a time
interval: in case A, ‘0’ erroneously arises, whereas in case B, ‘0’ is erroneously missed.
Yet, there is another important difference: in case A, the error could be avoided by
sufficient sampling granularity, whereas this does not apply for case B. In order to obtain
the desired stable relations, for situations such as case B, there are several possible
solutions. A reasonable one could be to introduce spatiotemporal limits or thresholds,
where stable relations occur whenever movements (such as relative speeds, or relative
directions) remain within these presumed thresholds. Depending upon the field of
application, suitable thresholds may be based on the precision or accuracy of the
trajectory data, the user-intended analysis granularity, the limits of the object’s (human,
animal, robot, etc.) perception, etc. This approach will also increase the chance of stable
relations actually occurring, where otherwise the usefulness of these so called
borderline cases is sometimes questioned (Gottfried 2008).
(e) While Assumptions 5.1-5.2 are applicable to every set of fixes, Assumption 5.3 is the
only one restricting the data collection method. In addition, this assumption may
sometimes be unrealistic, for instance when multiple unsynchronised sensors are used
or in case of missing fixes. However, as mentioned earlier, one can easily obtain
concurrency by applying first the Assumptions 5.1-5.2, and then resample the resulting
trajectory in order to fulfil Assumption 5.3.
Figure 5.7 – Trajectories of two objects k and l during a time interval for two situations (A and B)
according to two representations: realistic representation (a), (c); representation satisfying
Assumptions 5.1-5.3 (b), (d). Crosses represent fixes.
104 Chapter 5
(f) Throughout this chapter, the usual, absolute notion of time has been employed.
However, since QTC considers relative relationships, a relative time notion could be
useful in some cases, e.g. to compare movements with different time spans.
(g) In an applied setting, a user’s interest in QTC relations will be limited to those objects
that interact with each other. The assessment of interacting objects, i.e. the issue of
determining exactly which objects are interacting with each other, has been indicated as
an open problem (Andrienko et al. 2008). Hence, the advances made on that issue might
improve further implementations.
(h) The case studies in section 5.4 have shown that QTCAnalyst might be a useful tool for
the analysis of trajectory data within diverse contexts, such as traffic monitoring or
sports performance analysis. It is commonly accepted that qualitative and quantitative
formalisms should complement each other. This idea also underlies the results in section
5.4, were we analysed the quantitative properties of qualitative relations or patterns.
5.6 Conclusions and outlook
The contribution of this chapter is threefold:
A generic conceptual model for the analysis of moving point objects through a
Qualitative Trajectory Calculus was introduced.
An implementation methodology was proposed for the QTCB and QTCC calculi and a
prototype was developed (QTCAnalyst).
The applicability for a QTC-based information system was highlighted in two case
studies.
In future work, we intend to extend QTCAnalyst to other QTC calculi, especially QTCN. Since
QTCS abstracts from the actual moving objects, its implementation is not our first priority,
though there may be promising applications, e.g. in the field of trajectory similarity
measurement.
Also, a more advanced implementation may call for an explicit representation of conceptual
neighbourhood diagrams as well as composition tables.
As mentioned in the discussion, a fully fledged information system should implement
advanced mechanisms for post-processing and automatically interpreting raw QTC relations
and/or patterns. Moreover, its input modalities should go beyond conventional trajectory
data: whereas the intuitiveness and suitability for human decision making is a key motivation
for qualitative reasoning systems, possibilities for user-interactive and intuition-based data
input, such as query-by-sketch (Egenhofer 1996) should be considered. On the other hand,
these extensions would in turn benefit from increased output capabilities such as advanced
visualisation and communication means, e.g. animations.
Implementing a qualitative calculus to analyse moving point objects 105
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Implementing a qualitative calculus to analyse moving point objects 109
Appendix A
Consider two MPOs and with , , ,
, , , , , , , as defined in
section 5.2.2.
Let us consider a two-dimensional space, so that . Let t1, t2 be the time instants of
two consecutive fixes, so that:
,
,
.
Then, due to Assumptions 5.1-5.3:
with
– (5.13)
=
– (5.14)
(
) – (5.15)
(
– (5.16)
For ,
, , , , , , , as functions of , it follows from (5.13-5.16):
(5.17)
(5.18)
(5.19)
deg (5.20)
(5.21)
Hence, due to (5.17-5.21):
(5.22) (5.23) (5.24) (5.25)
110 Chapter 5
Appendix B
Relation Time Duration
unknown 28.00 (- + - +)
C21 0.04
(- 0 - +)C21 28.04
(- - - +)C21 0.16
(- 0 - +)C21 28.20
(- + - +)C21 0.20
(- 0 - +)C21 28.40
(- - - +)C21 0.12
(- 0 - +)C21 28.52
(- + - +)C21 0.68
(- + - 0)C21 29.20
(- + - -)C21 0.80
(- 0 - -)C21 30.00
(- - - -)C21 0.20
(- 0 - -)C21 30.20
(- + - -)C21 0.08
(- 0 - -)C21 30.28
(- - - -)C21 0.16
(- - 0 -)C21 30.44
(- - + -)C21 0.12
(- 0 + -)C21 30.56
(- + + -)C21 0.12
(- 0 + -)C21 30.68
(- - + -)C21 0.24
(- - + 0)C21 30.92
(- - + +)C21 0.28
(0 - + +)C21 31.20
(+ - + +)C21 0.40
(+ - 0 +)C21 31.60
(+ - - +)C21 0.12
(+ 0 - +)C21 31.72
(+ + - +)C21 0.24
(0 + 0 +)C21 31.96
(- + + +)C21 0.20
(- + + 0)C21 32.16
(- + + -)C21 0.36
(- + 0 -)C21 32.52
(- + - -)C21 0.16
(- + - 0)C21 32.68
(- + - +)C21 0.52
(0 + - +)C21 33.20
(+ + - +)C21 0.16
(0 + - 0)C21 33.36
(- + - -)C21 0.03
(0 + - -)C21 33.39
(+ + - -)C21 0.01
(0 + - -)C21 33.40
(- + - -)C21 0.28
(- 0 - -)C21 33.68
(- - - -)C21 0.24
(0 - - -)C21 33.92
(+ - - -)C21 0.16
(0 - - -)C21 34.08
(- - - -)C21 0.55
(0 - - -)C21 34.63
(+ - - -)C21 0.15
(+ 0 - -)C21 34.78
(+ + - -)C21 0.02
(+ + 0 -)C21 34.80
(+ + + -)C21 0.76
(+ 0 + -)C21 35.56
(+ - + -)C21 0.08
(0 - + 0)C21 35.64
(- - + +)C21 0.04
(- - 0 +)C21 35.68
(- - - +)C21 0.04
(- 0 - +)C21 35.72
(- + - +)C21 0.04
(- + - 0)C21 35.76
(- + - -)C21 0.16
(- 0 - -)C21 35.92
(- - - -)C21 0.59
(0 - - -)C21 36.51
(+ - - -)C21 0.02
(+ 0 - -)C21 36.53
(+ + - -)C21 0.31
(+ + - 0)C21 36.84
(+ + - +)C21 0.12
(+ 0 - +)C21 36.96
(+ - - +)C21 0.08
(+ - - 0)C21 37.04
(+ - - -)C21 0.24
(0 - - -)C21 37.28
(- - - -)C21 0.04
(- - 0 -)C21 37.32
(- - + -)C21 0.12
(- 0 + -)C21 37.44
(- + + -)C21 0.27
(- + 0 -)C21 37.71
(- + - -)C21 0.01
(- + 0 -)C21 37.72
(- + + -)C21 0.02
(- + 0 -)C21 37.74
(- + - -)C21 0.02
(- 0 - -)C21 37.76
(- - - -)C21 0.02
(- 0 - -)C21 37.78
(- + - -)C21 0.02
(- 0 - -)C21 37.80
(- - - -)C21 0.11
(- 0 - -)C21 37.91
(- + - -)C21 0.01
(- 0 - -)C21 37.92
(- - - -)C21 0.01
(- 0 - -)C21 37.93
(- + - -)C21 0.09
(0 + - -)C21 38.02
(+ + - -)C21 0.46
(+ 0 - -)C21 38.48
(+ - - -)C21 0.12
(0 - - -)C21 38.60
(- - - -)C21 0.60
(0 - - -)C21 39.20
(+ - - -)C21 0.12
(0 - - -)C21 39.32
(- - - -)C21 0.28
(- 0 - -)C21 39.60
(- + - -)C21 0.24
(0 + - -)C21 39.84
(+ + - -)C21 0.00
(0 + - 0)C21 39.84
(- + - +)C21 0.01
(0 + - +)C21 39.85
(+ + - +)C21 0.39
(+ 0 - 0)C21 40.24
(+ - - -)C21 0.04
(+ - - 0)C21 40.28
(+ - - +)C21 0.04
(+ - - 0)C21 40.32
(+ - - -)C21 0.04
(+ 0 - -)C21 40.36
(+ + - -)C21 0.36
(+ 0 - -)C21 40.72
(+ - - -)C21 0.08
(+ - 0 -)C21 40.80
(+ - + -)C21 0.12
(+ 0 + -)C21 40.92
(+ + + -)C21 0.12
(+ 0 + -)C21 41.04
(+ - + -)C21 0.09
(+ - + 0)C21 41.13
(+ - + +)C21 0.55
(0 - + +)C21 41.68
(- - + +)C21 0.08
(- - 0 +)C21 41.76
(- - - +)C21 0.20
(- - - 0)C21 41.96
(- - - -)C21 0.16
(0 - - -)C21 42.12
(+ - - -)C21 0.28
(+ 0 - -)C21 42.40
(+ + - -)C21 0.00
(+ 0 - -)C21 42.40
(+ - - -)C21 0.00
(+ 0 - -)C21 42.40
(+ + - -)C21 0.04
(0 + - -)C21 42.44
(- + - -)C21 0.31
Implementing a qualitative calculus to analyse moving point objects 111
(0 + - -)C21 42.75
(+ + - -)C21 0.05
(+ 0 - -)C21 42.80
(+ - - -)C21 0.12
(+ - - 0)C21 42.92
(+ - - +)C21 0.40
(+ - - 0)C21 43.32
(+ - - -)C21 0.08
(+ - 0 -)C21 43.40
(+ - + -)C21 0.56
(+ - 0 -)C21 43.96
(+ - - -)C21 0.08
(0 - - -)C21 44.04
(- - - -)C21 0.24
(0 - - -)C21 44.28
(+ - - -)C21 0.20
(0 - - -)C21 44.48
(- - - -)C21 0.16
(- 0 - -)C21 44.64
(- + - -)C21 0.00
(- 0 - -)C21 44.64
(- - - -)C21 0.00
(- 0 - -)C21 44.64
(- + - -)C21 0.10
(0 + - -)C21 44.74
(+ + - -)C21 0.02
(+ + - 0)C21 44.76
(+ + - +)C21 0.84
(+ + - 0)C21 45.60
(+ + - -)C21 0.04
(+ 0 - -)C21 45.64
(+ - - -)C21 0.04
(+ - 0 -)C21 45.68
(+ - + -)C21 0.20
(+ 0 + -)C21 45.88
(+ + + -)C21 0.08
(0 + + -)C21 45.96
(- + + -)C21 0.04
(- + 0 -)C21 46.00
(- + - -)C21 0.08
(- 0 - -)C21 46.08
(- - - -)C21 0.59
(- 0 - -)C21 46.67
(- + - -)C21 0.01
(- 0 - -)C21 46.68
(- - - -)C21 0.00
(- 0 - -)C21 46.68
(- + - -)C21 0.01
(0 + - -)C21 46.70
(+ + - -)C21 0.10
(+ + 0 -)C21 46.80
(+ + + -)C21 0.60
(+ 0 + 0)C21 47.40
(+ - + +)C21 0.04
(0 - + +)C21 47.44
(- - + +)C21 0.04
(- 0 0 +)C21 47.48
(- + - +)C21 0.04
(- + - 0)C21 47.52
(- + - -)C21 0.24
(- 0 - -)C21 47.76
(- - - -)C21 0.47
(0 - - -)C21 48.23
(+ - - -)C21 0.01
(+ 0 - -)C21 48.24
(+ + - -)C21 0.28
(+ + - 0)C21 48.52
(+ + - +)C21 0.36
(+ + - 0)C21 48.88
(+ + - -)C21 0.08
(+ 0 - -)C21 48.96
(+ - - -)C21 0.40
(0 - - -)C21 49.36
(- - - -)C21 0.08
(- - 0 -)C21 49.44
(- - + -)C21 0.05
(- - 0 -)C21 49.49
(- - - -)C21 0.11
(- 0 - -)C21 49.61
(- + - -)C21 0.03
(- 0 - -)C21 49.64
(- - - -)C21 0.02
(- 0 - -)C21 49.66
(- + - -)C21 0.02
(- 0 - -)C21 49.68
(- - - -)C21 0.12
(- 0 - -)C21 49.80
(- + - -)C21 0.00
(- 0 - -)C21 49.80
(- - - -)C21 0.02
(- 0 - -)C21 49.82
(- + - -)C21 0.02
(0 + - -)C21 49.84
(+ + - -)C21 0.08
(+ 0 - -)C21 49.92
(+ - - -)C21 0.03
(+ 0 - -)C21 49.95
(+ + - -)C21 0.01
(+ 0 0 -)C21 49.96
(+ - + -)C21 0.08
(+ 0 + -)C21 50.04
(+ + + -)C21 0.29
(+ + 0 -)C21 50.33
(+ + - -)C21 0.03
(+ + 0 -)C21 50.36
(+ + + -)C21 0.03
(+ + 0 -)C21 50.39
(+ + - -)C21 0.01
(+ + 0 -)C21 50.40
(+ + + -)C21 0.24
(+ + + 0)C21 50.64
(+ + + +)C21 0.24
(0 + + +)C21 50.88
(- + + +)C21 0.08
(- + 0 +)C21 50.96
(- + - +)C21 0.12
(- + - 0)C21 51.08
(- + - -)C21 0.08
(- 0 - -)C21 51.16
(- - - -)C21 0.81
(- 0 - -)C21 51.97
(- + - -)C21 0.03
(0 + - -)C21 52.00
(+ + - -)C21 0.00
(0 + - -)C21 52.00
(- + - -)C21 0.01
(0 + - -)C21 52.01
(+ + - -)C21 0.11
(+ + - 0)C21 52.12
(+ + - +)C21 0.30
(+ + - 0)C21 52.42
(+ + - -)C21 0.02
(+ + - 0)C21 52.44
(+ + - +)C21 0.40
(+ + - 0)C21 52.84
(+ + - -)C21 0.04
(+ 0 - -)C21 52.88
(+ - - -)C21 0.08
(+ - 0 -)C21 52.96
(+ - + -)C21 0.12
(0 - + -)C21 53.08
(- - + -)C21 0.04
(- - + 0)C21 53.12
(- - + +)C21 0.04
(0 - + +)C21 53.16
(+ - + +)C21 0.08
(+ - + 0)C21 53.24
(+ - + -)C21 0.12
(+ - 0 -)C21 53.36
(+ - - -)C21 0.24
(0 - - -)C21 53.60
(- - - -)C21 0.12
(- 0 - -)C21 53.72
(- + - -)C21 0.02
(0 + - -)C21 53.74
(+ + - -)C21 0.10
(+ 0 - -)C21 53.84
(+ - - -)C21 0.56
(0 - - -)C21 54.40
(- - - -)C21 0.04
(- - 0 -)C21 54.44
(- - + -)C21 0.16
(- 0 + -)C21 54.60
(- + + -)C21 0.06
(- + 0 -)C21 54.66
112 Chapter 5
(- + - -)C21 0.02
(- + 0 -)C21 54.68
(- + + -)C21 0.01
(- + 0 -)C21 54.69
(- + - -)C21 0.03
(- + 0 -)C21 54.72
(- + + -)C21 0.24
(- + 0 -)C21 54.96
(- + - -)C21 0.56
(- + - 0)C21 55.52
(- + - +)C21 0.04
(0 + - +)C21 55.56
(+ + - +)C21 0.08
(+ + - 0)C21 55.64
(+ + - -)C21 0.03
(+ + - 0)C21 55.67
(+ + - +)C21 0.01
(0 + - 0)C21 55.68
(- + - -)C21 0.12
(- 0 - -)C21 55.80
(- - - -)C21 0.12
(- 0 - -)C21 55.92
(- + - -)C21 0.04
(0 + - -)C21 55.96
(+ + - -)C21 0.04
(+ 0 0 -)C21 56.00
(+ - + -)C21 0.24
(+ - + 0)C21 56.24
(+ - + +)C21 0.12
(+ - + 0)C21 56.36
(+ - + -)C21 0.84
(+ - 0 -)C21 57.20
(+ - - -)C21 0.16
(0 - - -)C21 57.36
(- - - -)C21 0.60
(- 0 - -)C21 57.96
(- + - -)C21 0.24
(- + - 0)C21 58.20
(- + - +)C21 0.16
(0 + - +)C21 58.36
(+ + - +)C21 0.04
(+ + - 0)C21 58.40
(+ + - -)C21 0.12
(+ 0 - 0)C21 58.52
(+ - - +)C21 0.16
(+ - - 0)C21 58.68
(+ - - -)C21 0.16
(+ - 0 -)C21 58.84
(+ - + -)C21 0.56
(+ - + 0)C21 59.40
(+ - + +)C21 0.42
(+ - 0 +)C21 59.82
(+ - - +)C21 0.02
(+ - 0 +)C21 59.84
(+ - + +)C21 0.36
(0 - + +)C21 60.20
(- - + +)C21 0.28
(- 0 + +)C21 60.48
(- + + +)C21 0.52
(- + + 0)C21 61.00
(- + + -)C21 0.56
(- 0 + -)C21 61.56
(- - + -)C21 0.12
(- - + 0)C21 61.68
(- - + +)C21 0.08
(- - + 0)C21 61.76
(- - + -)C21 0.08
(0 - + -)C21 61.84
(+ - + -)C21 0.16
(+ - + 0)C21 62.00
(+ - + +)C21 0.12
(+ - + 0)C21 62.12
(+ - + -)C21 0.44
(+ - 0 -)C21 62.56
(+ - - -)C21 0.20
(0 - - -)C21 62.76
(- - - -)C21 0.23
(0 - - -)C21 62.99
(+ - - -)C21 0.29
(0 - - -)C21 63.28
(- - - -)C21 0.01
(0 - - -)C21 63.29
(+ - - -)C21 0.03
(0 - - -)C21 63.32
(- - - -)C21 0.06
(- 0 - -)C21 63.38
(- + - -)C21 0.10
(- + - 0)C21 63.48
(- + - +)C21 0.02
(0 + - +)C21 63.50
(+ + - +)C21 0.86
(+ 0 - +)C21 64.36
(+ - - +)C21 0.08
(+ - 0 0)C21 64.44
(+ - + -)C21 0.16
(0 - + -)C21 64.60
(- - + -)C21 0.08
(- - 0 -)C21 64.68
(- - - -)C21 0.24
(- 0 - -)C21 64.92
(- + - -)C21 0.08
unknown 65.00
Table 5.5 – Complete sequence, transition time and duration of QTC-C21 relations between two
squash opponents during a rally lasting 37 s.
Modelling moving objects in geospatial sketch maps 113
6 Modelling moving objects in geospatial sketch maps
Delafontaine M., Van de Weghe N.
in Tomko M., Richter K.-F. (Eds.): International Workshop on Adaptation in Spatial
Communication (2009)
Copyright © SFB/TR 8 Spatial Cognition
Abstract. Freehand sketching of spatial scenes is a natural way of everyday human
communication, and an important representation used in many geospatial reasoning
tasks. However, besides their spatial semantics, people tend to use sketch maps to
explain things that happen in time as well. Until now, this temporal aspect has been
neglected to a considerable extent. Motion – again a common aspect of human daily
life – is one such issue where time enters the picture. This chapter focuses on
opportunities for representing moving point objects (a specific subcategory of
motion) in geospatial sketch maps.
Keywords. Sketch maps – Moving point objects – Geospatial lifelines
6.1 Introduction
For a long time, sketch maps have appeared to be a powerful tool for recovering information
about spatial environments (Golledge & Stimson 1997), attracting attention from numerous
fields such as geography, planning and psychology. Nowadays, new technologies replace
traditional pencil-and-paper-based methods, creating new opportunities for data collection,
integration, and analysis (Huynh & Doherty 2007). Moreover, there is a need for computers
to be able to deal with sketch maps as people do.
To date, some researchers have been studying sketch maps in a geospatial context (Blaser
2000, Huynh & Doherty 2007, Huynh et al. 2008, Schlaisich & Egenhofer 2001, Sezgin,
Stahovich & Davis 2006, Egenhofer 1997a, Egenhofer 1997b, Davis 2007, Forbus, Usher &
Chapman 2003, Okamoto, Okunuki & Takai 2004), and a few systems have been developed
(Haarslev & Wessel 1998, Davis 2002, Forbus et al. 2008, Hammond & Davis 2005) (Figure
6.2 and Figure 6.4), allowing for basic reasoning and/or querying. These efforts share a
primary focus on spatial information, and hence somehow overlook the capabilities to
communicate temporal information by sketching as well. However, people tend to use
sketch maps to make inferences which involve information far beyond purely static spatial
scenes. Particularly, this applies to spatiotemporal phenomena, i.e. aspects that relate space
114 Chapter 6
to time or vice versa1. For example, consider the key role of temporal information in a soccer
coach’s sketch of an opponent attack or an eyewitness’s sketch of a car crash.
Note that both examples focus on moving objects which relate to the spatiotemporal
concept of motion. Over the past decade, the modelling of moving objects has been a hot
topic in fields as GIScience, Artificial Intelligence and Information Systems (Bitterlich et al.
2008). Recent technological advancements have enabled the low-cost capture of motion
data and thereby triggered the need for well-adapted analysis tools. In order to reflect this
tendency, sketch-based information systems should be able to represent and reason about
moving objects. In this chapter, we aim to contribute to this development by examining the
spatial and temporal properties of moving objects as represented in geospatial sketch maps.
The restriction to geospatial sketch maps thereby briefly implies the following three
assumptions:
Elements are drawn from a top view perspective.
Elements are drawn in a geographical space at an approximated spatial scale.
Moving objects can be represented as moving point objects (MPOs) at the approximated
scale of the geospatial sketch map.
The remainder of this chapter is structured as follows. In section 6.2, we explicate and
extend the concept of sketch maps and the related ontology of glyphs. In section 6.3, the
concepts of moving point objects and geospatial lifelines are first introduced, and then
utilised in order to determine the spatiotemporal characteristics of lifeline glyphs. Finally,
section 6.4 mentions conclusions as well as avenues for future research.
6.2 Extended Sketch Maps
Sketch maps can be defined from several perspectives, and according to different research
focuses. We will base on the alternative given by Forbus, Usher and Chapman (2003), where
sketch maps are considered to be “compact spatial representations that express the key
spatial features of a situation for the task at hand, abstracting away the mass of details that
would otherwise obscure the relevant aspects.” They consider sketch maps to be composed
of glyphs (entities) which on their turn consist of ink (drawing strokes) and content (the
conceptual entity that the glyph represents).
Sketch maps are maps in the sense that they depict features in their spatial context.
However, just as with cartography, where maps have evolved from paper to digital maps (1)
and from static to dynamic representations (2), we believe that sketch maps can be
extended in the same way. Although by definition, sketch maps are not precluded from
being paper maps, we assume them, according to contemporary standards, to be digital
1 Conversely, sketching would not make up a terribly good means to deal with abstract phenomena, i.e. aspects
that exist only in time, such as thoughts, feelings, and business relations.
Modelling moving objects in geospatial sketch maps 115
representations managed by an information system. Furthermore, we assume that they are
freehand drawn by means of a one-handed2 input device (e.g. a mouse, a touch pad or a
digital pen), with a standard click-and-drag line drawing tool in a two-dimensional space.
Though these assumptions are definitely constraining, they offer the same facilities as
common sketching with a pencil on a sheet of paper.
Concerning the evolution from static to dynamic representations, we basically agree with
Forbus et al.’s definition except for the spatial keyword, which we propose to replace with
spatiotemporal, in the sense that spatiotemporal features are features that exist both in
space and time (cf. 1). Temporality, in the sense of discourse sequentiality “controls an
assortment of media, art forms, representations”, quoting Sternberg (2004). This certainly
applies to sketching, which is as a kind of narrative or dialogue between the sketcher and its
audience, although an audience is not always required, for instance in design (Cross 1999).
The assumptions of one-handed input and line drawing mode inevitably impose an absolute
chronological order of drawing. Consequently, sketch maps are not to be restricted to spatial
knowledge, but should store temporal information as well. Hence follows the extended
sketch map ontology, where time has entered the picture in several ways, as shown in Figure
6.1.
First and foremost, there is temporal information involved with the ink concept. Like a
human observer, but vis-à-vis a conventional sheet of paper, an information system is able
to capture when a pen hits a tablet or when a mouse button is pressed or released. Each
stroke thus can be associated with a certain interval of drawing time (Figure 6.2 and Table
6.1). By consequence, ink can be considered as composed of spatial and temporal ink. On
the other hand, temporal knowledge can be associated with the content part. For instance,
this is the case when two or more separate glyphs model the consecutive states of one and
the same object. Next to temporal information, content may be characterised by spatial and
thematic, i.e. non-spatial and non-temporal, semantics.
While ink inherently constitutes what is sketched on a sketch map, content is usually
provided using a secondary modality (e.g. speech) or directly interpreted by the listener in
the case of a straight human-to-human communication. Consequently, in order to develop
intelligent and natural sketch interpretation systems, systems require the ability to interpret
glyphs as much automatically as possible, while avoiding significant error and/or information
loss. Therefore, this chapter will neglect the option of having additional content input. In
addition, the remainder of this chapter restricts to geospatial sketch maps, which are
considered to be sketch maps that represent features in a geographical space at an
approximated spatial scale from a top view perspective.
2 According to HCI research, it is natural to assume that only one (preferred) hand is used to draw sensu stricto,
while the other one performs complementary tasks such as leading and referencing (MacKenzie 2003).
116 Chapter 6
Figure 6.1 – ER diagram of the extended sketch map ontology.
6.3 Moving objects in geospatial sketch maps
6.3.1 Moving point objects and geospatial lifelines
From a geospatial background, a moving point object (MPO) is the most basic and commonly
used representation of motion (Laube 2005). This container concept can be used to
represent whatever individual object or subject moving in a geographical space, whether this
is a vehicle, an animal, a human being, or an earthquake epicentre. The most basic
conceptualisation of an MPO trajectory is the geospatial lifeline, or briefly lifeline (Laube
2005) (Figure 6.3). According to Mark (1998) a lifeline is a continuous set of positions that an
object occupies in space over a certain period of time. As a lifeline models a moving point, it
is equivalent to a continuous spatial curve which maps to a continuous time range.
However, in many, if not all cases, lifelines are approximated as a discrete set of space-time
locations, or fixes (Laube et al. 2007). A lifeline describes location as a function of time, and
hence, each time instant corresponds to a unique spatial location, while the reverse is not
true. In other words, for lifelines, time determines space.
Modelling moving objects in geospatial sketch maps 117
Note that the notion of geospatial lifelines drawn in top view (Figure 6.3) differs substantially
from approaches with other scientific backgrounds. In physics, for instance, a side view is
predominant, and the motion of objects is dictated by external forces, instead of being
predefined by a lifeline (Davis 2002), as illustrated in Figure 6.4.
Figure 6.3 – Map of a geospatial lifeline of a butterfly moving from A to B, passing flowers on its
way (own illustration after (Laube 2005)).
Ink Point X Ink Point Y Ink Point Timestamp (s)
1.22 1.10 130.44 1.39 1.03 130.50 1.44 0.98 130.53 1.44 0.81 130.59 1.42 0.64 130.62 1.34 0.47 130.65 1.27 0.34 130.69 1.24 0.23 130.72 1.24 0.13 130.75 1.27 0.02 130.78 1.36 -0.20 130.84 1.40 -0.32 130.87 1.40 -0.57 130.94 1.41 -0.66 130.97 1.41 -0.75 131.00 1.43 -0.83 131.03 1.43 -1.01 131.13 1.45 -1.06 131.15 1.46 -1.10 131.19 1.46 -1.10 131.22
Table 6.1. – Export of spatial and temporal ink of the
stroke in Fig. 2 as a set of timestamped polyline
vertices.
Figure 6.2 – Single-stroke glyph drawn in
CogSketch (Forbus et al. 2008).
118 Chapter 6
Figure 6.4 – A Shrewd Sketch Interpretation and Simulation Tool (ASSIST) (Davis 2002).
6.3.2 Lifeline glyphs
This section addresses the research question of how lifelines can be represented through
glyphs in geospatial sketch maps. To this end, a number of characterising binary distinctions
will be considered according to the relationship between these representations and the
lifeline of the underlying MPO they model. These distinctions can be regarded as
dichotomies for a user to choose from when sketching about an MPO in a geospatial
context. Of particular interest are the relationships which hold between the spatial and
temporal properties of a lifeline and respectively the spatial and temporal ink that
represents it.
Explicit vs. implicit
A major division can be made between implicit and explicit lifeline representations. Explicit
representations are glyphs that embody (part of) a lifeline, i.e. true lifeline glyphs. Implicit
representations are glyphs or groups of glyphs that do not directly represent a lifeline, but
instead imply one, just as road signs imply the route you should follow in the case of a traffic
diversion. Unless mentioned otherwise, the term lifeline glyph will refer to an explicit
representation in what follows. Examples of explicit representations are illustrated in Figure
6.5a-c; an implicit representation is shown in Figure 6.5d.
Single-stroke vs. multi-stroke
The most basic and uncomplicated lifeline glyphs consist out of one single drawing stroke
(Figure 6.5a). Otherwise, lifelines may be composed out of multiple strokes, for some
reasons (Figure 6.5b-d). The temporal (and perhaps spatial) gaps in between two successive
strokes may for instance model important breaks in the motion path of the underlying
object, e.g. stops, events, turning or decision points, etc. (Figure 6.5b). In addition, a single-
stroke approach will be inappropriate whenever the lifeline becomes too complex, e.g. when
the sketcher makes reflections about it while drawing. Also, glyphs with disconnected parts
must be multi-stroke (Figure 6.5c).
Modelling moving objects in geospatial sketch maps 119
Figure 6.5 – Sketch map representations of the butterfly lifeline in Figure 6.3: explicit single-stroke
lifeline glyph (a), explicit multi-stroke lifeline glyph (b), explicit multi-stroke lifeline glyph (c),
implicit representation by means of six flower glyphs and four arrow glyphs.
Continuous vs. Discrete
Although lifelines are by definition continuous spatiotemporal entities (section 6.3.1), their
representations may be either continuous or discrete. This distinction can be made both at
the spatial and the temporal level, i.e. with respect to spatial and temporal ink respectively.
Continuous spatial ink consists of one or more connected curves, whereas discrete spatial
ink comprises at least two disconnected elements. Due to the restrictions of conventional
sketching (assumptions of 6.2), there is always a temporal gap in between two successive
drawing strokes. Hence, for temporal ink, the continuous/discrete division is equivalent to
single-stroke/multi-stroke division. In addition, since discrete spatial ink has to be multi-
stroked, it cannot co-exist with continuous temporal ink. Thus, the following three
configurations are realisable according to this dichotomy:
Continuous spatial ink, continuous temporal ink, i.e. a simple single-stroke glyph (most
basic glyph, e.g. Figure 6.5a).
Continuous spatial ink, discrete temporal ink, i.e. a spatially connected multi-stroke
glyph (e.g. Figure 6.5b).
Discrete spatial ink, discrete temporal ink, i.e. a spatially disconnected multi-stroke
glyph (e.g. Figure 6.5c).
So far, one might ask for the difference between a discrete lifeline glyph (explicit
representation) and a (discrete) implicit representation. The spatial ink of a discrete lifeline
glyph is a discontinuous representation of a continuous curve, such as a dashed or dotted
line (Figure 6.5). By definition, an implicit representation has no lifeline glyph(s) but other
120 Chapter 6
glyphs that imply a lifeline instead: the flower and arrow glyphs in Figure 6.5d are
autonomous entities, whereas a dash segment in Figure 6.5c has no significance on its own.
Aligned vs. non-aligned
Without doubt one of the most valuable derivatives of temporal ink is the chronological
order of drawing. As is common in human communication, this communicative order often
reflects the chronology of the underlying content (Sternberg 2004). We will term this the
alignment relation: an element is aligned if its drawing chronology respects the order
(positive alignment) or reverse order (negative alignment) of the chronology in the
underlying content. Note that a negative alignment differs from the case of no alignment
which applies when neither the right nor the reverse order matches the actual chronology.
Three hierarchical levels of alignment can be distinguished within the context of sketch
maps: inter-glyph, inter-stroke, and intra-stroke alignment. Note that although Huynh et al.
(Huynh et al. 2008, Huynh & Doherty 2007) already emphasized the significance of drawing
sequences, they merely considered inter-glyph alignment.
The drawing evolution of aligned lifeline glyphs reflects the order of locations taken
chronologically by the underlying MPO. For lifeline glyphs, the alignment relations imply an
absolute ordering, i.e. a complete internal spatiotemporal chronology for properties such as
motion azimuth or events such as performing a specific movement pattern. Thereby, they
enable geospatial reasoning, allowing for making inferences like “the object took this bend
before heading north”. Note that single-stroke lifeline glyphs are always aligned, be it
positively or negatively.
Scaled vs. distorted
Alignment can be considered the key qualitative relationship between temporal ink and the
temporal semantics of the underlying content. Next to alignment, numerous quantitative
relationships may exist, which enable the extraction of high level information. However, it is
highly probable that quantitative relations – despite their existence – will not be intended by
the sketcher, and hence are meaningless. Nevertheless, the relationship of linear
proportionality (fixed scale) merits our specific attention for two reasons. First, a linear
proportionality is one of the simplest3 relationships between two quantitative variables.
Second, as elements in geospatial sketch maps are drawn at an approximated spatial scale,
then why would it not be straightforward and natural for people to be able to draw them at
an approximated time scale as well? Obviously, if intended so, perfect linear relationships
are unrealistic, instead of approximate correlations.
3 The simplest one would be the equality relation, which does not make sense, except for the trivial case where
the MPO of interest is the pointer of the input device at hand.
Modelling moving objects in geospatial sketch maps 121
As for the continuous/discrete dichotomy, the scaled/distorted division applies to both the
level of spatial and temporal ink. In geospatial sketch maps, spatial ink is believed to have an
approximate fixed scale. At the temporal level, alignment is a necessary condition for time-
scaled glyphs. Time-scaled lifeline glyphs, allow for inferences about relative speed and
travel time in statements such as “the object spent most of its time on this part of its
trajectory”, or “the speed of the object in the bends is half of its speed in the straight parts”.
At an intermediate information level, in between aligned and time-scaled representations,
temporal ink can be used to segment glyphs according to clearly distinguishable categories
such as slow, moderate, and rapid drawing speed. For lifeline glyphs, these categories, when
meaningful, reflect the actual speed of the modelled MPO.
6.3.3 Typology of lifeline representations
On the basis of the distinctions made in section 6.3.2, a typology of lifeline representations
in geospatial sketch maps can be deduced, as shown in Figure 6.6. A lifeline is modelled
through one or more glyphs (dashed relationship in Figure 6.6). These glyphs will be either
explicit, or implicit representations. Within both subtypes, aligned representations can be
distinguished from others (non-aligned). Explicit aligned glyphs can be further subdivided in
continuous and discrete types. Finally, continuous cases may be scaled or distorted.
Figure 6.6 – Typology of lifeline representations in geospatial sketch maps.
6.3.4 Multiple lifelines
So far, we considered the characterisation of individual lifeline glyphs. However, reasoning
about interactions between moving objects, requires relating multiple lifelines to each other.
These inter-lifeline relations have to be temporal, as time determines space for lifelines, and
not vice versa (section 6.3.1). In section 6.3.2 we have shown that alignment properties can
Lifeline
Glyph(s)
Explicit
Aligned
Continuous
Scaled Distorted
Discrete
Non-aligned
Implicit
Aligned Non-aligned
122 Chapter 6
be used to describe temporal relations within and among glyphs. However, lifeline
representations are not always aligned. More than that, the assumptions of one-handed
input and line drawing tool preclude a sketcher from drawing two separate elements
simultaneously. Hence, temporal inter-glyph relations cannot be expressed by means of
alignment, apart from the exceptional cases restricted to before and after relations.
Conversely, interactions between multiple MPOs will be especially relevant for lifelines that
happen simultaneously or at least have a temporal overlap. Therefore, these relationships
need to be imposed by means of additional content such as found in annotations, meta-
layers, or specialised interfaces, which is out of the scope of this chapter as stated earlier.
6.4 Conclusions and outlooks
In this chapter, we extended the static concept of sketch maps, and the related ontology of
glyphs carrying ink and content, to a dynamic framework. Temporal information – next to
spatial information – takes a key role in this renewed model, where it can be found in both
ink and content associations. In order to focus on the representation of MPOs in geospatial
sketch maps, we relied on the well-known notion of geospatial lifelines. The extended model
has then been utilised to elaborate a set of characteristic binary distinctions about lifeline
representations. They can be regarded as dichotomies for a sketcher to choose from when
sketching about MPOs in a geospatial context. Throughout the chapter, there is a focus on
the interrelations between the spatial and temporal properties of lifelines and the respective
spatial and temporal ink representing it. We believe that such interrelations are important
for information systems in order to improve the automatic interpretation of the content of
lifeline glyphs by their ink, thereby making extensive use of its temporal ink, next to its
spatial component.
Above all, this chapter can be considered a basis for further research. Its content has been
underpinned by theoretical concepts, literature and common sense arguments. Nonetheless,
the authors are well aware of the fact that empirical research is a necessary next step in
order to elucidate and assess how people do represent and reason about moving objects
through sketches. Therefore, we are planning to set up appropriate test cases and build a
tool to acquire and analyse the according sketch map data. This will enable to answer
questions such as to what extent do human sketchers respect alignment relations, or do
humans have the ability to reproduce time-scaled representations in sketch maps.
Sketching is often seen as a multi-modal, multi-domain and multi-disciplinary issue.
Consequently, in further stages, this work can be extended in numerous ways. To begin with,
special cases of lifelines have been overlooked. Examples are periodic displacements, e.g.
cycles and to-and-fro movements, and lifelines (partially) shared by multiple objects such as
for a herd of animals. In addition, several interpretative aspects are still to be assessed, such
as the abilities to inter- and extrapolate lifelines, or the integration of multiple lifelines and
their underlying interaction patterns. Furthermore, the restriction to MPOs and two-
Modelling moving objects in geospatial sketch maps 123
dimensional top view can be abandoned, and replaced with other motion concepts and
perspectives from different backgrounds. In time geography for instance, the predominating
perspective is that of a three-dimensional (two spatial and one temporal dimensions) space-
time cube (Kraak 2003), and, to our knowledge, the ability of people to draw sketch maps in
such setting has not been examined yet. In other future work, this work could be extended
beyond motion, to other concepts and applications which relate space and time such as
change assessment and physical planning. Finally, the applied data acquisition restrictions
may be adjusted. We have applied the constraints of conventional pencil-and-paper
sketching. Instead, different assumptions could be employed in order to reflect for instance
the opportunities offered by the latest or planned technological developments with respect
to multi-modal sketching interfaces.
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Travelling subjects
Part II
“Many a trip continues long after movement in time and space have ceased”
(John Steinbeck)
Analysing spatiotemporal sequences in Bluetooth tracking data 127
7 Analysing spatiotemporal sequences in Bluetooth
tracking data
Delafontaine M., Versichele M., Neutens T., Van de Weghe N.
in Environment and Planning B, submitted for publication
Abstract. The use of Bluetooth technology as a technique to collect data about the
movement of individuals is increasingly gaining attention. This chapter explores the
potential of sequence alignment methods to analyse data obtained from Bluetooth
tracking. To this end, an empirical case study is elaborated which applies sequence
alignment methods to examine the behavioural patterns of visitors tracked by
Bluetooth at a major trade fair in Belgium. The results and findings underline the
validity of Bluetooth tracking to collect data from visitors at mass events, as well as
the ability of sequence alignment methods to extract insightful information from
sequences within such data.
Keywords. Spatiotemporal patterns – Bluetooth tracking – Sequence alignment
7.1 Introduction
This chapter will use sequence alignment methods (SAM) to analyse patterns within tracking
data obtained from Bluetooth sensing. Although existing as a communication technology
since the mid-nineties, Bluetooth has only recently been employed for the tracking of
individual movement (O’Neill et al. 2006, Hermersdorf et al. 2006, Van Londersele,
Delafontaine & Van de Weghe 2009, Fallast, Scholz & Ekam 2008, Wasson, Sturdevant &
Bullock 2008, Bullock et al. 2010, Furbach, Maron & Read 2008). Despite its limited
positional accuracy, Bluetooth tracking is a low-cost alternative for true location-aware
technologies. A major advantage of this technique is that it allows for the distinction of
tracked subjects at the individual level. This is because Bluetooth-enabled devices broadcast
a unique MAC (48-bit physical address). Furthermore, due to its widespread standard
integration in nowadays personal wearable devices such as cellphones, PDA’s and headsets,
Bluetooth allows for unannounced tracking, i.e. tracking of subjects who are not aware of
being tracked. It therefore offers researchers valuable potential to conduct unbiased
experiments and gather uninfluenced observations of a large number of individuals.
In this chapter, we consider the most common approach to employ Bluetooth technology as
a tracking system. It consists of a number of Bluetooth access points, henceforth denoted as
nodes, installed at fixed strategic locations throughout the area of interest. Each node
continuously searches for nearby devices. Whenever a Bluetooth-enabled device enters the
128 Chapter 7
radio range of a node, its MAC address is logged, such that the presence of devices at nodes
can be recorded along the time line. From these records, the trajectory of an individual may
be approximated as the spatiotemporal sequence of node observations of the device (s)he is
carrying. In addition to this basic tracking system, optional supplementary attributes may be
logged such as the device class1 and the user-friendly name2, although these might demand
additional lookup time. To date, most Bluetooth tracking projects documented in the
literature have relied on this approach (e.g. Fallast, Scholz & Ekam 2008, Wasson, Sturdevant
& Bullock 2008, Van Londersele, Delafontaine & Van de Weghe 2009, Bullock et al. 2010). On
the other hand, apart from being robust and plain, the approach is appealing due to its easy
and low-cost implementation which requires merely a number of Bluetooth dongles3,
computational units and storage units. Furthermore, the approach is efficient in its passive
data collection as it does not set up true connections with devices, and thereby avoids any
interaction with the individuals being tracked.
In the large body of research on movement behaviour, considerable work has been
dedicated to the definition and extraction of patterns from movement data (e.g. Laube,
Wolle & Gudmundsson 2007, Gudmundsson, van Kreveld & Speckmann 2007, Dodge, Weibel
& Lautenschutz 2008, Dodge, Weibel & Forootan 2009). Most of these approaches stem
from a cross-pollination of GIScience, computational geometry, knowledge discovery in
databases, data mining, spatial cognition, and artificial intelligence (Laube, Wolle &
Gudmundsson 2007, Gottfried & Aghajan 2009, Miller & Han 2008). However, much of these
techniques may not be suitable to analyse Bluetooth tracking sequences. This is because
Bluetooth tracking sequences may be incomplete or inconsistent due to data failure of the
nodes (e.g. signal obstructions, data loss) and tracked devices (e.g. limited battery lives,
disabled by the carrier) on the one hand, and due to the limited coverage of the study area
in terms of node radio ranges on the other hand. REMO (Laube, van Kreveld & Imfeld 2005,
Laube, Imfeld & Weibel 2005), a generic geographic knowledge discovery approach to
describe relative motion patterns through a matrix, for example, would not be a suitable
formalism to represent and explore Bluetooth tracking sequences as it would require the
location of each device to be known at regular time stamps. Another example is the
Qualitative Trajectory Calculus (Delafontaine et al. 2010, Delafontaine, Cohn & Van de
Weghe 2011), which, despite its potential to handle incomplete information, is not eligible
for handling Bluetooth sequences as it builds on higher level motion attributes such as
motion azimuth and velocity.
This chapter will explore the potential of sequence alignment methods (SAM) for the
extraction of patterns within Bluetooth tracking sequences. SAM is a relatively new
1 The device class is a 3-byte value that describes a device by a hierarchical classification, e.g. Phone: Cellular,
Computer: Laptop. 2 A user-friendly name is an arbitrary word or phrase most often configurable by the user. 3 A Bluetooth receiver integrated into a USB stick.
Analysing spatiotemporal sequences in Bluetooth tracking data 129
technique in the research field focusing on movement patterns. In the next section, we will
briefly highlight the background and basics of SAM. Then, in section 7.3, we will apply SAM
to analyse Bluetooth tracking sequences gathered at a 5-day trade fair in Ghent (Belgium).
Finally, conclusions are drawn in section 7.4.
7.2 Sequence Alignment Methods
7.2.1 Background
Having a tradition in bioinformatics to measure the distance between DNA strings or protein
strands (Morrison 2010), SAM was first applied in social science by Abbott (1995) to analyse
career patterns. In turn, Abbott’s pioneer contribution has triggered an important body of
SAM studies within sociology (see (Abbott & Tsay 2000) for an overview). From that point
onwards, SAM has been considered a promising methodology to analyse the sequential
aspects of human space-time activities, which is evidenced, among others, through
contributions by Wilson (1998, 2001, 2008), Joh et al. (Joh, Arentze & Timmermans 2001a,
Joh, Arentze & Timmermans 2001b, Joh et al. 2002, Joh, Arentze & Timmermans 2007), and
Shoval et al. (Shoval & Isaacson 2007, Shoval et al. 2008).
Within the abundant research on human activity and travelling behaviour, SAM is usually
applied to data collected by means of questionnaires, activity-travel diaries and position-
aware devices. The application of SAM to empirical data obtained from passive wireless
tracking systems has, until present, not been scrutinised. An exception is the recent work of
Choujaa and Dulay (2009b, 2009a) who consider activity sequences inferred from cellphone
data. However, they employ SAM as a novel approach to predict gaps in the activity logs,
rather than to analyse these logs.
7.2.2 Methodology
Sequence alignment is the process of equating two or more sequences of elements of a well-
defined universe using a set of eligible operations (Morrison 2010). Sequence alignment
methods (SAM) seek for optimal alignments by employing dynamic programming algorithms
to either maximise a similarity measure, or to minimise a distance measure (Wilson 2008).
This distance measure is usually referred to as Levenshtein distance (Schlich 2003,
Levenshtein 1966) or biological distance (Shoval & Isaacson 2007, Bargeman, Joh &
Timmermans 2002). There exist two categories of SAM algorithms. Global alignment
methods force the alignment to span the entire length of the sequences, while local
alignment methods focus on the similar parts within sequences that may differ significantly
overall (Choujaa & Dulay 2009a).
The conventional operations eligible for a pairwise alignment, i.e. the alignment of two
sequences, are identity, substitution, insertion, and deletion. As they always occur together,
the latter two operations are known as indels and are accommodated by gaps in one of both
130 Chapter 7
sequences. Sequences are usually represented as a string of elements consisting of one or
more characters. A pairwise alignment of two single-character strings ‘Bluetooth’ and
‘Blåtand’4 is illustrated in Figure 7.1. It features three identities, four substitutions and two
indels. A multiple alignment, i.e. an alignment of three or more sequences, is usually
approximated by a procedure of multiple pairwise alignments, known as progressive
alignment (Wilson 2006).
To determine whether an alignment is optimal, the operations have to be weighted by a
priori defined similarity scores. Typically, some additive scoring scheme is adopted in which
the identity operation represents the highest similarity and is thus given the highest score.
Substitutions are mostly associated to zero scores and indels to penalties (negative scores).
However, depending on the nature of sequenced elements, combination-specific
substitution scores (or indel penalties) may be useful. For instance with respect to the
alphabet characters in the example (Figure 7.1), from an etymological-linguistic point of
view, the t-d substitution might be assigned a higher similarity score then the o-n
substitution. Specific similarity values are usually described by a scoring matrix which
contains all pairwise substitution scores.
Contrary to traditional measures such as Euclidean, Manhattan, or Hamming distances,
Levenshtein distances systematically capture the entire sequential dimension to assess the
similarity among two sequences (Shoval & Isaacson 2007). This is the principal advantage of
SAM with respect to other methods. In addition, the alignment process allows for
discovering hidden patterns buried within the dataset (Wilson 1998). This is a particularly
valuable characteristic within the context of this chapter, given the frequent gaps in
Bluetooth tracking logs.
According to Shoval and Isaacson (2007), two types of analysis can be conducted on the
basis of SAM. The most common one is an analysis of clusters of similar sequences and/or
representative sequences. Another possibility consists of detecting hypothetical behavioural
patterns within the sequence data at hand. The former use of SAM will be considered in the
next section of this chapter.
4 Bluetooth is named after the Danish king Harald Blåtand (940 – 981 A.D.).
B l u e t o o t h identity substitution
indel B l å t a n d
Figure 7.1 – Pairwise alignment.
Analysing spatiotemporal sequences in Bluetooth tracking data 131
7.3 Case study
In this case study we will apply sequence alignment methods to analyse the behavioural
patterns of visitors tracked by means of Bluetooth at the Horeca Expo in Ghent (Belgium).
The Horeca Expo is the most important annual trade fair for the hotel and catering industry
in Belgium, and it is particularly well-chosen as a setting for the examination of visitor
movement patterns for several reasons. To begin with, the fair is a well-organised and
controlled indoor event which is exclusively accessible for paying visitors, exhibitors and
crew members. This strongly limits the potential interference and data noise due to all kinds
of passers-by devices out of the study scope, which is, for instance, less evident in outdoor
environments (e.g. Van Londersele, Delafontaine & Van de Weghe 2009, Furbach, Maron &
Read 2008). Secondly, the fair organisers allowed us to passively track participants without
their prior knowledge5, such that the experiment is by no means biased in that sense. In
addition, the daily variation and extent of additional smaller events that may cause
temporary deviant behaviour of visitors during the fair is strongly limited. The data
collection, preparation and results are discussed in depth in the remainder of this section.
7.3.1 Data collection
The data for this case study have been collected during the 21st edition of the Horeca Expo
(November 22-26, 2009). This edition has counted 53 146 visitors, most of them being
professionals in the catering industry, for 607 exhibition stands. The Horeca Expo takes place
at the Flanders Expo exhibition centre in Ghent (Belgium). The centre has eight exhibition
halls over an area of about 56 000 m² (Figure 7.2). Each hall groups exhibition stands of a
specific theme (e.g. hall 1: breweries, hall 5: kitchen contractors). 22 Bluetooth nodes,
denoted A – T6, have been discreetly installed throughout the entire site. The nodes are
equipped with power class 2 Bluetooth dongles which are developed to cover a radio range
of about 20m, although experiments have shown that this range may vary substantially,
among others due to indoor reflections. Given this presumption, it follows that the study
area is not completely covered by all nodes, and that some node pairs have an overlap in
their covered areas (Figure 7.2).
The Bluetooth nodes continuously scan for nearby devices and log all discovered MAC
addresses with the timestamp of discovery. Over the entire 5-day course of the fair, 14 498
unique devices have been observed, most of which are mobile phones and the like (95%)
(Figure 7.3). Although at most 2%7 of the observed devices are not wearable (e.g. desktop
computers), these will be detained for further analysis since their tracking logs are not
5 Unanounced Bluetooth tracking complies with the statutory privacy legislation on personal information
protection imposed by the Belgian Privacy Commission (http://www.privacycommission.be), since Bluetooth tracking data relates to devices and as such does not allow for identifications at the individual level. 6 Node H has been left out as it is located out of the study area in this case study.
7 Including the devices for which the class is unknown.
132 Chapter 7
expected to reflect visitor movements. 89% of all devices have been observed only on one
day (Figure 7.4), which suggests a large majority of one-day participants. In terms of unique
devices per day, the dataset consists of 20 148 device-days. A histogram of device-day
duration, i.e. the duration between the first and last node observation of a device on a day,
is depicted in Figure 7.5. Two notable remarks can be drawn. First, almost 20% of the device-
days have been observed for less than fifteen minutes. This can be explained among others
by a quick disabling of devices of persons entering the fair and by short Bluetooth-enabled
episodes of devices of people who intentionally make use of the Bluetooth functionality.
Since this case study aims to analyse visitor behavioural patterns, such fragmented device-
day observations can be considered unrepresentative and have therefore been excluded.
Second, over 10% of the device-days have observations that cover over eight hours, which is
about the daily opening duration of the fair. Since these devices most probably accrue to
exhibitors, crew members and/or are non-wearable, they have been excluded as well.
Figure 7.2 – Schematic map of Flanders Expo with indication of entrances and exits for visitors
(arrows), exhibition halls (H1-H8, black rectangles), and Bluetooth nodes (A-T, x-marks) with 20m
radio range (black circles).
Analysing spatiotemporal sequences in Bluetooth tracking data 133
Figure 7.3 – Distribution of Bluetooth device classes across observed devices
Figure 7.4 – Histogram of observed days per device
Figure 7.5 – Histogram of device-day duration.
7.3.2 Data preparation
For each remaining device-day we have determined the chronological sequence of node
observations. To filter for noise in the data, subsequent observations by the same node that
are less than one minute apart have been concatenated to one observation lasting over the
entire interval. Some additional preparative steps have been taken to extract representative
2%
76%
1% 18%
0%0% 1% 0% 2%
Audio
Cellular phone
Cordless phone
Smart phone
Desktop computer
Handheld computer
Laptop computer
Palm sized computer
Unknown
0
2000
4000
6000
8000
10000
12000
14000
1 2 3 4 5
Freq
uen
cy o
f o
bse
rved
dev
ices
Days
0
500
1000
1500
2000
2500
3000
3500
4000
< 0.25 0.25 - 1 1 - 2 2 - 3 3 - 4 4 - 5 5 - 6 6 - 7 7 -8 > 8
Freq
uen
cy o
f o
bse
rved
dev
ice
-day
s
Observation duration (h)
134 Chapter 7
sequences for visitors and to exclude as much as possible the sequences of exhibitors, crew
members and outlier sequences. The following restrictions have been imposed:
The first and last observations in the sequence are observed at node P or R which are
located near the visitor entrances and exits (Figure 7.2);
The time span of a sequence is within the official opening hour intervals of the fair, i.e.
each day from 10:30 AM to 7:00 PM;
The time gaps in between two subsequent observations in the sequence have a
maximum duration of 15 minutes;
The sequence contains observations of at least eight different nodes.
Further, the observation sequences that satisfy the above restrictions have been transcoded
to single-character sequences to facilitate sequence alignment. To this end, a temporal unit
of 3 minutes has been postulated as being the minimum duration for visiting a certain
location within the fair. Hence, the observation sequences have been divided into 3-minute
episodes, each of which has been allocated a character according to the following rules:
If more than 50% of an episode is covered by observations of the same node, the node’s
character is allocated to the episode;
If more than 50% of an episode is covered by observations of two nodes, the character
of the node which observations cover the greater share is allocated to the episode;
If an interval has observations of three or more nodes, a character V is allocated to the
interval;
In all other cases a gap character (-) is assigned.
Figure 7.6 presents some of the resulting sequences. The interpretation of sequence
characters is as follows. A node character represents a visiting event in the neighbourhood
of the corresponding node; a V character represents a travelling episode, i.e. a visitor
travelling through the fair (e.g. in between two visiting events); and gaps represent the
unknown information. Note that SAM are – more than any other methodology – able to
handle gaps which are interpreted as indel operations (section 7.2.2). Given the above
constraints and the strategic dispersion of nodes across the study area (Figure 7.2), it is
probable that visitors remain near to the node of their last observation during gaps. As the
interpretation of gaps and V episodes may depend on neighbouring characters, sequences
consisting for more than 50% of gaps or V episodes have been excluded.
Analysing spatiotemporal sequences in Bluetooth tracking data 135
Figure 7.6 – Extract of transcoded Bluetooth sequences.
7.3.3 Sequence alignment
The area covered by a node’s radio range contains multiple fair stands which hampers the
analysis of visiting patterns at the stand-level. Therefore, we will rely on the thematic
grouping of stands within the exhibition halls (see section 7.3.1) to define the mutual
similarity of sequence characters. Node character episodes of nodes within the same hall can
be considered more similar than those of nodes in different halls. Figure 7.7 displays the
considered scoring matrix. An exact character match (identity) is assigned a similarity score
of 10 (maximal similarity). A mismatch (substitution) is given a similarity score of 7 in the
case of characters of nodes in the same hall, and 0 (maximal dissimilarity) otherwise. An
exception has been made for the substitutions A-K, A-M, J-M, B-L, B-P, and L-P which have
been allotted lower similarity scores due to the greater distances between the
corresponding nodes. Also, alternative scores apply for the identity and substitution of V
characters in order to lower the priority of matching V episodes in the alignment process. To
this end, the identity value for V characters is set to 3 and the substitution value with respect
to all other characters to 1 (not to 0 as V characters are related to at least three different
nodes, see section 7.3.2). Finally, separate indel penalties have been considered for gap
openings and for gap extensions; respectively -5 and -3.
510 sequences were found to validate the restrictions imposed by the data preparation
(section 7.3.2). Using the yet specified similarity scores and penalties, a multiple alignment
of these sequences has been generated within the ClustalTXY software package (Wilson
2008) by means of a progressive alignment procedure which consists of (i) a pairwise
alignment of all sequence pairs (i.e. 129 795 pairs) using a local alignment algorithm (Smith
& Waterman 1981), (ii) a neighbour-joining process (Saitou & Nei. 1987), and (iii) a multiple
alignment using a global alignment algorithm (Needleman & Wunsch 1970). The neighbour-
joining process aims to structure the sequence data by joining similar sequences on the basis
136 Chapter 7
of their pairwise alignment score such that a guide tree is derived which determines the
optimal order for adding sequences to the multiple alignment by proceeding from the leaves
to the root of the tree.
A B C D E F G I J K L M N O P Q R S T V
A 10 0 0 0 0 0 0 0 7 5 0 3 0 0 0 0 0 0 0 1
B 0 10 0 0 0 0 0 0 0 0 5 0 0 0 5 0 0 0 0 1
C 0 0 10 0 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0 1
D 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 1
E 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 7 0 0 0 1
F 0 0 0 0 0 10 0 0 0 0 0 0 7 0 0 0 0 0 0 1
G 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0 1
I 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 7 1
J 7 0 0 0 0 0 0 0 10 5 0 0 0 0 0 0 0 0 0 1
K 5 0 0 0 0 0 0 0 5 10 0 7 0 0 0 0 0 0 0 1
L 0 5 0 0 0 0 0 0 0 0 10 0 0 0 5 0 0 0 0 1
M 3 0 0 0 0 0 0 0 0 7 0 10 0 0 0 0 0 0 0 1
N 0 0 0 0 0 7 0 0 0 0 0 0 10 0 0 0 0 0 0 1
O 0 0 7 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 1
P 0 5 0 0 0 0 0 0 0 0 5 0 0 0 10 0 0 0 0 1
Q 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 10 0 0 0 1
R 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 0 0 1
S 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 10 0 1
T 0 0 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 10 1
V 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3
Figure 7.7 – Sequence alignment scoring matrix.
7.3.4 Results
Three results are obtained from the threefold sequence alignment process described in
section 7.3.3: (i) a square matrix with pairwise alignment scores, (ii) a neighbour-joining
guide tree, and (iii) a multiple alignment. The alignment matrix is the most raw and low-level
result which will not be further considered as the information it contains is also captured by
the other results.
The guide tree obtained from the neighbour-joining process (section 7.3.3) is shown in
Figure 7.8. It totals 509 hierarchical clusters of similar sequences. The clusters observed in
this guide tree may assist in the determination of a typology of different visitor behavioural
patterns. The number of members in a cluster can thus be considered an indicator for the
importance of the corresponding behavioural pattern. Sequences in smaller clusters,
however, tend to have more elements in common. In SAM literature regarding activity
patterns, it is usually considered up to the analyst to determine the number and
interpretation of clusters in the guide tree. This can be facilitated by means of the multiple
Analysing spatiotemporal sequences in Bluetooth tracking data 137
Figure 7.8 – Multiple alignment guide tree with clusters and subclusters labeled at their root node.
alignment. To enable a visual exploration of patterns in the multiple alignment, we have
sorted the aligned sequences according to the leave order of the guide tree. In addition, the
node characters in the multiple alignment have been colour coded according to the
exhibition hall where they are located. A fragment of this representation is displayed in
Figure 7.8. It illustrates a clear pattern of related sequences with predominant episodes at
the exhibition halls 8, 7 and 1 (see further cluster 1.1).
Within the guide tree, three major clusters (Figure 7.8, 1-3) can be observed at the top of the
hierarchy. At this level, the aligned sequences hardly share common characteristics, if at all.
On the basis of visual supervision of the sorted and colour coded multiple alignment (Figure
7.9) an exhaustive subdivision has been made into 21 non-overlapping subclusters (Figure
138 Chapter 7
7.8, 1.1-3.8). For each subcluster the number of members and the shared pattern has been
summarised in Table 7.1. In addition, the subcluster median and average sequences have
been listed in Table 7.2. The median and average sequences are representative sequences of
a cluster (Wilson 2008). In analogy to the homonymous descriptive statistics, these
sequences respectively minimise the sum Levenshtein distances and the sum of squared
Levenshtein distances to all other members of the cluster.
Figure 7.9 – Extract of the sorted and colour coded multiple alignment (colour legend: hall 1,
hall 2, hall 3, hall 4, hall 5, hall 6, hall 7, hall 8).
The results in Table 7.1 and Table 7.2 reveal some interesting aspects about the behaviour of
Horeca Expo visitors. First of all, they reflect a large heterogeneity of visiting patterns in
terms of visit duration, the number of visited locations, and in particular the order of visiting
these. Notwithstanding that the fair can be entered and left from only two locations, the
variety of tracking sequences emphasizes the lack of one or a few predominant
spatiotemporal behavioural patterns of Horeca Expo visitors. Inferences can be made
concerning the attractiveness of locations, although these might be misleading given that
not all exhibition halls have been equally covered by Bluetooth nodes (e.g. hall 6). The
abundant hall 1 episodes reflect that the main hall is also the most important one in terms of
visits, as could be expected given its size and central location. More than that, it can be
observed that most sequences visit the main hall more than once, whereas other halls are
usually visited once at most. 12 of the 21 common cluster patterns in Table 7.1 feature two
disjoint episodes at hall 1, whereas none of them features repetitive visits of other halls
(except for hall 7 in cluster 2.1). Thus, people most often tend to benefit maximally from
their visit by passing as much locations as possible, thereby avoiding revisiting halls, which is
inevitable for the main hall. Merely one cluster (2.2) seems to represent an exhaustive visit
to the fair, i.e. calling at all exhibition halls. However, given that only shared episodes have
Analysing spatiotemporal sequences in Bluetooth tracking data 139
been listed in Table 7.1, many other clusters may encompass such visits as well (e.g. see
Table 7.2, Figure 7.9).
Cluster Members Common pattern Legend
1.1 47
1.2 17 predominant episode
1.3 17 frequent episode
1.4 18 occasional episode
1.5 29 disjunction of episodes
1.6 19 1 – 8 exhibition halls
1.7 49 A – T Bluetooth nodes
1.8 17
2.1 38
2.2 15
2.3 44
2.4 14
2.5 34
3.1 26
3.2 26
3.3 5
3.4 22
3.5 19
3.6 7
3.7 38
3.8 9
Table 7.1 – Number of members and common patterns per cluster. Pattern episodes are colour
coded to hall location and annotated with hall numbers or node characters. Hollow episode
symbols represent episodes at one of the eight exhibition halls.
Other inferences can be made regarding the chronology of hall visits. Most sequences
consist to a considerable extent of logically structured chains of subsequent episodes at
neighbouring locations. The common patterns of clusters 1.1, 1.3, 1.7, 2.2-3.3, and 3.4-3.8
consist entirely of such chains. The most frequently combined exhibition halls are halls 1-2,
2-4, and 1-7 (in both directions). When considering the Flanders Expo map, the first two
combinations seem straightforward for visitors entering the fair at node R (Figure 7.2). The
third combination, on the other hand, is particularly reasonable for visitors who have
reached the end of the main hall (and its adjacent halls) and want to make the bridge to hall
8. When looking into more detail, such combinations may give insights into the importance
of different connections. The concatenation of D and Q episodes, for instance, underlines
the significance of the direct passage which connects both halls (Figure 7.2). Finally,
8 7 1 2 4
1 8 1
4 1
1 5 3 7 1
7 3 8 A
A J A J 7
3 2 8 1 1 4
7 1 6 4 2 1
4 6 2 1 1
6 1 7 3
2 7 4 6 M 3 5
5 1 4 2 8 1
B 1 7 1 7
M 2 4 6 8 1
8 5 7 4
T 1 1
K 5 3
1 2 4 6 1 7 5 3
7 J 4 2
2 3 J 4
2 4 6 1 7 8 3 5 1
M
140 Chapter 7
concerning the time passed beyond visiting exhibition halls, it can be observed that visitors
tend to spend more time at the entrance than at the exit (e.g. see Table 7.2, Figure 7.6,
Figure 7.9). This can be explained by typical entrance activities such as registering, informing
and depositing luggage in a cloakroom, which do not or to a lesser temporal extent apply for
visitors leaving the fair.
Median and average sequence
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
2.1
2.2
2.3
2.4
2.5
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
Table 7.2 – Median (top) and average (bottom) sequence per cluster (colour legend: hall 1, hall
2, hall 3, hall 4, hall 5, hall 6, hall 7, hall 8).
7.4 Conclusions
In this chapter, we have explored the potential of sequence alignment methods to analyse
data obtained from tracking individuals by means of Bluetooth. After a brief introduction on
Bluetooth tracking and sequence alignment methods (SAM), an experimental case study has
been presented on the sequence alignment analysis of spatiotemporal patterns of visitors
tracked by Bluetooth nodes at the Horeca Expo fair in Belgium. The contribution of this work
is original since, until present, SAM have not been applied to analyse Bluetooth tracking
data. This study is also important in light of the growing attention to Bluetooth as a novel
technology to track people at mass events. We have shown that, provided that the
necessary steps have been taken to filter raw Bluetooth tracking data, SAM can be
successfully adopted to analyse Bluetooth tracking sequences. The results of the case study
Analysing spatiotemporal sequences in Bluetooth tracking data 141
have revealed some important and plausible insights about the behaviour of visitors at the
Horeca Expo. In particular, the study has disclosed the existence of a large variety of visiting
patterns especially with respect the number of and order of visited locations.
Notwithstanding this considerable heterogeneity, we have demonstrated the ability of SAM
to detect and extract the sequential structures hidden in the tracking data. The vast majority
of tracking sequences respects a reasonable chronological concatenation of visited locations,
which in turn confirms the ability of, in essence, simple Bluetooth tracking systems to
capture the spatiotemporal behaviour of large crowds of individuals at a mass event. The
results of our study may be insightful to the planners and organisers of such events in
keeping track of and exploring the behaviour of participants over the course of an event.
Despite the above contributions, some aspects still limit the potential of sequence alignment
methods for the analysis of tracking data. Unlike the structure of nucleotides in a strand of
DNA, spatiotemporal sequences within tracking data might differ very much amongst
tracked individuals, both with respect to sequence composition as with respect to the
number of elements (duration). In sequence alignment, the latter aspect may cause a large
number of gaps, for which there is yet no consensus on their interpretation (Wilson 2006).
Shoval and Isaacson (2007) recognize the lack of a solid method to assess the reliability of
alignments, as well as the lack of knowledge on the impact of the spatial and temporal scale
on the results. Other issues relate to the shortcoming of SAM as an exact science, or to
quote Morrison (2010, p. 363): “The basic problem with sequence alignment is that it seems
to be more an art than a science”. For example, there is no consensus method or standard
calibration procedure for the setting of sequence alignment parameters such as indel
penalties. Regarding tracking data, even common practices are lacking in this respect. Future
progress on these issues will enable more refined analysis configurations and support
stronger and more detailed interpretations of alignment results.
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Modelling potential movement using rough obstacle-constrained space-time prisms 145
8 Modelling potential movement in constrained travel
environments using rough space–time prisms
Delafontaine M., Neutens T., Van de Weghe N.
in International Journal of Geographical Information Science, forthcoming
Copyright © Taylor & Francis
Abstract. The widespread adoption of location-aware technologies (LATs) has
afforded analysts new opportunities for efficiently collecting trajectory data of
moving individuals. These technologies enable measuring trajectories as a finite
sample set of time-stamped locations. The uncertainty related to both finite sampling
and measurement errors makes it often difficult to reconstruct and represent a
trajectory followed by an individual in space–time. Time geography offers an
interesting framework to deal with the potential path of an individual in-between
two sample locations. Although this potential path may be easily delineated for
travels along networks, this will be less straightforward for more non-network-
constrained environments. Current models, however, have mostly concentrated on
network environments on the one hand and do not account for the spatiotemporal
uncertainties of input data on the other hand. This chapter simultaneously addresses
both issues by developing a novel methodology to capture potential movement
between uncertain space–time points in obstacle-constrained travel environments.
Keywords. Qualitative calculus – Moving point objects – Implementation
8.1 Introduction
Recent years have seen the development of a range of widely and readily available tracking
technologies, such as location-aware technologies (LATs) (Schiller & Voisard 2004) and
geosensor networks (Stefanidis & Nittel 2003). These technologies are revolutionising the
ways in which data about spatial behaviour is acquired by enabling researchers to collect
massive volumes of trajectory data of mobile objects and individuals in real-time. Tracking
data, however, are affected by at least two important sources of spatiotemporal uncertainty.
First, trajectories are typically approximated by a sequence of locations pinpointed at
discrete timestamps. Due to finite sampling, the uncertain positions of an individual have to
be interpolated between successive sample points. While uncertainty about an individual’s
trajectory increases if sampling intervals are larger, higher sampling frequencies result in
finer granularity and more spatiotemporal detail (Hornsby & Egenhofer 2002). The sampling
146 Chapter 8
frequency may be inherent to the tracking device at hand or may result from an incomplete
spatial coverage of a geosensor network (i.e. the position of an individual is not recorded in
areas outside the radio range of the sensors). In addition, sampling frequency can be
influenced by system failures. For example, the sampling rate of GPS measurements may
decrease in urban locales if the signal is blocked by obstructions (e.g. buildings). A second
source of uncertainty arises from the fact that sample points themselves are prone to
measurement inaccuracy depending on the spatial resolution of the tracking technique used.
While individuals may be traced with an acceptable accuracy using GPS, the accuracy of
short-range, wireless radio-communication technologies is often much lower and may
depend upon the radio range and power class of the sensors and the amount of overlap
between their radio ranges. Both finite sampling and measurement errors often hamper a
straightforward reconstruction of individual trajectories on the basis of tracking data.
To cope with the problem of finite sampling in moving object databases (MODs), several
researchers, among them Sistla et al. (1998), Moreira et al. (1999), Trajcevski et al. (2004),
and Pfoser et al. (2005) have sought to delineate and query the unknown path between two
observed locations given a presupposed maximum travel velocity in an unconstrained
isotropic travel environment. In line with the advances in MODs, time geographers have also
studied the sampling problem extensively using time geography’s key concept, i.e. the
space-time prism (Hägerstrand 1970, Yu & Shaw 2008, Miller 1991, Kwan & Hong 1998).
However, while the sampling problem is well-studied in time geography, the equally
important problem of how this sampling problem interferes with the imperfect observation
of sample points has received far less attention (Miller 2005). A notable exception is Neutens
et al. (2007) who, relying on the basic principles of rough set theory (Pawlak 1982), provide a
conceptual framework to analyse how spatial and temporal uncertainty about the sample
points propagates through a space-time prism by specifying lower and upper approximations
of the prism dimensions. While conceptually appealing, their model has limited applicability
since it assumes that travel occurs in an environment without any obstacles. The aim of the
present chapter is to enhance the applicability of this conceptual model to constrained travel
environments and put it into practice by proposing and implementing a formal theoretical
framework for defining and constructing rough space-time prisms in planar space with
obstacles. The framework is particularly useful for modelling non-network-constrained
phenomena (e.g. pedestrian movement in urban and built environments) and accounts for
both finite sampling and measurement errors.
The remainder of this chapter is organised as follows. Since our approach relies on time
geography, the next section introduces the key concepts of time geography and documents
the geocomputational models that have been developed in recent years for analysing an
object’s uncertain position between two fixed sample points. Section 8.3 discusses the
formal definition and representation of a traditional space-time prism. This definition is
extended in section 8.4 toward the case of uncertain travelling constraints, i.e. uncertainty
Modelling potential movement using rough obstacle-constrained space-time prisms 147
about an individual’s departure time, arrival time, and potential travel speed. In section 8.5,
this case is further extended toward travel environments populated with obstacles. Then in
section 8.6, both approaches are combined, and an algorithm to derive obstacle-constrained
space-time prisms with uncertain constraints is presented. An example case within a simple
environment is used throughout the chapter to clarify the methodology. Finally, in section
8.7, we draw conclusions and outline avenues for future research.
8.2 Background
Time geography
Back in the 1960s, Torsten Hägerstrand (1970) and his associates at the University of Lund
(Sweden) developed a worldview for understanding the interdependencies between human
beings, nature and technology, known as time geography. Time geography provides a
conceptual perspective to analyse spatiotemporal patterns of human movement. In
particular, the time-geographical approach articulates the scarcity of space and time, and
emphasizes the importance of the constraints an individual is faced with when moving
through geographical space (Lenntorp 1978, Pred 1977). Three types of constraints are
distinguished: (i) Capability constraints refer to an individual’s cognitive limitations and
physiological necessities such as eating or sleeping; (ii) Coupling constraints restrict travel
and activity participation by dictating where, when, and for how long individuals have to join
other people, tools, or material artefacts in space and time; (iii) Authority constraints refer to
the institutional and societal context including laws, rules, norms and other regulations
implying that specific areas are only accessible at specific times for specific persons. These
three constraints are interrelated and manifest themselves by dictating the time budget
during which activities can be undertaken to achieve a project (i.e. a series of sequential
tasks necessary to the completion of any intention-inspired or goal-oriented behaviour), as
well as the individual’s travelling restrictions (e.g. travel velocity) (Carlstein, Parkes & Thrift
1978, Pred 1981).
The basic tenet of time geography is the space-time path which represents the
uninterrupted string of movements of an individual in space-time. The course of a space-
time path results from the interaction between constraints and projects and is typically
visualised in a three-dimensional framework in which time is integrated orthogonally to a
flattened topography. In this representation, an individual’s travel speed is inversely
proportional to the slope of its space-time path, where more horizontal paths represent
moves at higher speed, whereas vertical paths (infinite slope) express stationarities (zero
speed). Another key concept is the space-time prism which demarcates the envelope of all
space-time paths an individual might have drawn during the time budget between two
successive timestamps. It is important to note that while a space-time path represents
revealed spatial behaviour, space-time prisms capture potential spatial behaviour.
148 Chapter 8
Implementations of time-geographical concepts
In the past two decades, the time-geographical approach has regained attention in
geographical information science and transportation geography. Technological advances in
geographical information systems (GISs) as well as the increased availability of
georeferenced trajectory data have opened up new opportunities to enhance the realism of
the time-geographical entities and to apply these in empirical studies regarding individual
accessibility (Schwanen & de Jong 2008, Miller 1991, Yu & Shaw 2007, Kwan & Hong 1998).
Renewed interest in time geography also dovetails with the paradigm shift in transportation
policy towards travel demand management and the activity-based approach to travel
forecasting that has increasingly gained momentum since the mid-70s (Axhausen & Gärling
1992, Timmermans, Arentze & Joh 2002, Dong et al. 2006).
Modelling heterogeneous travel environments
In recent years, there has been a flurry of geocomputational methods to model the unknown
position of an individual during the time budget between two time-stamped positions. These
methods have sought to improve the classical representation of the space-time prism to deal
with the complexities of real-world travel environments. An important accomplishment is
the calculation of potential path areas within transportation networks. Following the seminal
work of Miller (1991), a number of authors have specified GIS-based algorithms to derive the
paths that an individual could have taken between two discrete locations within a road
network (e.g. Kwan & Hong 1998, Miller & Wu 2000, Wu & Miller 2001, Weber & Kwan
2002, Kim & Kwan 2003). These network-based approaches offer only a static synopsis of an
individual’s travel possibilities but do not account for the spatial variation in travel
possibilities during a time budget. Therefore, some authors have proposed algorithms to
derive the full three-dimensional, network-based space-time prism based on shortest path
algorithms within road networks (Neutens et al. 2007, Kuijpers & Othman 2009). Despite the
proliferation of methods to delineate travel possibilities within transportation networks,
only few studies have been concerned with modelling non-motorised, non-network yet
spatially constrained movements through space-time prisms. A recent example is given by
Miller and Bridwell (2009). They introduced an analytical theory to derive field-based space-
time paths and prisms using velocity fields. A velocity field is a smooth differential function
that assigns a velocity to each location in continuous space (Puu & Beckmann 1999).
Although this method allows examining theoretical conjectures about accessibility in
continuous space, a spatial decomposition into a lattice is required to use the approach in
empirical research. A drawback of this decomposition is that it introduces errors that cannot
be resolved by increasing the lattice density (see Miller & Bridwell 2009, Goodchild 1977).
Modelling travel constraint uncertainty
Another line of scientific inquiry concerns the implications of spatiotemporal uncertainty
about the prism properties (i.e. maximum travel velocity, origin and destination point) for
Modelling potential movement using rough obstacle-constrained space-time prisms 149
the prism dimensions. For example, several researchers have examined the ways in which
prism-based accessibility is affected by uncertainty in travel time caused by unreliable
transportation or systematically recurring congestion (e.g. Schwanen & de Jong 2008, Hall
1983, Ettema & Timmermans 2007). Hendricks et al. (2003), for their part, have proposed a
sequential partitioning method to model a wayfinder’s indiscernibility between future travel
possibilities. Neutens et al. (2007) have furthered this approach and sought to calculate and
represent the three-dimensional prism if its origin and destination points are not known
exactly. They introduced the concept of a rough space-time prism to model the potential
movement between two uncertain sample points through the prism’s lower and upper
approximation. Although a conceptually elegant solution to deal with both finite sampling
and measurement errors, the application of the approach is currently limited to
unconstrained travel environments. Furthermore, it does not explicitly address how
measurement uncertainty about sample points intertwines with uncertainty about the
maximum travel velocity.
The present chapter contributes to these lines of inquiry in at least two ways. First, we
complement existing network-based methods with a novel approach to model non-network-
constrained phenomena, including pedestrian movements in urban and built environments.
Drawing on research in computational geometry (e.g. Inkulu & Kapoor 2009, Kapoor,
Maheshwari & Mitchell 1997, Hershberger & Suri 1999), we propose a methodology to
construct space-time prisms in planar space with obstacles. Our approach does not require a
discretisation of space and time. Rather than approximating space-time prisms as a set of
contours at discrete moments in time using a field-based lattice, space-time prisms are
modelled and implemented as solid objects in continuous space. This eliminates errors
resulting from discretisation and avoids the storage and processing of large amounts of voxel
data. Second, the approach allows gaining insights into how combinations of uncertainty
about the maximum travel velocities and the spatiotemporal uncertainty about sample
points affect an individual’s travel possibilities.
8.3 A space-time prism in an unconstrained travel environment
A space-time prism measures the ability to reach locations in space and time in between two
locations separated in time, respectively denoted as origin and destination. Origins and
destinations may be derived from the locations of fixed activities reported in travel diaries
(e.g. Cullen & Godson 1975, Weber & Kwan 2003), or they can be estimated using stochastic
frontier modelling (e.g. Kitamura et al. 2006, Pendyala, Yamamoto & Kitamura 2002). As in
(Shoval & Isaacson 2007, Miller 2005), this chapter will take the viewpoint of origins and
destinations sampled through a tracking system, although our method can be applied to
spatiotemporal data obtained from other observation or estimation techniques as well. In
classical time geography, a space-time prism is determined by its origin, destination, and a
finite maximum velocity in an unconstrained isotropic travel environment (Miller 2005).
150 Chapter 8
Given these constraints, a space-time prism is obtained from the intersection of two cones
(Figure 8.1). The forward cone encloses all space-time points that can be reached from the
origin, while the backward cone captures all space-time points where an individual could
have come from when (s)he is to arrive at the destination. In the remainder we will refer to
these cones as reachability cones. The height of the reachability cones corresponds to the
time budget that results from the origin and destination temporal coupling constraints. Their
side slopes and aperture correspond to the maximum travel velocity that an individual may
attain.
Figure 8.1 – Space-time prism obtained from the intersection of a forward cone and a backward
cone.
More formally, a space-time prism in an unconstrained isotropic travel environment can be
defined as follows. Let be the set of real numbers, the set of positive real numbers,
and the two-dimensional real plane with metric being the Euclidean distance. Though
any metric space with metric would be possible, we will consider travel in the -
plane and represent this movement in -space , where represents time.
Let denote the origin, the destination,
the time budget, and the maximum velocity.
Definition 8. 1. The forward cone with origin , time budget , and
maximum velocity is the set of all space-time points that satisfy:
The forward cone has its apex at the origin and is oriented forward in time.
Modelling potential movement using rough obstacle-constrained space-time prisms 151
Definition 8.2. The backward cone with destination , time budget , and
maximum velocity is the set of all space-time points that satisfy:
The backward cone has its apex at the destination and is oriented backward in time.
A space-time prism can now be defined as the intersection of a forward and a backward
cone:
Definition 8. 3. The space-time prism with origin , destination , and
maximum velocity is given by:
Figure 8.1 shows how for is
obtained from the intersection of reachability cones. In the remainder, we will extend the
space-time prism to cope with uncertain origins, destinations, and maximum velocities, and
with obstacle-constrained travel environments.
8.4 A rough space-time prism in an unconstrained travel environment
In order to model the uncertainty of an individual’s travelling constraints, each space-time
prism will be represented as a rough set through its lower and upper (approximation)
prism. The upper prism includes all space-time locations that are potentially reachable.
is delimited by the least restricted space-time paths in terms of accessibility, i.e. what is
reachable in the best case. Suppose that there is uncertainty about the departure time
(temporal coupling constraint) of an individual. Then will be bounded by space-time
paths that assume the earliest possible departure time. Analogously, the lower prism
represents the space-time points that are reachable in all cases. It consists of all feasible
space-time paths in the potentially most constrained situation (e.g. assuming the latest
possible departure time). The uncertain part of a rough space-time prism is the boundary
body , which equals . Hence, three parts can be distinguished: what is certainly
reachable ( ), what may be reachable ( , and what is certainly not reachable ( .
Though this distinction has to be kept in mind, we will not explicitly consider any
further, due to its dependency on and . In the remainder of this chapter, we will use
the term rough to refer to the dual representation of a lower and upper approximation.
Rough space-time prisms can deal with three types of uncertainty, i.e. spatial, temporal and
velocity uncertainty (Neutens et al. 2007). In the context of tracking systems, there is spatial
and temporal uncertainty stemming from the measurement inaccuracy of the tracking
technology. Wireless tracking technologies such as Bluetooth and WiFi employ a certain
spatial radio range and temporal scanning interval. Although uncertainty may differ in space
and time, for many tracking data it makes sense to presume a constant spatial and temporal
152 Chapter 8
uncertainty related to the accuracy of the technology at hand. The maximum velocity, on the
other hand, cannot be directly related to measurement accuracy and is often approximated
by means of a lower and an upper estimate (e.g. maximum velocity on a road during
respectively peak and off-peak hours).
Consider origin , destination , time budget , spatial accuracy , temporal accuracy ,
maximum velocity , lower maximum velocity , and upper maximum velocity , with
, and .
Definition 8.4. The lower forward cone is the set of all space-time
points that satisfy:
Definition 8.5. The upper forward cone is the set of all space-time
points that satisfy:
Definition 8.6. The lower backward cone is the set of all space-time
points that satisfy:
Definition 8.7. The upper backward cone is the set of all space-time
points that satisfy:
In analogy to Definition 8.3, lower and upper space-time prisms can be determined from the
intersection of respectively the lower and upper forward and backward cones:
Definition 8.8. The lower space-time prism is given by:
Definition 8.9. The upper space-time prism is given by:
The following property expresses the relationship between a classical space-time prism
(Definition 8.3) and its corresponding rough approximations (Definitions 8.8-8.9):
Property 8.1.
Modelling potential movement using rough obstacle-constrained space-time prisms 153
That is, for each space-time prism and for each set of valid rough maximum velocities,
spatial accuracy, and temporal accuracy, there exist a lower space-time prism and an
upper space-time prism , such that contains , and contains . Note that might
be the empty set independent of the uncertainty parameters, whereas can never be an
empty set whenever one of these parameters is strictly positive. The model of rough space-
time prisms also generalises the classical model, which is obtained from the special case
where accuracies are negligible ( and rough maximum velocities are considered
equal ( . Therefore, the boundary body dissolves and, according to Definition 8.9,
the attained upper and lower prisms both equal the classic prism. In addition, note that,
according to the first equation of Definition 8.5, the upper forward cone has its apex at time
. However, due to the second equation, only time stamps higher than
or equal to are valid. Analogous reasoning applies for the upper backward cone, and
therefore, upper reachability cones are flattened at the top over a circular area with radius
which reflects the underlying spatial uncertainty.
The example approximation prisms and
are illustrated in Figure 8.2 (with , as in Figure 8.1).
Figure 8.2 – An uncertain space-time prism modelled by its lower (grey), and upper (black outlines)
approximation.
8.5 A space-time prism in an obstacle-constrained travel environment
Until now, movement has been considered to happen in an unconstrained travel
environment. Though this assumption underlies traditional time geography, it is hardly
tenable and most often highly unrealistic for true geographical spaces. This assumption has
been abandoned in later work, as discussed in section 8.2. In addition to these approaches,
we present an alternative considering an isotropic travel space populated with obstacles.
Obstacles can be any kind of inaccessible areas, as are building blocks, water bodies and
highways to pedestrians. The space in between the obstacles is assumed to be
154 Chapter 8
unconstrained and isotropic, which enables us to preserve the maximum velocity constraint
and thereby support the well-studied time-geographical entities introduced earlier.
We will clarify our approach using a simple example case. Figure 8.3 shows a map of three
buildings , , and at university campus ‘De Sterre’ in Ghent (Belgium). The area
surrounding the buildings can be assumed open and accessible to pedestrians. Two positions
are located at building entrances, for which we assume they are a student’s origin and
destination in between two subsequent lectures. Let us consider a time budget of two
minutes for the student to walk from to , with a maximum walking velocity of 2m.s-1 as
an educated guess. Our aim is now to construct the student’s space-time prism according to
these constraints, taking account of the obstacles blocking his/her passage.
Figure 8.3 – Travel environment constrained by university buildings A, B, and C.
As follows from section 8.3, reachability cones provide an answer to two fundamental
questions: (i) which locations are reachable for the individual within the given time budget if
(s)he starts at the origin; (ii) from which locations is the destination reachable within the
given time budget. Assessing the accessibility from (to) a certain location requires knowledge
about all shortest paths from (to) this place. In an unconstrained isotropic space, all
reachable locations lie within a certain radius from the origin or destination, as all shortest
paths are simply the straight beeline connectors. To construct space-time prisms in obstacle-
constrained environments, however, shortest paths are to be calculated that avoid the
obstacles.
In computational geometry and geographical information systems, obstacles such as
buildings and impassable areas are generally modelled as regions using a polygonal
geometry. Research in computational geometry has offered efficient algorithms to compute
the shortest paths in a Euclidean plane in the presence of such polygonal obstacles. To this
end, there have been two fundamentally different approaches. The visibility graph method
Modelling potential movement using rough obstacle-constrained space-time prisms 155
(Kapoor & Maheshwari 1988, Kapoor, Maheshwari & Mitchell 1997), on the one hand, and
the wavefront method (Mitchell 1993, Hershberger & Suri 1999) on the other hand. Some
(e.g. Inkulu & Kapoor 2009) have also considered combinations thereof. For exact algorithms
and computational details, we refer to the specialised literature. We may employ such an
algorithm in order to determine all necessary shortest paths within an obstacle-constrained
travel environment in case of obstacles modelled as polygons, as we will further assume
according to its generality in GIS. It is important to note that only the shortest paths to
polygon vertices have to be calculated, due to the following reasoning. Whenever an
obstacle blocks the straight connection from to any other point, the corresponding
shortest path(s) from will pass along an extreme (i.e. a tangential point) of when
observed from . This extreme will always be a vertex in the case of a polygonal obstacle.
Let be a set of obstacles, and be the set of
vertices of obstacle . Let denote the shortest path from to avoiding the
obstacles in . Let denote the parent, i.e. the preceding vertex, of vertex along
.
Definition 8.10. The shortest path tree from with respect to the obstacles in
is given by:
An is a tree in which each vertex is a node that is associated with its parent along the
shortest path from the root parent to . Given a shortest path tree , the shortest
path can be easily determined as the ordered sequence of parent vertices
starting from the root parent to the destination vertex Figure 8.4 and Figure 8.5
respectively show a map of and for the example case.
Each vertex can be associated with a shortest path distance. Let denote the distance
to vertex along shortest path . A vertex is reachable if its shortest path distance is smaller
than or equal to the distance budget, i.e. the product of time budget and maximum velocity.
Based on the shortest path distance and the time budget, we may define the set of
reachable vertices:
Definition 8.11. The reachable set is the set of all vertices of obstacles in
that lie within distance budget from origin along a shortest path avoiding the
obstacles in :
156 Chapter 8
Figure 8.4 – Shortest paths (black lines) from the origin (big dot) to all obstacle vertices (small
dots).
Figure 8.5 – Shortest paths (black lines) from the destination (big dot) to all obstacle vertices (small
dots).
All parent vertices in the reachable set act as wavefront propagators that induce separate
reachability cones according to the time budget that remains at the time they are reached.
Given a set of obstacles , a time budget , and a maximum velocity , the forward
cone at parent vertex is denoted as with
. Analogous reasoning applies for the backward cone and
.
Not all parts of the yet obtained cones are reachable within the remaining time budget
. Only the directly reachable parts, i.e. parts accessible by a straight path from the
Modelling potential movement using rough obstacle-constrained space-time prisms 157
parent concerned, are to be selected, as the other parts will either be directly reachable
from other parent vertices, or they will not be accessible within . Hence, the non-
directly reachable parts are to be subtracted from the cone. The spatial footprint of these
parts belongs to either areas that overlap with an obstacle (i), or to areas that are obscured
by one or more obstacles (ii). The directly reachable parts of a parent’s reachability cone can
be obtained by extruding the spatial zones (i) and (ii) vertically along the time axis, and
subtracting these volumes from the cone. As thereafter, the resulting body is no longer a
true cone, we will term it a reachability body, i.e. forward body and backward body. The
respective reachability bodies for a parent vertex can be defined as follows:
Let denote the straight spatial connection line segment from to .
Definition 8.12. The parent forward body for a parent with respect
to origin , obstacle set , time budget , and maximum velocity is given by:
Definition 8.13. The parent backward body for a parent with
respect to destination , obstacle set , time budget , and maximum velocity is
given by:
Figure 8.6 and Figure 8.7 illustrate the reachability bodies for a parent vertex of building ,
according to origin, destination and time budget specified earlier. The figures also indicate
the footprint of the obstructed zones to be extruded ((i) and (ii)). Note that the reachability
bodies are situated at different time intervals, due to their different temporal orientation as
well as to the temporal difference corresponding to the respective shortest path distances
from to and from to .
A parent reachability body delimits the potential path space at a parent vertex, according to
the remaining time budget at that vertex. The overall reachability bodies are now obtained
from the union of all reachability bodies, either the forward or the backward bodies, over all
parents in the reachable set.
Definition 8.14. The forward body with origin , obstacle set , time
budget , and maximum velocity is given by:
158 Chapter 8
Definition 8.15. The backward body with destination , obstacle set ,
time budget , and maximum velocity is given by:
In analogy to Definition 8.3, the obstacle-constrained space-time prism is obtained from the
intersection of the forward and backward bodies (Figure 8.8):
Definition 8.16. The obstacle-constrained space-time prism with origin ,
destination , obstacle set , and maximum velocity is given by:
The yet obtained obstacle-constrained space-time prism demarcates the potential path
space for an individual travelling from origin to destination, respecting a given maximum
velocity, and avoiding the obstacles in his/her enviroment.
Figure 8.6 – Parent forward reachability body (grey) with indication of the parent vertex (black dot)
and the spatial extrusion zones (black outlines).
Modelling potential movement using rough obstacle-constrained space-time prisms 159
Figure 8.7 – Parent backward reachability body (grey) with indication of the parent vertex (black
dot) and the spatial extrusion zones (black outlines).
8.6 A rough space-time prism in an obstacle-constrained travel environment
8.6.1 Combination of approaches
This section concerns the integration of the approaches of sections 8.4 and 8.5. Whereas in a
classical unconstrained environment, space-time prisms follow from the intersection of two
reachability cones (Definitions 8.1-8.3), two sets of parent reachability bodies are to be
intersected, when accounting for obstacles (Definitions 8.14-8.16). These reachability bodies
are geometrically equivalent to cones with subtracted vertical extrusions (section 8.5). The
constraints that determine these underlying cones, however, are not affected by the further
subtraction of parts (Definitions 8.13-8.14), and subsequent union with other bodies
(Definitions 8.15-8.16). Therefore, we may preserve the methodology of section 8.5 and
adopt Definitions 8.13 and 8.14, in order to obtain rough parent reachability bodies.
Subsequently, the Definitions 8.15-8.17 can be adapted analogously in order to construct the
rough reachability bodies and space-time prisms for an environment constrained by
obstacles. Hence, for a given origin , destination , obstacle set , time budget , spatial
160 Chapter 8
accuracy , temporal accuracy , lower maximum velocity , and upper maximum
velocity , we obtain:
Figure 8.8 – Obstacle-constrained space-time prism (grey) with indication of obstacles (black).
Definition 8.17. The lower parent forward body for a parent is
given by:
Definition 8.18. The upper parent forward body for a parent
given by:
Definition 8.19. The lower parent backward body for a parent
is given by:
Modelling potential movement using rough obstacle-constrained space-time prisms 161
Definition 8.20. The upper parent backward body for a parent
is given by:
Definition 8.21. The lower forward body is given by:
Definition 8.22. The upper forward body is given by:
Definition 8.23. The lower backward body is given by:
Definition 8.24. The upper backward body is given by:
Definition 8.25. The lower obstacle-constrained space-time prism is
given by:
Definition 8.26. The upper obstacle-constrained space-time prism
is given by:
8.6.2 Algorithm
Based on the methodology of section 8.5 and the Definitions 8.17-8.26, we have
implemented an application program that takes an origin, a destination, a set of obstacles, a
spatial accuracy, a temporal accuracy, a lower maximum velocity, and an upper maximum
velocity as input parameters, and returns the corresponding rough obstacle-constrained
space-time prisms. The resulting prisms are then visualised as 3D solids by means of a CAD
162 Chapter 8
system. A description of the application’s main algorithm is given in pseudo-code in
Algorithm 8.1.
Input:
Output:
Algorithm:
01: [shortest paths from o]
02: [shortest paths from d]
03: [time budget]
04: [lower forward reachability set]
05: [lower backward reachability set]
06: [upper forward reachability set]
07: [upper backward reachability set]
08: for each parent in
09: [lower parent forward cone]
10: [extrusions]
11: [lower parent forward body]
12: [lower forward body]
13: next
14: for each parent in
15: [lower parent backward cone]
16: [extrusions]
17: [lower parent backward body]
18: [lower backward body]
19: next
20: for each parent in
21: [upper parent forward cone]
22: [extrusions]
23: [upper parent forward body]
24: [upper forward body]
25: next
26: for each parent in
27: [upper parent backward cone]
28: [extrusions]
29: [upper parent backward body]
30: [upper backward body]
31: next
32: [lower obstacle-constrained prism]
33: [upper obstacle-constrained prism]
34: return
Algorithm 8.1 – Main algorithm for computation of rough obstacle-constrained space-time prisms.
Modelling potential movement using rough obstacle-constrained space-time prisms 163
The algorithm first computes the shortest paths from the origin ( and from the
destination , relying on an existing algorithm as discussed in section 8.5. Next, these
shortest path sets are used to compute the reachable sets of obstacle vertices according to
the time budget and maximum velocity for the lower and upper
approximations. Then, the reachability bodies
corresponding to the four reachable sets are calculated. According to Definitions 8.21-8.24,
this is achieved as the union of the respective parent bodies over all parents in the reachable
set. As follows from Definitions 8.17-8.20, these parent bodies are obtained as cones
subtracted with extrusions of obstructed areas. Finally, forward and backward bodies are
intersected to achieve the overall lower and upper space-time prisms .
From an application point of view, the algorithm will have to be reasonably efficient when
dealing with massive datasets consisting of numerous origins, destinations, time budgets,
and obstacles. The efficiency of Algorithm 8.1 highly depends on the following subroutines:
The calculation of shortest paths avoiding polygonal obstacles (Algorithm 8.1, lines 1-2).
According to Inkulu et al. (2009), the known lower bound on time complexity for finding
such a path is , with the number of obstacles, and the number of
vertices of all obstacles together. Given this complete dependency on the amount of
vertices and obstacles, we note that for large datasets these amounts may be reduced in
preprocessing, by means of shape approximation algorithms.
The subtraction of a body from another body (Algorithm 8.1, lines 11, 17, 23, 29).
The union of two bodies (Algorithm 8.1, lines 12, 18, 24, 30).
The intersection of two bodies (Algorithm 8.1, lines 32, 33).
For the latter three subroutines, the computational efficiency will further depend on
whether or not the resulting bodies are to be represented visually, such as with the 3D solids
returned in our CAD implementation.
8.6.3 Example
To illustrate our methodology, we will reconsider the university campus example with a
student having two minutes to travel from to (Figure 8.3). Suppose that (s)he was
tracked at and with a spatial accuracy of 10m and a temporal accuracy of 5s.
According to Bohannon (1997), reliable estimates for an adult’s maximum gait speed range
from 1.749m.s-1 to 2.533m.s-1 when considering differences in sex and age class. Let us take
this as lower and upper approximation maximum velocity respectively. The lower and
upper prisms corresponding to these constraints are presented in Figure 8.9 and Figure 8.10.
A cross section through both prisms along the origin-destination axis is shown in Figure 8.11.
Note that, according to the definitions and properties of section 8.4, the temporal extremes
of the prism lie strictly within the time budget for the lower approximation, whereas they
164 Chapter 8
exceed the time budget in the upper approximation. Also, the upper prism is flattened out at
its origin and destination, due to the spatial uncertainty.
It appears that there is a large difference between the lower and upper prisms in this case:
whereas the student might have easily passed along all sides of all buildings in the upper
prism, (s)he is restricted to an almost linear course passing north of the buildings in the
lower approximation scenario. Hence, it would have been a harmful limitation not to
consider the given spatial and temporal accuracy for this case. However, beyond this
example, this reasoning may apply for many real-world applications, as similar or even lower
accuracies may be obtained from existing tracking technologies. Further, we note that only a
limited part of the lower prism intersects with the beeline connector from to (Figure
8.11), which contrasts sharply with the case of an unconstrained environment emphasizing
the impact of accounting for intermediate obstacles.
The resulting lower and upper prisms can be considered a basis for further analysis. The
volume of a space-time prism, for instance, may be used as a measure of general
accessibility (Lenntorp 1978, Villoria 1989, Burns 1979). Let us apply this measure in order to
illustrate the impact of our approach. Table 8.1 presents the resulting volumes for all four
scenarios that arise from taking into account or otherwise neglect the uncertainty and/or
the obstacles. We obtain significantly smaller volumes when accounting for the uncertainty
and for the obstacles. Ignoring uncertainty, we find a restriction to 68% when taking account
of the obstacles. Analogously, when considering uncertainty, we achieve restrictions to 13%
and 82% for lower and upper approximations respectively. Hence, with respect to the prism
volume, we may conclude that, for this case, considerable overestimates are to be made
whenever we neglect either the uncertain constraints, or the obstacles.
Without uncertainty* With uncertainty
Unconstrained 1 908 020 (lower) 185 383
(upper) 5 423 407 Unconstrained with removal of obstacle extrusions
1 675 617 (lower) 144 206
(upper) 5 032 327
Obstacle-constrained 1 297 306 (lower) 24 125
(upper) 4 432 806
* taking
Table 8.1 – Space-time prism volumes in m².s according to four different scenarios.
To isolate the effect of travel restrictions induced by the obstacles in their surrounding
environment from the obstacles themselves being inaccessible, Table 8.1 additionally
specifies the volumes of the unconstrained prisms after removal of the obstacle extrusions.
For the case without uncertainty, we observe that 62% of the total volume reduction is due
to this effect, whereas only 38% is caused by the inaccessible obstacles themselves. For the
lower and upper prisms, we find respective shares of 74% and 60%. These findings
Modelling potential movement using rough obstacle-constrained space-time prisms 165
Figure 8.9 – Obstacle-constrained lower space-time prism (grey) with indication of obstacles
(black).
Figure 8.10 – Obstacle-constrained upper space-time prism (grey) with indication of obstacles
(black).
166 Chapter 8
demonstrate that merely removing the obstacles from unconstrained prisms causes
considerable overestimation of an individual’s travel possibilities in obstacle-constrained
environments.
Figure 8.11 – Cross section through time of lower (dark grey) and upper (light grey) prisms along
the axis origin (o) – destination (d), with indication of vertical obstacle extrusions (white
rectangles).
8.7 Conclusions
Taking the viewpoint of nowadays tracking technologies, our contribution to time geography
is twofold. First, it was shown how classical time-geographical concepts can be redefined in
order to model the uncertainty associated with their underlying constraints (section 8.3).
Typically with tracking data, uncertainties will arise from inaccuracies, errors and noise
associated with the technology at hand. Relying on the basic principles of rough set theory,
we have formally elaborated how space-time prisms under uncertainty can be described as
rough sets with lower and upper approximations. Not only are these approximations
conceptually appealing, they are also robust as they allow an easy integration of different
sorts of uncertainty. In addition, rough approximations are efficient when it comes to
computation and interpretation, as they abstract from a mass of numerical details that may
otherwise increase the computational load and blur the complex results in alternative
approaches.
Secondly, we have proposed an alternative to the assumption of unconstrained travel
environment by assuming an isotropic space studded with impassable obstacles (section
8.4). A comprehensible methodology for the construction of space-time prisms according to
this alternative assumption was elaborated. We may find many kinds of environments, both
indoors and outdoors, that might be acceptably abstracted to isotropic spaces with
impassable obstacles. Pedestrian precincts in urban environments, among others, are usually
open, freely accessible and populated with discrete obstacles such as buildings, monuments,
fenced or hedged areas, etc. Our approach complements earlier studies that have modelled
space-time prisms within transportation networks. It also adds to the recent work by Miller
Modelling potential movement using rough obstacle-constrained space-time prisms 167
and Bridwell (2009) who propose a field-based representation implemented as a lattice
approximation. Although their approach allows for a complete relaxation of the uniform
velocity assumption, it will be a less efficient solution, in terms of both storage and
computation, in case of isotropic environments with obstacles. Our approach avoids the
elongation and deviation errors related to a lattice approximation, and offers a valuable
alternative if the necessary data is lacking to build a reliable and fully covering velocity field.
Both contributions, when integrated (section 8.5), offer a framework for time geography to
represent and analyse uncertain spatiotemporal data in an environment constrained by
obstacles. The yet obtained rough and obstacle-constrained space-time prisms allow for the
assessment of the impact of different spatial and temporal uncertainty factors as well as
various configurations of obstructs on accessibility. Rough obstacle-constrained prisms, and
by extension the chains (or necklaces) of chronologically successive prisms, are powerful
tools for accessibility analysis. The approach presented will be particularly effective for
micro-scale applications because the smaller the travel environment and time budgets, the
more impact spatiotemporal uncertainty will have and the less acceptable will be the
ignorance of obstacles. While it may be acceptable to neglect uncertainty and abstract entire
cities or urban districts as network-constrained spaces at a macro or meso scale (e.g. Kwan &
Lee 2004, Kwan 1999, Weber 2003, Weber & Kwan 2002), this reasoning may not apply
when focusing on city centres and urban neighbourhoods at a micro scale. Therefore, we
believe that our approach may provide increased insights into various micro-scale
applications, including monitoring tourists or mass event visitors, crowd management, crime
scene analysis, disaster management and evacuation planning.
Several extensions and refinements of our model should be addressed in future work. From
a computational perspective, as reported in section 8.6.2, challenges lie in a more detailed
elaboration, and eventually optimization of the complete approach in terms of
computational complexity. Further, since the concept of a space-time prism has now gained
an acceptable degree of realism in order to analyse common tracking data in obstacle-
constrained environments, we are planning to validate our methodology by means of
extensive data sets. Particular emphasis will be placed on how to employ the proposed
concepts to infer additional knowledge about trajectories and to measure the accessibility in
space and time (Dijst, de Jong & van Eck 2002, Shoval & Isaacson 2007, Schwanen & de Jong
2008, Neutens et al. 2008, Berger et al. 2009). Furthermore, we could consider alternatives
to modelling uncertainty. Detailed and abundant numerical uncertainty data, if available,
may validate the calculation of presence probabilities or membership functions. These
functions, however, may significantly complicate the proposed methodology, especially
when it comes to the combination of different sorts of uncertainty. Concerning the
environmental constraints, an appealing extension could be to consider time-varying
constraints. Instead of permanent obstacles, this would allow for handling temporary objects
such as those associated to temporary events (e.g. stages, tents, and stands during a
168 Chapter 8
festival). Another challenge is the relaxation of the assumption of an isotropic travel
environment in between obstacles, and the associated maximum speed. For example, it may
well be that another maximum travel speed applies in the direct neighbourhood of an
obstacle. Also, we might consider obstacles with passable interiors for which then different
constraints apply. Lawn and bushes patches in a park, for example, could, instead of
isotropic space, be considered permeable obstacles with a deviant maximum velocity with
respect to pedestrian visitors.
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Reconciling place-based and person-based accessibility: a GIS toolkit 171
9 Reconciling place-based and person-based accessibility: a
GIS toolkit
Delafontaine M., Neutens T., Van de Weghe N.
in International Journal of Geographical Information Science, submitted for
publication
Abstract. This chapter introduces a novel GIS toolkit for measuring and mapping the
accessibility of individuals to services. The toolkit contributes to earlier
implementations by combining the benefits of both place-based and person-based
accessibility measures. To this end, place-based accessibility measures are derived
from a person-based framework by considering space-time prisms which are centred
at service facilities rather than individual anchor points. The implementation is also
innovative by explicitly accounting for the opening hours of service delivery in its
accessibility measurement. In addition, the toolkit is aimed to be user-friendly and to
generate insightful and comprehensible results for non-technically-oriented users,
which is illustrated in a brief case study.
Keywords. Accessibility – Place-based measures – Person-based measures – GIS
9.1 Introduction
Accessibility is a fundamental concept in transport geography and urban planning. It refers
to individuals’ ability to travel and participate in activities given the available transport and
land use system (Pirie 1979). To assess accessibility, researchers have relied on various
accessibility measures. Roughly, these can be divided into place-based and person-based
measures (Miller 2007). Place-based measures associate a level of accessibility to a location
or spatial unit of analysis (e.g. census tract, ward, traffic analysis zone, etc.). They express
accessibility as the proximity to desired activity locations from key locations in an individual's
daily life, such as the residence or workplace. The family of place-based measures includes
such well-known and commonly applied accessibility measures as the travel time or distance
to the nearest opportunity and the number of opportunities within a particular area or
within a specific cut-off distance from a given location. While rigid, easily implementable and
insightful, place-based measures have often been criticised as they tend to reduce individual
travel behaviour to a (set of) key location(s), while ignoring important temporal constraints
on activity behaviour such as the opening hours of urban opportunities and the limited
availability of discretionary time on the part of an individual (Neutens et al. 2010a). This
172 Chapter 9
criticism has called a foundation for the development of person-based accessibility
measures.
Person-based measures are specified at the lowest level amenable to the social sciences, i.e.
the individual. Drawing on concepts of time geography (Hägerstrand 1970), these measures
express accessibility on the basis of detailed observations of the spatiotemporal constraints
individuals are faced with. Hence, person-based measures allow accessibility to fluctuate
during the course of the day as well as across persons, as has been shown by, among others,
Kwan (1998, 1999), Kim and Kwan (2003), Miller (2007), Casas (2007), Schwanen en de Jong
(2008), Yu and Shaw (2008), Neutens et al. (2010a, 2010b), and Páez et al. (2010). Whilst
more subtle and detailed, they are also more complex to calculate than place-based
measures – although this difficulty has been increasingly overcome by the growing
capabilities of geographical information systems (GIS) in recent years. In addition, the
assessment of person-based accessibility requires dedicated and representative information
about the activities (e.g. travel diaries) of sampled individuals, which is in many cases
unavailable or at least difficult to collect.
From a cartographical point of view, person-based measures have an important
disadvantage over place-based measures, since they cannot simply be summarised into a
single map. This is because a person-based accessibility value should be seen as an attribute
of a travelling individual who may visit multiple activity locations over the course of a day,
and therefore cannot be simply mapped onto a single location such as an individual’s
residence. Maps can be drawn representing the accessibility of locations according to a
person-based measure (e.g. potential path areas). Such maps, however, only express the
accessibility of one or ultimately a few individuals, instead of providing an area-
representative map. Place-based measures, on the other hand, can easily be represented
synoptically in conventional maps as they directly capture the physical proximity of urban
opportunities at a particular location (e.g. see some recently published maps El-Geneidy &
Levinson 2007, Nettleton et al. 2007, Drew & Rowe 2010, Colclough & Owens 2010,
Achuthan, Titheridge & Mackett 2010, Lei & Church 2010).
This chapter attempts to make the link between place-based and person-based tools by
introducing PrismMapper, a GIS toolkit for measuring and mapping accessibility to service
facilities. To this end, we will consider place-based measures that implement person-based
concepts building on time geography. In addition, the toolkit is intended to be simple, robust
and comprehensible such that it is directly appealing to all kinds of end-users. The toolkit is
available from (http://users.ugent.be/~mdlafont/PrismMapper). The remainder of the
chapter is structured as follows. The next section discusses related tools to measure
accessibility and provides a motivation for the current contribution. Section 9.3 presents the
toolkit and the accessibility measures it implements. An example case is illustrated in section
9.4, followed by conclusions in section 9.5.
Reconciling place-based and person-based accessibility: a GIS toolkit 173
9.2 Related tools
Many existing information systems and services implement accessibility measures in various
ways. GPS devices and routing systems, for instance, are able to calculate travel times and
distances, i.e. simple place-based accessibility measures. These implementations, however,
are usually a black box to the user. Web services, such as the OpenRouteService
(http://openrouteservice.org) (Neis, Dietze & Zipf 2007), the National Accessibility Map of
the Netherlands (http://www.bereikbaarheidskaart.nl), and the recently released
Mapnificent (http://www.mapnificent.net), offer explicit tools to map accessibility
measures. While useful, such services are often limited on two aspects: (i) meta-information
on the data sources and exact methods they use to assess the implemented accessibility
measures, and by consequence (ii) the freedom they offer the user to manipulate or
configure these data and methods.
On the other hand, dedicated GIS packages or extensions, most of which implement place-
based measures, can cope with the above shortcomings. Travel times and distances, are
supported in many GIS packages either by the calculation of beeline distances in
unconstrained space or shortest paths within a geographic network. A noteworthy toolkit
going beyond these simple measures is Flowmap (http://flowmap.geog.uu.nl), developed at
the University of Utrecht and released in 1990. Flowmap has been specifically designed to
handle spatial flow patterns, but also supports computing travel costs along a network, and
modelling the market areas of existing or planned facilities. In 1998, the Environment
Systems Research Institute (ESRI®) introduced the Network Analyst extension of its
ArcView™ software (nowaydays ArcGIS™). The Network Analyst allows calculating and
mapping shortest paths, nearest facilities, and service areas over a given network (N. N.
1998). O’Sullivan et al. (2000) have described a desktop GIS application to map isochrones
for accessing facilities trough public transport. In 2004, Liu and Zhu (2004) have presented
their own ArcGIS accessibility extension. Although their implementation includes tools that
are currently also covered by the Network Analyst extensions, such as origin-destination
travel cost matrices, it additionally supports more complicated place-based measures such
as gravity- and utility-based measures and catchment profiles. Despite the authors’
argumentation that the extension is available to a wide range of users, any further reference
on how to obtain and use their toolkit is regretfully missing. The same is true for the recent
Urban.Access tool (Benenson, Martens & Rosenthal 2010) which allows for the mapping of
place-based measures based on detailed car and bus travel times.
Regarding person-based accessibility, especially in the field of exploratory spatiotemporal
data analysis and visualisation, several implementations exist which may assist the
assessment of person-based accessibility measures without explicitly operationalizing these
(e.g. Andrienko & Andrienko 1998, Yu 2006, Andrienko et al. 2009, Andrienko et al. 2010,
Kraak 2003). In addition, there have been early operationalizations of fundamental time-
geographical concepts such as (Kitamura, Kostyniuk & Uyeno 1981, Lenntorp 1976, Villoria
174 Chapter 9
1989, Nishii & Kondo 1992, Kondo & Kitamura 1987, Landau, Prashker & Alpern 1982). These
have been characterised by an unrealistic modelling of the travel environment as they ignore
the transportation network (Kwan & Hong 1998). This shortcoming has been addressed later
in both theoretical (Neutens et al. 2007, Kuijpers et al. 2010, Miller & Bridwell 2009) and
empirical work (Kim & Kwan 2003, Kwan & Weber 2003, Kwan & Hong 1998, Kwan 1998,
Neutens et al. 2010a, Neutens et al. 2010b).
In 2000, Miller and Wu (2000) introduced the first true person-based toolbox which allows
mapping three different benefit measures for an individual to participate at discretionary
activities in space and time. This prototype is a user-friendly front-end / back-end application
for measuring an individual’s accessibility. Recently, Neutens, Versichele and Schwanen
(2010) presented a stand-alone person-based accessibility toolkit for assessing the
opportunities for joint activity participation. Their toolkit provides a dynamic and animated
view of the activity locations that are accessible to a person or group during the course of
the day. Both toolkits are characterised by a sincere demand for detailed input data about
the individual activity schedules. Not only is such information merely occasionally available
for a sample of individuals, it is also questionable whether this sample data is representative
in all its dimensions for the associated population. This delicate issue has never been
profoundly addressed in studies on person-based accessibility, which questions the extent to
which the results of the related tools may be extrapolated. More than that, given that the
necessary information would be available, these toolkits are unable to generate maps of the
accessibility of an entire population - let alone area covering maps - thereby passing over the
synoptic power of maps. This could be considered a significant inadequacy in the eyes of
decision makers or urban planners dissuading them from using these tools. In addition, both
toolkits implement comparable benefit measures which are obtained from complex utility
functions. The complexity of these functions obscures the interpretation for end-users who
do not have prior knowledge about time geography and accessibility modelling. Finally, a last
and perhaps most poignant point of critique is that the toolkits nullify the added value of
person-based measures since they only account for the spatiotemporal constraints on the
part of the individual while neglecting the time constraints on the part of the urban facilities
(e.g. opening hours, waiting times).
The PrismMapper tool introduced in this chapter aims to contribute to the set of existing
implementations in at least three ways. First, it tends to support a comprehensible set of
simple and rigid place-based accessibility measures. That is to say, the measures should be
meaningful, interpretable and self-evident, even for end-users who are not acquainted with
the accessibility literature. Second, while taking advantage of the mapping opportunities of
place-base measures, it implements key properties of person-based measures by considering
reverse space-time prisms (see section 9.3). This enables making true accessibility maps
representative for an entire region or population instead of a single (or a few) predefined
Reconciling place-based and person-based accessibility: a GIS toolkit 175
individual(s). Third, PrismMapper accounts for the space-time constraints of service delivery
by explicitly considering the opening hours of service facilities.
9.3 PrismMapper
This section will first describe the person-based accessibility measures implemented in
PrismMapper and then give an overview of the system.
9.3.1 Accessibility measures
Most person-based measures rely on the well-known time geographical framework
originated in the 70’s by Hägerstrand (1970). The basic unit of analysis in time geography is
the space-time path, i.e. an individual’s daily trajectory in space and time. Space-time paths
comply with three types of constraints: (i) an individual’s physiological capabilities (capacity
constraints), (ii) an individual’s commitments that bind him/her to specific locations and
time budgets (coupling constraints), and (iii) rules stemming from norms and laws (authority
constraints). These constraints delineate a set of space-time points accessible, i.e. physically
reachable, by the individual. The subset of this set which corresponds to an individual’s
space-time budget that is available between an origin and a destination is referred to as a
space-time prism (STP). The origin and destination are denoted as the anchor points of the
STP. STPs are typically represented in a 3D space-time cube where a vertical time axis is
integrated with a flattened topography (Figure 9.1, Figure 9.2a). The STP of an individual
with a time budget from to between an origin and a destination can be formally
described as:
– (9.1)
with the travel time from to , the travel time from to . In the case of
an isotropic travel environment with a constant finite maximum velocity – as has been
understood in Figure 9.1 and Figure 9.2 – a STP is obtained from the intersection of two
cones (Miller 2005). While the STP is a powerful concept to model a global level of an
individual’s physical accessibility, many studies have used it to assess, specifically, individual
accessibility to services by simply considering a service accessible on the basis of the
presence of its location within the space-time prism. Hence, they overlook the time
constraints of services, which are however only delivered and thus accessible within well-
defined opening hours. Therefore, in addition to STPs, PrismMapper will account for the
opening hour regimes of facilities in order to decide on their accessibility (see equation (9.2)
further on in this section).
The cartographical equivalent of a space-time prism, i.e. its spatial footprint, is called a
potential path area (PPA) (Figure 9.1). Since one individual may have more than two anchor
points in the course of a day and thus multiple STPs, the mapping of PPAs across multiple
individuals soon becomes cluttered. To overcome this cartographical problem, PrismMapper
176 Chapter 9
considers ‘reverse’ STPs which are centred at service facilities instead of anchor points, and
which we will refer to as reverse space-time prisms (RSTPs). The RSTP for a facility and its
opening hour time slot from to with respect to a time budget from to is given by:
(9.2)
Figure 9.1 – Space-time prism and related concepts.
The RSTP comprises all space-time points from which an individual may travel to a facility in
order to visit it at some time point during the given opening hour such that the individual
may return back to his/her origin within the given time budget. The difference between a
STP and a RSTP is illustrated in two cross sections through space-time shown in Figure 9.2.
Instead of looking at the accessible locations in between two anchor points, the PPAs of
RSTPs capture all anchor points that may be interpreted as valid pairs of a coinciding origin
and destination, such that an individual can travel from the origin to visit the facility and
return to the destination within the time budget at hand. The interpretation of such back-
and-forth trips is straightforward, since they are common in daily life, especially with respect
to residential locations, i.e. home-facility-home trips.
RSTPs have an essential property:
(9.3)
Thus, for each space-time point of a RSTP, all coinciding earlier space-time points within the
time budget belong to the RSTP as well. In other words, from each anchor location within a
RSTP, an individual may always leave earlier within the time budget. Hence, the space-time
points will determine the PPA of the RSTP, with the earliest possible departure
time.
Reconciling place-based and person-based accessibility: a GIS toolkit 177
Figure 9.2 – Cross section through space-time of a STP (left) and a RSTP (right).
RSTPs differ significantly from traditional STPs in being independent from individual anchor
points. This is advantageous in several respects. First, it takes away the requirement of high-
level individual activity/travel data. Second, this is also desirable from a computational point
of view, since RSTPs have only to be calculated once in total, instead of once for each
individual. Finally, yet most importantly, given a set of facilities with their opening hours and
a presumed time budget, RSTPs are representative for all anchor locations. Thus, area-
covering maps may be produced to represent the PPAs of RSTPs, which will be
PrismMapper’s core functionality. In addition to calculating and mapping the PPAs of RSTPs,
the toolkit implements three optional user-defined cut-off criteria to further refine the set of
valid anchor locations: (i) a maximum travel time , (ii) a minimum activity duration , and
(iii) a minimum number of accessible facilities . Hence, whether a facility with opening
hours is accessible to an individual at location with a time budget from to within a
total travel time of at most for a duration of at least can be expressed by a function
:
(9.4)
with
(9.5)
Given a set of facilities and the cut-off criteria , and , PrismMapper implements six
accessibility measures (equations (9.6-9.11)) with respect to an individual at a location
with a time budget from to . Building on equations (9.4) and (9.5), these measures are
defined as follows:
(9.6)
(9.7)
178 Chapter 9
(9.8)
(9.9)
(9.10)
(9.11)
In other words, for an individual with a time budget from at location :
returns a boolean value which expresses whether (true) or not (false) there
exists a facility in s(he) can visit respecting and ;
returns an integer value which represents the number of facilities s(he) can visit
respecting and ;
returns a ratio value which indicates the minimum total travel time that is
required for visiting a facility , respecting and , and returning to ;
returns the facility which corresponds to the minimum travel time specified by
;
returns a ratio value which indicates the maximum feasible duration for visiting a
facility in respecting and ;
returns the facility which corresponds to the maximum feasible duration
specified by .
The parameter expressing the minimum number of accessible facilities has been absent in
the accessibility measures’ formulas, but it can be easily implemented by considering in
equations (9.6-9.11) only these for which is true for at least
facilities.
Unlike the implicit assumption of an isotropic travel environment and a constant maximum
travelling velocity underlying Figure 9.1 and Figure 9.2, PrismMapper implements all above
accessibility measures within a much more realistic network-based travel environment with
maximum travelling velocity varying all across the network (see Computational module,
section 9.3.2).
Reconciling place-based and person-based accessibility: a GIS toolkit 179
9.3.2 System
An overview of the system architecture of PrismMapper is presented in Figure 9.3. It has
three main components: (i) a GIS component, (ii) a computational module (CM) and (iii) a
graphical user interface (GUI).
Figure 9.3 – PrismMapper system architecture.
PrismMapper is embedded in ESRI’s ArcGIS Desktop GIS software as an ArcMap project
template. The choice for ArcGIS Desktop is due to several reasons. To begin with, ESRI has
been considered the global market leader in GIS software, ever since it was established in
1969 (Scott & Janikas 2010, Qingquan et al. 2010). Moreover, ESRI’s GIS products have been
especially developed to serve spatial planners, decision makers, and (local) government
officials (e.g. see Greene 2000, O'Looney 2000, Huxhold, Fowler & Parr 2004, Thomas &
Humenik-Sappington 2009, Scott & Janikas 2010), which is the target group of PrismMapper.
ArcGIS Desktop is ESRI’s most important desktop GIS product, offering a comprehensive set
of tools to manipulate, analyse, visualise and store geospatial data in general and network-
based data, including some place-based accessibility measures (see section 9.2), in
particular. Embedding PrismMapper as a project template thus enables the integration of its
functionality with the yet extended set of ArcGIS tools. This allows PrismMapper to be
incorporated with one’s yet existing ArcGIS projects on the one hand, and to further
manipulate, analyse, store or export PrismMapper results in a rich and well-known GIS
environment on the other hand. The form of a project template ensures an easy distribution
of the toolkit, as well as no installation requirements.
For more information about the GIS component of PrismMapper, the reader is referred to
the official ArcGIS documentation (http://www.esri.com/products/index.html#desktop_gis).
GIS
• database
• visualisation
• analysis
CM
• ACCESS
• CUMF
• MINT
• MINTF
• MAXD
• MAXDF
GUI
• load network
• load facility locations
• add/edit opening hours
• set individual parameters
• set accessibility measure
• solve
ArcGIS Desktop
PrismMapper ArcMap project template
180 Chapter 9
The remainder of this section will respectively discuss the computational module and
graphical user interface components, respectively.
Computational module
The core of the PrismMapper toolkit is the computational module (CM). It consists of several
code modules written in Visual Basic using the ArcObjects object model (Burke 2003). CM is
responsible for the calculation and mapping of the accessibility measures presented in
section 9.3.1. To this end, CM relies on the following input data:
Network dataset ;
Travel time attribute ;
Set of facilities ;
Time budget ;
Maximum travel time ;
Minimum activity duration ;
Minimum number of accessible facilities ;
Accessibility measure .
The network dataset represents a transportation network which delineates the
considered travel environment and thus the area of potential anchor locations. Networks are
supported by ArcGIS through the Network Dataset data type. Network Datasets may be
created from all kinds of data sources that participate in a transportation network such as
road segments, junctions and turns. The data type implements an advanced connectivity
model to handle complex issues such as multimodality. Also, it may carry a number of
attributes in order to model travelling impedances, restrictions, and hierarchy within the
network. These attributes are accounted for within PrismMapper, which requires at least
one travel time attribute to enable the calculation of travel times.
A facility dataset consists at least of the location and the opening hours for a set of service
facilities. Facility locations are obtained from a point data source and they should be covered
by the study area delimited by the . The opening hours are represented by their weekly
schedules given that this is the most general manner to express the opening hours of service
facilities.
The individual parameters time budget , maximum travel time , minimum activity
duration , minimum number of accessible facilities , and the accessibility measure
are all obtained from manual user input. The computation of proceeds as follows. First,
the set is filtered to which includes only those facilities of which the opening hours have
a temporal overlap with the time budget . Second, for each facility in , the module
calculates all shortest paths within the threshold time according to the attribute .
This is done once for all paths towards and once for all paths from in order to obtain all
necessary travel costs and required in equations (9.2) and (9.5). For each
Reconciling place-based and person-based accessibility: a GIS toolkit 181
location which is on at least of such shortest path pairs, the module proceeds with
assessing using one of the equations (9.6-9.11). The final accessibility results are
spatially summarised to the level of network segments and stored as a polyline shapefile.
After the computation of the accessibility results, these are mapped within the ArcGIS map
environment. Map type and symbolisation are chosen according to the data type of (see
section 9.3.1):
The network locations for which is true are represented through a single-value
map by means of a simple solid bright green line symbol;
, , and are mapped onto a chloropleth map with in between five
and seven equal interval classes. These classes are symbolised through solid line
symbols with colours ranging from bright green for the network locations in the most
accessible class to bright red for the locations in the least accessible class;
and are expressed by a chorochromatic map using a random color
ramp to associate each facility with a unique color.
Graphical user interface
According to the underlying objective, the PrismMapper’s GUI is kept particularly simple. The
associated workflow for using the toolkit is depicted in Figure 9.4. After opening the
PrismMapper project template, the user can launch the application by clicking a command
button which opens up the toolkit’s main window (Figure 9.5). From this window, the user
proceeds with three tasks, in arbitrary order, to load the necessary data sources and set the
required parameters for the accessibility computations (see Computational module).
First, a network dataset should be loaded. This is done by clicking the associated button
(Figure 9.5) and selecting an appropriate data source within a file system browser window.
After loading a network dataset, the user may choose the desired travel time attribute from
a drop-down list (the application automatically restricts this list to impedance attributes
expressed in temporal units). In addition, some network analysis settings can be configured.
These include a threshold distance for matching facility locations to the network, and
restrictions to account for when calculating shortest paths (e.g. one-way traversable
network segments). Second, the user can specify a facility dataset, which can be done by
either loading a point dataset from a physical source as for the network dataset, or by
manually picking facility locations on screen. In the latter case, the main window disappears
and the user is subsequently prompted to click a facility location within the ArcGIS map view
and specify a name for this facility. Once facility locations are loaded or picked, the user can
add or modify their weekly opening hours within a separate window. Opening hours can be
manually added or deleted per facility, copied from one facility to others, or they can be
loaded from a text file. Finally, the user should set the remaining parameters, i.e. the
individual time budget, maximum travel time, minimal activity duration, and minimum
number of facilities.
182 Chapter 9
Figure 9.4 – PrismMapper workflow.
Having configured all required input settings, the user can pick one of the six accessibility
measures and start the solving process by clicking the ‘Solve’ button (Figure 9.5). A small
status window informs the user about the major steps within the calculation process (see
Computational module). After a successful calculation, the accessibility results are stored in a
shapefile and a map of the requested accessibility measure is drawn in the ArcGIS map view.
Load
network
dataset
Solve
Set travel
time attribute
Set time budget
Set maximum travel
time
Set minimum activity
duration
Accessibility map
Results
shapefile
Open project template
Launch
Set accessibility
measure
Load facility
locations
Pick facility
locations
Set minimum number
of accessible facilities
Add opening hour
Delete opening
hour
Copy opening
hours
Load opening
hours
Edit network
analysis
settings
Reconciling place-based and person-based accessibility: a GIS toolkit 183
Figure 9.5 – PrismMapper main application window.
9.4 Example case
In this section, we will elaborate a brief case study to illustrate the application of the
PrismMapper toolkit. In this study, we will examine the daily variability in individual
accessibility to the public libraries in Ghent (Belgium). A transportation network for Ghent
has been compiled from TeleAtlas MultiNet® road network data. Car travel times can be
estimated from this network using the shortest travel time attribute. The locations,
presented in Figure 9.6, and weekly opening hours of Ghent’s municipal libraries have been
obtained from the official city website (http://www.gent.be). Ghent has one central main
library and fifteen smaller branch libraries dispersed across the city.
In order to configure the individual settings, we will consider persons who would like to
make an evening library visit of at least half an hour in between 6:20 PM and 8:00 PM, and
who do not want to travel by car for more than 15 minutes in total. We will compare the
accessibility results for these settings on Monday to those obtained on Tuesday. The latter
configuration is also shown in Figure 9.5. The resulting accessibility maps are shown in
Figures 9.7-9.16. The maps for MAXDF have been left out in this case, since these maps are
of the same type as the MINTF maps.
184 Chapter 9
Figure 9.6 – Public libraries in Ghent (Belgium).
The ACCESS maps, which show the anchor locations from which individuals may access one
or more facilities respecting the specified constraints, indicate considerable difference
between the situation on Monday and Tuesday. On Monday, individuals may pay an evening
visit to a library from practically everywhere in Ghent, whereas on Tuesday this is not
feasible in the more peripheral parts of the city, especially in the northern area. Note that, in
spite of their proximity to accessible libraries, some network locations offer no library access
in both maps because they are located within a tangle of one-way streets, or they are
prohibited for cars (e.g. the pedestrian precinct in downtown Ghent).
The CUMF maps show that on a Monday evening, more than nine libraries can be visited in
downtown Ghent – even over a dozen within some areas. In peripheral areas this reduces to
less than three. On Tuesday evening, merely two libraries can be visited from the city centre
and at most three libraries within two zones south and west of the city centre.
The MINT maps clearly illustrate the decay of travel times near accessible facilities. It follows
that, on Monday the minimum total travel times to the nearest accessible facility are fairly
limited (below 9 minutes on most locations), given the number of accessible facilities and
their spatial dispersion across the city. On Tuesday, many of the locations outside the city
centre feature travel durations of at least 12 minutes. Note that MINT maps reflect physical
Reconciling place-based and person-based accessibility: a GIS toolkit 185
Figure 9.7 – Map of ACCESS on Monday.
Figure 9.8 – Map of ACCESS on Tuesday.
186 Chapter 9
Figure 9.9 – Map of CUMF on Monday.
Figure 9.10 – Map of CUMF on Tuesday.
Reconciling place-based and person-based accessibility: a GIS toolkit 187
Figure 9.11 – Map of MINT on Monday.
Figure 9.12 – Map of MINT on Tuesday.
188 Chapter 9
Figure 9.13 – Map of MINTF on Monday.
Figure 9.14 – Map of MINTF on Tuesday.
Reconciling place-based and person-based accessibility: a GIS toolkit 189
Figure 9.15 – Map of MAXD on Monday.
Figure 9.16 – Map of MAXD on Tuesday.
190 Chapter 9
proximity in a nuanced manner as they neatly articulate the discordance between network-
based proximity, captured by back-and-forth shortest paths, and beeline proximity read
from the map. Especially the directional nature of network segments may cause higher
travel times than expected. Many locations in the vicinity of the main library, for instance,
have rather high travel times due to the predominance of one-way streets in that area.
The MINTF maps complement the MINT maps in indicating by colour which facility is
accessible in the least travel time from which anchor location. Each thus obtained connected
zone can be considered a facility’s catchment area in terms of minimum travel time. On
Monday, catchment areas are unequally distributed both in terms of size and road density.
In this respect, it is remarkable that – notwithstanding its high road density – by far the
smallest catchment area accrues to the main library. Since fewer libraries are accessible on
Tuesday evening, their catchment areas are significantly larger than on Monday and of
comparable size.
Finally, the MAXD maps depict the maximum feasible activity duration at each possible
anchor location. These durations depend on the time budget, the travel time and the facility
opening hours. Since all libraries that are accessible on Monday and Tuesday have the same
evening closing hour, the maps in this case capture the effects of minimum travel time on
potential activity duration and therefore mirror the patterns observed in Figure 9.11 and
Figure 9.12 in this case.
9.5 Conclusion
This chapter has introduced a novel toolkit, named PrismMapper, for measuring and
mapping the accessibility of individuals to services. The toolkit aims to combine the benefits
of both place-based and person-based accessibility measures basing on the time
geographical concept of a space-time prism. Through the consideration of reverse space-
time prisms, potential path areas can be derived that are representative for individual
anchor locations in general rather than for one individual in particular. Thus, a foundation
for deriving place-based measures has been obtained on the basis of person-based
constraints including individual time budget, maximum travel time, and minimum activity
duration. The toolkit is also innovative in explicitly accounting for the opening hours of
service delivery in its assessment of accessibility.
Beyond the original integration of place-based and person-based aspects and the
incorporation of opening hours, PrismMapper has been developed with the eye on two main
objectives which have hitherto often been ignored in existing related tools. First, the toolkit
should be simple and user-friendly. This has been achieved through an undemanding user
interface and an easy workflow which consists, in essence, of three steps: launching the
application, configuring the input data and parameters, and processing the results. In
addition, the embedding of the toolkit within an ArcGIS project template allows an easy
Reconciling place-based and person-based accessibility: a GIS toolkit 191
distribution and takes away any installation requirements. A second objective has been that
the results generated by the toolkit should be useful and in as much as possible
comprehensible for end-users who are not acquainted with the accessibility literature. We
believe to have demonstrated this through the case study presented in section 9.4, although
a genuine user study may support a further evaluation of this objective.
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The relationship between opening hours and accessibility of public service delivery 195
10 The relationship between opening hours and accessibility
of public service delivery
Neutens T., Delafontaine M., Schwanen T., Van de Weghe N.
in Journal of Transport Geography, forthcoming
Copyright © Elsevier Science
Abstract. In the past two decades urban time policies have been proposed and
implemented in many European cities as a complement to traditional spatial planning
methods. Such policies seek to provide an answer to the growing number of people
facing time problems as a result of an erosion of collective time rhythms and a
desynchronisation of different time structures of urban life. Particular emphasis is
being placed on the reconciliation of opening hours of public service facilities with
the travel and activity patterns of citizens in order to increase individual accessibility
to urban services. In spite of the increasing relevance of time policies, only limited
quantitative research has been conducted about the relationships between opening
hours and accessibility. This chapter seeks to extend this line of inquiry by exploring if
and to what extent the accessibility of public facilities can be ameliorated by
redesigning the timetables of service delivery. A method is proposed to optimise the
temporal regime of public service delivery in terms of accessibility. The method is
illustrated in a case study of accessibility of government offices within the city of
Ghent (Belgium). Our findings suggest that by rescheduling the opening hours of
public service facilities individual accessibility to service delivery can be improved
significantly. Our study may support urban service deliverers, policymakers and urban
planners in assessing timetables for a better ‘accessible’ service provision.
Keywords. Accessibility – Opening hours – Public services – Time geography
10.1 Introduction
In recent years there has been increasing awareness about the impact of urban time policies
on people’s quality of life. Especially in Europe, several projects have been launched, seeking
to improve the temporal organization of public service provision (Mückenberger & Boulin
2002, Boulin 2006). These temporal policies concentrate on the ways in which the opening
hours of urban service delivery can be better attuned to the activities and travel patterns of
citizens. Due to the erosion of collectively maintained time rhythms and the fragmentation
of activities in space and time, people’s time use patterns are becoming increasingly
individualized and intensified (Breedveld 1998, Couclelis 2004, Glorieux, Mestdag and
196 Chapter 10
Minnen 2008). Public service administrations try to respond to these trends by rescheduling
the opening hours of public service facilities in order to increase the accessibility of services
to particular constituencies and to improve the quality of urban life. As such, temporal
planning is increasingly becoming a critical aspect of city government (Boulin 2005).
Micro-economists have extensively studied the strategic aspects of opening hour decisions,
but have primarily focused on the provision of private services with price competition. A
number of authors have concentrated on the implications of changes to temporal
regulations and the liberalization of service hours (e.g. Clemenz 1994, Thum and
Weighenrieder 1997, Rouwendal & Rietveld 1999, Jacobsen & Kooreman 2005). Others have
sought to derive the socially optimal service hours by specifying utility-theoretic models that
maximise both consumer surplus and industry profits (Shy & Stenbacka 2006, 2008). While
insightful, these studies fail to address the heterogeneity of consumers’ space-time activity
patterns and travel behaviour. This is to be considered a critical inadequacy in the context of
public service provision because, in the absence of price competition, consumer surplus
primarily relates to consumers’ accessibility benefits at public service locations and thus
indirectly to their space-time behaviour (Miller 1999). Transport geographers have long since
stressed the importance of individual-specific space-time constraints on activity participation
when evaluating individual accessibility to urban services. In particular, a large number of
researchers have shown that individual accessibility is shaped by inter alia an individual’s
mandatory activity schedule, trip chaining behaviour and transport mode availability (see
e.g. Weber & Kwan 2002; Kim & Kwan 2003; Kwan & Weber 2008; Schwanen & De Jong,
2008, Neutens et al. 2010a, 2010b, Neutens, Versichele & Schwanen 2010).
The present chapter examines how such individualized aspects of accessibility can be
considered in the determination of optimal opening hours of public services in terms of
accessibility. Rather than to look for the most cost-efficient regime of opening hours, we
want to examine the ways in which opening hours can be amended to improve individual
accessibility. More specifically, a novel, sample-based geocomputational procedure is
developed that determines the collectively optimal regime of opening hours of a network of
service facilities by maximising the overall accessibility of citizens. The proposed procedure
can aid urban service deliverers, policymakers and urban planners in defining optimally
accessible timetables of service provision. It is applied in a case study of government offices
in the city of Ghent, Belgium. These government offices take care of the administration
concerning marriage, cohabitation, birth, death, residential moves, elections, and so on. The
case study is particularly timely because local authorities are currently seeking to reschedule
and curtail the historically emerged opening hours of the government offices and to tailor
these to the daily activity patterns of the citizens.
The chapter is organized as follows. The next section reviews prior research on the
relationships between space-time demands, opening hours and accessibility, and identifies
The relationship between opening hours and accessibility of public service delivery 197
relevant research gaps. Section 10.3 presents a measure of the space-time accessibility of
public services based on the concept of locational benefits, and discusses a method to
optimise service opening hours in terms of accessibility. The methodology is illustrated in a
case study. Data and data preparation are described in section 10.4.1 and 10.4.2. Results are
reported in section 10.4.3. Finally, we conclude with the major findings and outline some
avenues for future work.
10.2 Space-time demands, opening hours and accessibility
In the past two decades, lack of time has become an important social issue that is felt in
virtually all strata of society (Glorieux, Mestdag & Minnen 2008). More and more people
seem to have become caught up in a ‘temporal treadmill’ (Law & Wolch 1993, Jarvis 2005,
Szollos 2009), experiencing competing claims on their time-space resources by different
responsibilities. Negative effects of continued time shortage on well-being can be profound
and can include work-life imbalance, lower family satisfaction and such health issues as
stress, over-fatigue and burn-out (Pelfrene et al. 2001, Ritsema van Eck, Burghouwt & Dijst
2005). People’s experience of time shortage seems to be exacerbated by malfunctioning
urban infrastructures, exemplified by road congestion and delays in public transport
systems. Further, transport and communication technologies, which are often believed to
accelerate activity patterns and make them more efficient (e.g. by reducing travel time),
seem to have complex and contradictory effects in practice. While technologies such as the
Internet and mobile phone imply that people can be reached more easily anywhere, anytime
and that home-work boundaries become more blurred for many (Schwanen & Kwan 2008),
transport infrastructures intended to speed up daily travel are often used to travel longer
distances rather than shorter times (Harris et al. 2004, Metz 2008). As a result, individual
activity patterns are frequently stretched out across multiple geographical scales, exceeding
the administrative boundaries of cities and regions.
Activity patterns have also become more fragmented over time. Recent years in particular
have witnessed a tendency towards a desynchronisation of social times, and more diverse
and complex activity schedules due to the increase of temporal constraints imposed by daily
obligations (e.g. paid labor, childcare, etc.) and limited mobility resources. Given the large
and growing number of women entering the European labour market and the concomitant
decay of the traditional male breadwinner model, scheduling incompatibilities are emerging
most strongly within dual-earner families – families with young children in particular – who
are juggling employment, housework, care-giving and leisure activities (e.g. Kwan 1999,
Jarvis 2005, McDowell et al. 2006, Schwanen 2007).
The above and related developments imply that the demand for urban services fluctuates
and shows irregular patterns over time and that individual accessibility can no longer be
measured straightforwardly in terms of physical proximity to the residence or workplace
(Weber & Kwan 2003). Rather, accessibility has become a matter of connectivity, which
198 Chapter 10
implies that access to places and services not only depends on spatial proximity but also on
the tense interface between individual daily time schedules and the temporal rhythms of the
city.
The increasing importance of connectivity in relation to time problems is currently
challenging the efficiency of traditional planning methods such as zonal land-use plans which
are largely focused on improving accessibility on the basis of stationary populations within
administrative boundaries (Zandvliet et al. 2008). Recently, however, a number of scholars
have expressed their concern about the a-temporal nature of spatial planning policies and
have called for more attention to the distributional effects of temporal practices (see e.g.
Moccia 2000, Hajer & Zonneveld 2000, Nuvolati 2003, Healey 2004, Deffner 2005, Zandvliet
& Dijst 2005). Their concern develops in tandem with a growing number of initiatives in
European cities for harmonizing the time structures of urban environments with the needs
and the desires of the inhabitants (for overviews see e.g. Mückenberger & Boulin 2002,
Boulin 2006).
While interest in temporal planning is starting to grow, only few studies have been carried
out about the ways in which opening hours can be amended to enhance individual
accessibility to services and to foster the quality of life in cities. Research that has made ex-
ante and ex-post evaluations of temporal regimes of opening hours by means of accessibility
measures is virtually non-existent. This may in part be attributed to the paucity of
accessibility measures that can adequately capture the temporal dimension of individuals’
mobility patterns. The majority of accessibility measures proposed to date does not explicitly
consider the potential temporal mismatch between individuals’ mandatory activity schedule
and the opening hours of services.
An exception to the neglect of temporal connections in accessibility research lies in the
strand of literature that has evolved around time geography. Time geography (Hägerstrand
1970) is a conceptual framework for analyzing spatiotemporal activity patterns and
individual accessibility on the basis of a set of space-time constraints. The nature of these
constraints is threefold: (i) capability constraints are linked with physiological capabilities
such as the need or wish to sleep and eat, (ii) coupling constraints refer to the need to join
other people or material artefacts in space-time, and (iii) authority constraints are imposed
by laws, norms and regulations such as the opening hours of public services and the
timetables of public transport. A key concept within time geography is the space-time prism
which delineates all possible space-time points that an individual can reach within a given
time budget (i.e. the time available for travel and discretionary activity participation
between two mandatory activities). The spatial footprint of the space-time prism is called
the potential path area.
Relying on these time-geographical concepts, various so-called space-time accessibility
measures (STAMs) have been proposed that incorporate the performance of the transport
The relationship between opening hours and accessibility of public service delivery 199
network (Miller 1991, Kwan 1998, Neutens et al. 2008b, Miller & Bridwell 2009, Kuijpers et
al. 2010). Spurred on by the developments in geographical information systems (GIS) and
the availability of disaggregate travel data, the use of network-based STAMs has developed
rapidly in the past decade. Within the STAM tradition, at least three studies are important
for evaluation of accessibility along the temporal dimension. Weber and Kwan (2002) have
calculated various STAMs for 200 individuals in Portland (OR, USA) such as the number of
accessible opportunities and the total length of accessible road segments and shown that
ignoring the effects of traffic congestion and opening hours of opportunities may produce
spatially uneven reductions in individual accessibility. Their work has been continued in the
ethnographic space-time accessibility analysis by Schwanen & De Jong (2008) who have
demonstrated that extending the opening hours of childcare centres can help to improve the
work-life balance of dual-earner families. Finally, Neutens et al. (2010a) have shown that
individuals with certain personal and household attributes are affected differently by
changes to the temporal regime of public service facilities.
While previous research has clearly foregrounded the ramifications of opening hours for
individual accessibility, no attempt has been made thus far to explore the ways in which
opening hours can be amended to achieve a higher accessibility of urban services. In what
follows, we will extend accessibility research in this direction.
10.3 Method
10.3.1 Measuring accessibility
The point of departure of our method is an accessibility measure that takes into account the
spatial and temporal dimensions of people’s daily activity paths. The measure presented
here is based on Burns’ (1979) utility-theoretic framework that assesses accessibility in terms
of the benefits accruing to individuals at particular activity locations – henceforth termed
locational benefits. Burns’ framework has been extended to transport networks and
reconciled with consumer surplus approaches by Miller (1999). Ever since, the approach has
received increased attention in the transport modelling field, which is exemplified by the
various extensions to the framework that have been proposed in recent years, including
Ashiru, Polak and Noland (2003), Hsu and Hsieh (2004), Ettema and Timmermans (2007),
Neutens et al. (2008a), and Neutens, Schwanen & Miller (2010).
A central assumption of the Burns/Miller framework is that, when seeking to perform a
discretionary activity, individuals are both spatially and temporally constrained by a set of
fixed activities that bind them to particular places at specific times of the day (Cullen &
Godson 1975, Schwanen, Kwan & Ren 2008). Fixed activities are mandatory commitments
that are difficult to reschedule in the short run and include such activities as paid labour and
fetching children.
200 Chapter 10
For an individual , let denote the chronologically ordered set of fixed
activities, where each activity has a location and a time span from to .
Between each pair of subsequent fixed activities and , there is an amount of space
and time available for discretionary activities, denoted as . Each is constrained by the
compulsory trip from at to at . In line with time geography, we will
refer to this space-time volume as a space-time prism (Miller 2005) (Figure 10.1). Let
denote the chronologically ordered set of opening hour intervals
of a service facility . Then, the potential activity window (PAW) for individual
to participate in a discretionary activity at a facility between two fixed activities and
and during the opening interval is given by (Figure 10.1):
– (10.1)
with the travel time from to ;
the travel time from to .
Figure 10.1 – Cross section through space (horizontal axis) and time (vertical axis) of the space-time
prism (grey) between fixed activities xj and xj+1 of an individual i, with the indication of the PAW
with respect to the opening hour interval hk of service facility f.
A is located between two fixed activities of and within the opening hours of
. Each PAW can be assigned a utility value expressing the benefit an individual enjoys from
participating in an activity at a service facility over the time span of the PAW. For a given
, this utility value, henceforth termed locational benefit, can be
specified as:
The relationship between opening hours and accessibility of public service delivery 201
(10.2)
with attractiveness of service facility ;
benefit resulting from attractiveness ;
benefit resulting from the activity duration ;
travel cost to facility ;
disutility resulting from travel cost .
A locational benefit measures the benefit that an individual derives from participating in an
activity at a certain facility as a function of the facility’s attractiveness, the duration of the
activity and the physical separation to/from this facility. To determine the different
components in equation (10.2), we follow earlier specifications by Burns (1979). For the
attractiveness and activity duration components, we use a simple linear function to express
that benefits increase proportionately to the attractiveness of and the activity duration at a
service facility. The advantage over other functional forms (e.g. a positive power function) is
that the linear function does not require complex parameter estimation procedures and
dedicated data collection methods. For generality, a minimum required activity duration
threshold will be left unspecified; this and other refinements (such as delay times) should be
accommodated in future work. The multiplicative functional form of equation (10.2) ensures
that an individual will not derive any utility if a service facility is not attractive or if an
individual cannot spend time at the service facility. For the disutility component associated
with the travel costs, we adopt a negative exponential function with parameter . This
function implies that the willingness to travel to services decays most rapidly at low travel
costs. Since the negative exponential form declines more gradually relative to power
functions, it is better suited to express travel impedance for shorter trips such as those to
the government offices considered in our case study (Ianoco et al., 2010). Incorporating the
above assumptions in equation (10.2) yields:
(10.3)
The travel cost in equation (10.3) can be calculated as the detour travel costs
for to travel to in between the first and the second fixed activity instead of travelling
directly in between both fixed activities:
(10.4)
The locational benefit for an individual over an arbitrary time window (ATW) can
then be expressed as:
(10.5)
Based on equation (10.5), we can specify the locational benefit of a network of service
facilities to an individual over a given time interval. When considering public facilities that
202 Chapter 10
offer highly comparable services – as is the case for the government offices (see section
10.4.1) – an individual may not benefit from having a larger set of facilities to choose from.
In other words, it is assumed that an individual is a rational decision maker who patronizes
the service facility that yields the largest locational benefit. Therefore, when calculating an
individual’s accessibility to a network of facilities, we will assume that an individual
maximises the locational benefits over the available facilities during the considered ATW.
More formally, the accessibility of a network of service facilities to individual over an ATW
is specified as:
(10.6)
For clarification, a simple example of how a locational benefit is calculated over a time
interval is given in Table 10.1.
Example
Consider a person , for whom we would like to assess his/her locational benefit
over the time interval from 8.00 AM to 9.00 AM with respect to the
facility . Suppose that is opened over a time interval from 8.00 AM to 12.00 AM,
and that has two fixed activities and with ending at 8.10 AM and
starting at 10.10 AM. Also, it is given that it takes 25 minutes to travel from to ,
15 minutes to travel from to , and 20 minutes to travel from to . Then, from
equation (10.1) it follows that , of which 35
minutes are within . We then calculate the locational benefit of to
over using equations (10.3-10.5) as .
Table 10.1 – Locational benefit calculation example
10.3.2 Optimising opening hours in terms of accessibility
Having introduced a measure for evaluating the accessibility of a network of service facilities
to a population of individuals over a time interval, we now propose a method for identifying
the opening hours that would generate the highest total accessibility for a given population.
It should be noted that we will only seek to optimise along the temporal dimension of
service delivery; spatial relocations of service facilities will not be considered in this chapter
(i.e. facility locations will be assumed fixed during the optimisation procedure).
In our approach, the study period at hand (e.g. one week) is subdivided into a discrete
sequence of non-overlapping time intervals (e.g. hours). These minimum time intervals
(MTIs) are the basic temporal units of analysis. We will refer to an MTI during which a service
facility is open as a minimum opening interval (MOI) and denote it as a pair (facility, MTI).
The complete schedule of opening hours of a set of service facilities can be represented as a
set of MOIs, henceforth termed a regime. Starting from an empty regime (zero MOIs),
then, of all possible MOIs not in , the MOI returning the highest additional benefit for the
The relationship between opening hours and accessibility of public service delivery 203
entire population with respect to the benefit of , can be iteratively assessed using equation
(10.6) and added to . This best-first selection procedure is presented in Algorithm 10.1.
Input:
set of individuals
set of service facilities , with denoting the set of MOIs allocated to facilities in
set of all possible MOIs of facilities in covering the study period
number of MOIs
Output:
-MOI regime (ordered set of MOIs) with maximal total locational benefit
total benefit associated with
Algorithm:
01: set to ,
02: for to
03: set to
04: set to ,
05: for each in subtract from
06:
07: add to
08: for each in
09:
10: end for
11: if then
12:
13: set to
14: end if
15: subtract from
16: end for
17: add to
18:
19: end for
20: return
Algorithm 10.1 – Computational procedure to determine the optimal n-MOI regime.
The algorithm takes as input a population of individuals with their fixed activities, a set
of service facilities , a set of all possible MOIs of facilities in over the entire study
period, and the number of requested MOIs in the resulting regime. Obviously, is limited
to the number of MOIs in . The output is the -MOI regime (i.e. regime consisting of
MOIs) that yields the maximal total locational benefit, which is returned as well. The
algorithm consists of two major nested iterations. The inner iteration (lines 5-16) runs
through all remaining MOIs in the study period that are not yet included in the optimal
204 Chapter 10
regime so far. Each of these MOIs is alternately added to the set of facility opening hours in
the regime in order to assess the total locational benefit of its addition by cumulating all
individual benefits using equation (10.6) at line 9. From this inner iteration, the algorithm
holds back the MOI whose addition returns the highest total benefit and adds it to the
regime in the outer iteration (lines 2-19). The latter is done until the regime contains the
requested number of MOIs.
At this stage we are able to derive the optimal -MOI regime in terms of total accessibility.
However, since no conditions have been specified concerning the internal consistency of a
regime, it may well be that a regime consists of combinations of non-contiguous MOIs
scattered across the study period, which may be impracticable and undesirable to
implement by local authorities. In an attempt to derive the -MOI regime that accounts for
continuity of service delivery, we propose a second algorithm using a penalty and a reward
parameter, denoted and respectively. The idea is that the locational benefits for an
added MOI have to be valued higher (multiplied by ) when an MOI connects with one of the
previously selected MOIs of the same facility, whereas they have to be devaluated
(multiplied by ) if the MOI is not temporally adjacent with a yet included MOI. This
extended approach has been pseudo-coded in Algorithm 10.2.
Although it would be straightforward to choose symmetric (i.e. inverse) values for and ,
i.e. , we have intentionally introduced them as two different parameters, because
they have different effects on the allocation of MOIs across facilities. On the one hand,
rewarding contiguous opening hours ( ) will favour a regime consisting of
contiguous opening hours for a limited set facilities. Penalising ( ), on the other
hand, will favour a regime with contiguous opening hours for multiple facilities.
Both parameters can be adjusted by policymakers at will in order to derive meaningful
regimes. It should be noted, however, that temporal contiguity may come at the expense of
accessibility: the more contiguity is aimed for (i.e. the more and deviate from 1), the less
optimal in terms of the number of people who can access the evaluated facility or facilities a
resulting regime may be.
The relationship between opening hours and accessibility of public service delivery 205
Input:
see Algorithm 10.1
penalty factor
reward factor
Output:
connected -MOI regime (ordered set of MOIs) with (sub)optimal total locational
benefit
total benefit associated with
Algorithm:
01: set to
02: for 1 to n
03: set to
04: set to
05: for each in subtract from
06:
07: add to
08: if adjacent then
09: q = r
10: else
11:
12: end if
13: for each in
14:
15:
16: end for
17: if then
18:
19:
20: set to
21: end if
22: subtract from
23: end for
24: add to
25:
26: end for
27: return
*The Boolean function returns true if the MOI is temporally adjacent
with an existing MOI of facility in regime ; otherwise false is returned
Algorithm 10.2 – Computational procedure to determine the (sub)optimal connected n-MOI
regime.
206 Chapter 10
10.4 Case study
In order to illustrate the applicability of the method described in section 10.3, we will now
elaborate a case study. In this case study we will try to find the optimal regime of opening
hours for the government offices in the city of Ghent (Belgium). The input data, data
preparation and results will be discussed below.
10.4.1 Data
The study area is the city of Ghent, which is the third largest city in Belgium and capital of
the province of East-Flanders. Ghent has a population of approximately 240 000 inhabitants
and an area of almost 160 km² (Figure 10.2). The northern part of the study area is sparsely
populated and known for its flourishing industrial and harbour activities.
For this case study, we rely on the following data sources:
Individuals
The first data source is an activity/travel data set consisting of a two-day consecutive diary of
out-of-home activities of persons aged five or more living in the Ghent region. The data set
was collected in 2000 within the framework of the SAMBA project (Spatial Analysis and
Modelling Based on Activities) (see Tindemans et al. 2005). Reported activity locations were
geocoded at the street level. Individuals sampled at the same day of the week are grouped
and their fixed activities are considered representative for the type of activities that they
usually undertake on this day of the week. Since no fixity levels are available for the reported
activities, fixed activities were determined on the basis of the activity purpose. The
categories “work”, “school”, “pick up/drop off” and categories closely related to these were
considered fixed, given that it is generally difficult to conduct them at other places and
times. In total 3 047 person-days were selected, ranging from Monday to Saturday. Sunday
openings will not be considered in this case study as they relate to different societal
constraints and are not considered by the local authorities. Given that households were
randomly sampled within the SAMBA project, we will assume that the spatial distribution of
the home locations of the selected individuals mirrors the general distribution of the actual
population (Figure 2).
Service facilities
The second source of data comprises information about the government offices in Ghent.
The addresses, opening hours and services offered are obtained for each government office
from the official city website (http://www.gent.be). Two types of government offices are
distinguished: head and branch offices (Figure 10.3). The centrally located head office forms
the core of the municipal service delivery network. In addition to the conventional
administrative services delivered at all branch offices, the head office offers few additional
though rather exceptional formalities. Furthermore, this office is generally able to process
administrative documents (e.g. identity cards) quicker than the branch offices.
The relationship between opening hours and accessibility of public service delivery 207
Figure 10.2 – Study area and sampled households.
Figure 10.3 – Spatial distribution of government offices.
208 Chapter 10
The current regime of opening hours is given in Table 10.2 (opening hours are grey-
coloured). The opening hours of government offices 4-15 exhibit a lot of overlap, while the
opening hours of offices 1-3 in the northern part of the city are very limited.
Monday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 8 AM- 9 AM
9 AM-10 AM 10 AM-11 AM 11 AM-12 AM 12 AM- 1 PM 1 PM- 2 PM 2 PM- 3 PM 3 PM- 4 PM 4 PM- 5 PM 5 PM- 6 PM
Tuesday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 8 AM- 9 AM
9 AM-10 AM 10 AM-11 AM 11 AM-12 AM 12 AM- 1 PM 1 PM- 2 PM 2 PM- 3 PM 3 PM- 4 PM 4 PM- 5 PM 5 PM- 6 PM
Wednesday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 8 AM- 9 AM
9 AM-10 AM 10 AM-11 AM 11 AM-12 AM 12 AM- 1 PM 1 PM- 2 PM 2 PM- 3 PM 3 PM- 4 PM 4 PM- 5 PM 5 PM- 6 PM Table 10.2 – Current regime of opening hours for the government offices in Ghent (1-15).
The relationship between opening hours and accessibility of public service delivery 209
Thursday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 8 AM- 9 AM
9 AM-10 AM 10 AM-11 AM 11 AM-12 AM 12 AM- 1 PM 1 PM- 2 PM 2 PM- 3 PM 3 PM- 4 PM 4 PM- 5 PM 5 PM- 6 PM
Friday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 8 AM- 9 AM
9 AM-10 AM 10 AM-11 AM 11 AM-12 AM 12 AM- 1 PM 1 PM- 2 PM 2 PM- 3 PM 3 PM- 4 PM 4 PM- 5 PM 5 PM- 6 PM
Saturday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 8 AM- 9 AM
9 AM-10 AM 10 AM-11 AM 11 AM-12 AM 12 AM- 1 PM 1 PM- 2 PM 2 PM- 3 PM 3 PM- 4 PM 4 PM- 5 PM 5 PM- 6 PM Table 10.2 (continued)
Transport system
The third data source is TeleAtlas® MultiNetTM (version 2007.10) road network data for
Belgium. Based on this data set, travel times were estimated using ESRI®’s Network Analyst
(ArcGIS 9.3.1). Two predominant transport modes in Ghent will be considered in this case
study: car and bicycle. Local public transportation is not addressed in this study because it
would significantly increase the computational intensity and requires among others
information about the location of stops and the time tables for each of the different public
transportation alternatives (i.e. trains, trams and buses) in Ghent, which were not available
to us.
210 Chapter 10
To compute travel times by car, we have manipulated our data set in order to account for
congestion. Therefore, we relied on a recent report prepared for the Federal Government
Service for Mobility and Transport (Maerivoet and Yperman, 2008), where average travel
times are reported for Ghent and its conurbation for three different road classes at four
different times of the day for both weekdays and weekends. A factor for each of these
categories has been determined (Table 10.3). As expected, the highest congestion (i.e.
highest factor) is found during weekday mornings and weekday evening peaks, while the
lowest congestion (i.e. lowest factor) occurs during weekend middays and nights. These
Road type Morning
6 AM – 9 AM Midday
9 AM – 4 PM Evening
4 – 7 PM Night
7 PM – 6 AM
Weekday Highways and ring roads 1.062 1.057 1.065 1.029
Regional and main connection roads 1.202 1.117 1.249 1.117
Other paved roads 1.118 1.094 1.196 1.094
Weekend Highways and ring roads 1.013 1.000 1.026 1.007
Regional and main connection roads 1.000 1.025 1.037 1.025
Other paved roads 1.060 1.000 1.036 1.000
Table 10.3 – Congestion factor according to day type, day time and road class.
congestion factors allow us to estimate time-varying travel times by car as the weighted
product of the uncongested travel time (based on TeleAtlas® MultiNetTM) with the
corresponding factors in Table 2. If the uncongested travel time covers different congestion
periods, factors are weighted accordingly.
Specific information about specialized bicycle facilities (e.g. dedicated bicycle paths) was not
readily available for the city of Gent. Hence, in order to compute travel times by bicycle, we
had to adopt a compromise solution following Ianoco, Krizek & El-Geneidy (2010). This
compromise solution consisted of excluding highways and other exclusive motorways from
the transport network and allowing travel directions for non-motorized travelers – one-way
streets for motorized vehicles passable in both directions for bicyclists are common in
Ghent. Travel times by bicycle were estimated as the product of the shortest path distance
and a mean travel speed of 15 km/h (El-Geneidy, Krizek & Iacono 2007). Note that these
estimates are rather coarse; they could have been refined given that recent empirical studies
about pedestrian and bicycle travel have shown travel times and speeds to vary with micro-
level characteristics of the built environment (Krizek & Roland 2005; Krizek, Handy & Forsyth
2009) and according to age and gender (Wendel-Vos et al. 2004; Gomez et al. 2005).
However, we believe that the current estimations are accurate enough for testing our
method and leave such refinements for future work.
The relationship between opening hours and accessibility of public service delivery 211
10.4.2 Data preparation
Prior to the optimisation, the input data needs to be adapted. The following issues have
been dealt with. First, all necessary detour travel costs have been calculated as described in
section 10.4.1. To account for mobility resources, we have assumed that car-owners with a
driving license are able to travel by car, whereas others are assumed to travel by bicycle.
Second, the attractiveness value af (see equation (10.3)) was determined for each
government office. On the basis of the number of extra services provided at the head office
and in consultation with the local authorities, we have specified the attractiveness difference
between the head office and the branch offices at the proportion of 1 for the central office
to 0.8 for the other offices. Third, the decay parameter α of the negative exponential
deterrence function (see equation (10.3)) was estimated for car and bicycle separately, using
the observed cumulative distribution of service trips according to travel time (Figure 10.4).
Similar decay parameters were found across both travel modes: = 0.081 ( = 0.97) and
= 0.092 ( = 0.98).
Figure 10.4 – Estimation of distance decay parameters.
Finally, the Algorithms 10.1 and 10.2 presented in section 10.3.2 have been implemented in
a Visual Basic module.
0
10
20
30
40
50
60
70
80
90
100
0 10 20 30 40 50 60
Trip
like
liho
od
(%
)
Travel time (min)
Observed (car)
Observed (bicycle)
Exponential decay (car)
Exponential decay (bicycle)
y = e-0.081x
y = e-0.092x
212 Chapter 10
10.4.3 Results
Optimal temporal regimes by number of opening hours
We start our analysis by examining if and to what extent the accessibility of Ghent’s
government offices can be improved by rescheduling the current opening hours using
Algorithm 10.1. We will consider MTIs of one hour, which reflect the minimal time span for a
government office to be open as can be found in the current regime (office 3 on Wednesday,
Table 10.2). Given that people generally have less time constraints resulting from fixed
activities on Sundays and in the evening (Neutens et al. 2010a), it is rather self-evident that
citizens’ accessibility will be improved significantly by shifting the current opening hours
towards these time periods. Therefore, we will restrict our analysis to the accessibility gains
that can be made by applying the optimisation algorithm within the current range of
opening hours (i.e. 8 AM to 6 PM and Monday to Saturday). Within this range, the fifteen
government offices can maximally cover 900 possible opening hours (MOIs) – each office can
be open for ten hours between 8 AM to 6 PM for six days a week (Monday to Saturday).
Currently, the government offices cover 405 of these 900 possible opening hours.
Using Algorithm 10.1, we have assessed the 900 optimal regimes ranging from one to all 900
opening hours in the study period. Figure 10.5 shows that the accessibility increases with the
number of opening hours at a decreasing rate. The accessibility values on the vertical axis of
this diagram have been calculated as a trade-off between attractiveness, possible activity
duration and travel costs (equation (10.7)) and express how well the complete set of
individuals is able to access the network of government offices during a given regime of
opening hours. Figure 10.5 offers a yardstick regarding the number of opening hours to be
included in a temporal regime. One can see that accessibility increases quite rapidly for the
first, say, 150 opening hours. Hence, a curtailment of these hours would considerably harm
the overall accessibility of government offices. Beyond this value the marginal utility of
adding extra opening hours declines until 820 opening hours. From that point on, expanding
the opening hours does not increase the total accessibility anymore because none of the
added opening hours is able to attract (i.e. offer higher benefits to) individuals from
government offices with concurrent opening hours that were already included in the optimal
regime. In other words, for the remaining 80 opening hours – covered only by the peripheral
government offices no. 1 and 2 – people are better off, if they go to surrounding offices.
Evaluating the current temporal regime
We have also calculated the total accessibility of the current regime of 405 opening hours
and have positioned this regime into the diagram depicted in Figure 10.5. Vertical
movements in the diagram represent gains or losses in accessibility caused by rescheduling
the current number of opening hours; horizontal movements represent curtailing or
expanding the opening hours. Clearly, the current regime is suboptimal in terms of
accessibility since the same level of accessibility can be achieved with merely 98 opening
The relationship between opening hours and accessibility of public service delivery 213
hours if the optimal regime is adopted. In other words, Figure 10.5 indicates that significant
improvements in the total accessibility can be made by simply reconfiguring the existing
opening hours without expanding them.
Figure 10.5 – Total accessibility for all 900 optimal regimes with indication of the current regime.
Improving accessibility by rescheduling opening hours
To improve the accessibility of the government offices, a suitable strategy would be to
reschedule the current 405 opening hours within the current range of opening hours. The
regime that yields the maximum total accessibility with 405 opening hours has been
calculated using Algorithm 10.1 and is depicted in Table 10.4. At least two characteristics of
this optimal regime can be identified. First, a relatively large share of government offices
have been allocated opening hours between 4 PM and 6 PM on weekdays, reflecting that
many individuals in the sample have time available for accessing a government office upon
completing (mandatory) paid work activities. Second, opening hours tend to be allocated to
government offices that are located centrally within the city – offices 5, 8 and 15 in
particular. This can be explained by the high concentration of residences and employment
(or other fixed activity) locations within this area from which people tend to access the
government offices. The optimal regime of 405 opening hours also implies that the small
demand for branch offices 1 and 2 can easily be taken over by the other offices. Also, it
appears that in the optimal regime the head office (no. 15) is continuously open on each day
of the study period. This could have been expected since this office was assigned a larger
attractiveness and is located centrally.
214 Chapter 10
Monday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 8 AM- 9 AM
9 AM-10 AM 10 AM-11 AM 11 AM-12 AM 12 AM- 1 PM 1 PM- 2 PM 2 PM- 3 PM 3 PM- 4 PM 4 PM- 5 PM 5 PM- 6 PM Tuesday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
8 AM- 9 AM 9 AM-10 AM 10 AM-11 AM 11 AM-12 AM 12 AM- 1 PM 1 PM- 2 PM 2 PM- 3 PM 3 PM- 4 PM 4 PM- 5 PM 5 PM- 6 PM Wednesday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
8 AM- 9 AM 9 AM-10 AM 10 AM-11 AM 11 AM-12 AM 12 AM- 1 PM 1 PM- 2 PM 2 PM- 3 PM 3 PM- 4 PM 4 PM- 5 PM 5 PM- 6 PM Thursday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
8 AM- 9 AM 9 AM-10 AM 10 AM-11 AM 11 AM-12 AM 12 AM- 1 PM 1 PM- 2 PM 2 PM- 3 PM 3 PM- 4 PM 4 PM- 5 PM 5 PM- 6 PM Table 10.4 – Optimal 405-hour regime.
The relationship between opening hours and accessibility of public service delivery 215
Friday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 8 AM- 9 AM
9 AM-10 AM 10 AM-11 AM 11 AM-12 AM 12 AM- 1 PM 1 PM- 2 PM 2 PM- 3 PM 3 PM- 4 PM 4 PM- 5 PM 5 PM- 6 PM Saturday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
8 AM- 9 AM 9 AM-10 AM 10 AM-11 AM 11 AM-12 AM 12 AM- 1 PM 1 PM- 2 PM 2 PM- 3 PM 3 PM- 4 PM 4 PM- 5 PM 5 PM- 6 PM Table 10.4 – (continued)
While, compared to the current regime, the total accessibility can be increased by 70%
without expanding the number of opening hours, the optimal 405-hour regime is rather
impracticable as it contains 13 discontinuities (gaps within an office’s day schedule) with
nine isolated hours (offices opened for only one hour). To overcome this issue, we have
computed the (sub)optimal 405-hour regime using Algorithm 10.2 with symmetric reward
and penalty parameters (i.e. ). In order to limit the impact of connectedness on
the total accessibility, we have gradually increased the impact of both factors simultaneously
(increased and decreased by increments of 0.1 to the same extent), starting from = =
1. We found that for and regimes without any discontinuities are
obtained. The results of the adjusted regime are depicted in Table 10.5. Since 96% of the
opening hours of the optimal regime are preserved in the adjusted contiguous regime, the
total accessibility has diminished by less than 1% compared to the optimal regime with 405
hours. In other words, by adjusting the reward and penalty parameters, we are able to
develop a regime consisting of contiguous blocks of opening hours that offers high levels of
accessibility among the population. This regime may be used by local authorities as a basis
for amending the opening hours of their network of service facilities.
216 Chapter 10
Monday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 8 AM- 9 AM
9 AM-10 AM 10 AM-11 AM 11 AM-12 AM 12 AM- 1 PM 1 PM- 2 PM 2 PM- 3 PM 3 PM- 4 PM 4 PM- 5 PM 5 PM- 6 PM Tuesday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
8 AM- 9 AM 9 AM-10 AM 10 AM-11 AM 11 AM-12 AM 12 AM- 1 PM 1 PM- 2 PM 2 PM- 3 PM 3 PM- 4 PM 4 PM- 5 PM 5 PM- 6 PM Wednesday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
8 AM- 9 AM 9 AM-10 AM 10 AM-11 AM 11 AM-12 AM 12 AM- 1 PM 1 PM- 2 PM 2 PM- 3 PM 3 PM- 4 PM 4 PM- 5 PM 5 PM- 6 PM Thursday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
8 AM- 9 AM 9 AM-10 AM 10 AM-11 AM 11 AM-12 AM 12 AM- 1 PM 1 PM- 2 PM 2 PM- 3 PM 3 PM- 4 PM 4 PM- 5 PM 5 PM- 6 PM Table 10.5 – Contiguous sub-optimal 405-hour regime.
The relationship between opening hours and accessibility of public service delivery 217
Friday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 8 AM- 9 AM
9 AM-10 AM 10 AM-11 AM 11 AM-12 AM 12 AM- 1 PM 1 PM- 2 PM 2 PM- 3 PM 3 PM- 4 PM 4 PM- 5 PM 5 PM- 6 PM Saturday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
8 AM- 9 AM 9 AM-10 AM 10 AM-11 AM 11 AM-12 AM 12 AM- 1 PM 1 PM- 2 PM 2 PM- 3 PM 3 PM- 4 PM 4 PM- 5 PM 5 PM- 6 PM Table 10.5 – (continued)
10.5 Conclusion and avenues for future work
The purpose of this chapter has been to study the relationship between opening hours and
accessibility in the context of public service delivery. More specifically, a method has been
presented and implemented that allows optimising the opening hours of public service
delivery in terms of the accessibility experienced by a city’s population with heterogeneous
activity and travel patterns. Accessibility has been specified by means of locational benefits
which express the desirability for an individual to participate in an activity at a certain service
facility on the basis of the facility’s attractiveness, the potential activity duration and the
travel costs involved. The proposed method has been illustrated for a case study of public
service delivery in the city of Ghent, Belgium. Our initial findings have shown that substantial
improvements in total accessibility can be made by rescheduling instead of expanding the
existing opening hours of service facilities. We believe that the current study is relevant in
light of the growing attention to time problems and the increasing relevance of urban time
policies. Optimal temporal regimes in terms of accessibility offer policymakers a useful
benchmark to identify the margins within which access to services can be improved by
temporal changes to service delivery.
Although our optimisation method has a sound and generic theoretical basis, a number of
refinements could and should be made in future work. The first and perhaps most important
issue from a policy point of view concerns the absence of equity considerations in our
218 Chapter 10
algorithm. While we were able to identify the regime of opening hours that maximises the
accessibility over the entire (sample of the) population, we did not account for the
inequalities in the distribution of individual accessibility that may ensue from this regime.
One way to promote a more equitable distribution of individual accessibility would be to
weight the individual benefits in the optimisation procedure, such that larger weights are
assigned to the benefits of individuals with less discretionary time available. In this way
policymakers could give priority to the preferences of those persons who are most
vulnerable to a modification of opening hours. For example, local authorities may want to
‘humanize’ the timetables of public service delivery by making these more compatible with
the activity schedules of those constituencies who generally face considerable space-time
demands in their daily lives, such as dual earner households or young women with children.
Policymakers may also want to alter the temporal regime of public service delivery to attract
more visitors from particular socioeconomic groups. Visitor surveys of library use, for
example, have already provided initial support that the opening hours of public libraries
affect the social composition of their visitor populations (Glorieux, Kuppens & Vandebroeck
2007).
Second, the realism of the space-time accessibility measure used in this chapter can be
improved further. Some temporal aspects warrant more attention including the
incorporation of delay times, waiting times, a minimum activity duration and local changes in
travel times due to a rescheduling of opening hours. The valuation of attractiveness and
possible activity duration also deserves more attention. Whereas both components have
currently been assumed directly proportionate to individual accessibility, behaviourally more
appealing functions have been proposed to express this relationship (see e.g. Joh, Arentze &
Timmermans 2001; Ettema, Ashiru & Polak 2004). Increasing the behavioural realism of
space-time accessibility can also be achieved by accounting for dependencies between
household members with respect to car allocation, ride sharing and task re-allocation
strategies (Zhang & Fujiwara 2006, Soo et al. 2009). Since these aspects may impose
additional coupling constraints on activity participation, they should be incorporated in
future work. Finally, given that our approach is sample-based, it is important to point out
that the resulting optimal regime highly depends on the size and the accuracy of the travel
diary data at hand. This is because activities reported in a travel diary on a particular day
may not be representative for the type of activities that an individual is likely to regularly
engage in that day. Ideally, longitudinal data covering multiple days or even weeks should be
used to verify the consistency of activity patterns over a longer time horizon.
Third, at a more general level, rescheduling of the operational hours of public services,
commercial activities and employment may have downsides for family and social life and at
some point begin to reduce social welfare. This is because those services, commercial
activities and firms whose operational hours are to be rescheduled will demand that at least
some of their employees will have to come to work at the rescheduled hours. These hours
The relationship between opening hours and accessibility of public service delivery 219
may well coincide with times that children, spouses, friends and others will experience fewer
space-time constraints and are available for social and leisure activity participation. This
situation of rescheduled employment hours is most likely to occur for people with low levels
of sovereignty over their employment hours, many of whom will occupy the lower steps on
the occupational ladder, hold less secure jobs, and will be lowly educated and female
(Breedveld 1998, Hildebrandt 2006). Hence, the disadvantages that a large-scale
rescheduling of opening hours would have for family and social life will be distributed
unevenly across socio-economic groups in society (Mills & Taht 2010). There are at least two
ways to account at least to some extent for the negative effects of a rescheduling of
operational hours on family and social life. One, which has also been adopted here, is to a
priori determine a time window, during which opening hours can be rescheduled. Certain
periods of time, such as late evenings and Sundays, could in this way be excluded from the
rescheduling process. Second, it would be possible to incorporate people’s time-of-day
preferences regarding when they would like to participate in certain types of activities into
the Burns/Miller accessibility measures (see also Neutens et al. 2010a). Blocks of time during
the week and during which large groups of people would prefer to engage in social activities
with family, friends and others rather than visit a public service or commercial activity would
then have a lower weight in the calculation of the optimal regime of opening hours. The
value of this second approach could be explored in future work.
Despite these refinements to be made, we believe that the proposed method can be a
valuable instrument aiding policymakers, facility managers and others to explore different
configurations of opening hours that maximise potential visitors’ opportunity to pursue
activities at facilities across cities and regions.
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The impact of opening hours on the equity of individual space-time accessibility 225
11 The impact of opening hours on the equity of individual
space-time accessibility
Delafontaine M., Neutens T., Schwanen T., Van de Weghe N.
in Computers, Environment and Urban Systems, forthcoming
Copyright © Elsevier Science
Abstract. While many studies have concentrated on the effects of the spatial
distribution of services on individual accessibility, only little is known about the ways
in which equity of individual accessibility is affected by the temporal organisation of
service delivery. This chapter seeks to deepen our understanding about the
relationship between accessibility, equity and the opening hours of public service
facilities on the basis of space-time accessibility measures. Three approaches based
on different equity principles are presented to schedule the opening hours of public
service facilities: a utilitarian, an egalitarian and a distributive approach. A case study
of public libraries in Ghent (Belgium) demonstrates the relevance of these
approaches for amending the opening hours of public services to control the equity
of accessibility levels across individuals.
Keywords. Accessibility – Equity – Opening hours – Public services – Time geography
11.1 Introduction
Achieving a higher and more equitable level of access to essential public services has been
an issue of major concern in the urban service delivery and social exclusion literature for at
least three decades (Bigman & ReVelle 1979, Schönfelder & Axhausen 2003, Cass, Shove &
Urry 2005, Hero 1986, Mclafferty 1982, Talen & Anselin 1998, Tsou, Hung & Chang 2005,
Miller 2006). Within this well-developed and active line of research, attention has primarily
been directed toward the variations in service levels between geographic subunits or social
groupings as a consequence of an uneven spatial distribution of public services and
transportation facilities within a city (Scott & Horner 2008). Not only are local authorities
and policymakers concerned with maximising the accessibility of public services, they are
also sensitive to the degree to which the spatial configuration of service allocation favours
particular constituencies over others.
While numerous studies have sought to analyse the distributional effects of the spatial
configuration of public services, far less attention has been paid to the ways in which
accessibility and equity of accessibility is influenced by amending the opening hours of
226 Chapter 11
service facilities. This may in part be a corollary of the fact that accessibility to services has
traditionally been analysed as a static spatial phenomenon and measured through indicators
based on spatial proximity (for reviews about accessibility measures, see e.g. Pirie 1979, Guy
1983, Handy & Niemeier 1997, Kwan et al. 2003, Neutens et al. 2010a).
Recently however, Neutens et al. (2010b) have begun to substantiate empirically the
importance of accounting for opening hours of service delivery in evaluative studies of
accessibility. Relying on earlier empirical contributions in the realm of time geography (Kim
& Kwan 2003, Schwanen & de Jong 2008, Weber & Kwan 2002, Kwan & Weber 2008), they
have shown that, since people differ much in terms of the location, number, duration and
timing of their mandatory activities, changes to opening hours may remediate or exacerbate
individual disparities in accessibility as much as do amendments to the spatial distribution of
public service facilities.
With this in mind, we have developed a method to identify the temporal regime that
maximises person-based accessibility over (a sample of) a population, which has been
documented in Neutens et al. (2011). While this method provides insights into the margins
within which the overall accessibility can be improved by rescheduling the hours of service
delivery, it neglects the equity of accessibility levels across individuals. In other words,
regimes that maximise the overall accessibility may unintentionally favour particular groups
within the population over others.
In this chapter, we will extend our approach by explicitly introducing equity considerations in
the scheduling process. It is however not our intention to derive directly implementable time
schedules for a network of cooperating service facilities. Rather, the aim is to gain insights
into how and to what extent equity of individual accessibility to public services can be
influenced by amendments to the opening hours of service delivery. To this end, we will use
a person-based measure of space-time accessibility since previous research (Neutens et al.
2010a) has shown that such measures are more appropriate (and more conservative) to
assess equity than traditional place-based measures. This is because person-based measures
are premised on multiple reference locations, reveal interpersonal variations in time
budgets, recognize trip-chaining behaviour, and require only a single run to articulate
variations in accessibility across the diurnal cycle.
Three approaches based on different equity principles are presented to schedule the
opening hours of public service facilities: a utilitarian, an egalitarian and a distributive
approach. These scheduling approaches will be illustrated and validated in a case study on
the public libraries in the city of Ghent, Belgium. As with many other public services, public
libraries want to offer a high and equitable level of access to a large and socially diverse
public. Also, prior research (Cole & Gatrell 1986, Grindlay & Morris 2004, Glorieux, Kuppens
& Vandebroeck 2007, Loynes & Proctor 2000) has repeatedly shown that reduced
The impact of opening hours on the equity of individual space-time accessibility 227
accessibility through inadequate opening hours is one of the most important causes of a
decline in annual book issues per capita. Furthermore, local authorities are currently re-
examining the regimes of opening hours of public services within the city of Ghent to better
attune these to the activity patterns of the active citizens. These aspects make our case
study particularly relevant and timely to demonstrate our method.
11.2 Method
11.2.1 Scheduling procedure
All approaches to be presented in this chapter rely on the same core scheduling procedure,
which has been generalised from Neutens et al. (2011) and is pseudocoded in Algorithm
11.1. The basic unit of analysis in the procedure is a minimum opening interval (MOI), i.e. a
Input:
set of individuals
set of all candidate MOIs
accessibility function for an individual with respect to a regime
evaluation function for all accessibility values of all individuals in with respect
to a regime
requested number of MOIs
Output:
output regime (ordered set of MOIs)
Additional variables:
, MOIs
, numerical values
Algorithm:
01:
02: for to
03: for each in
04:
05:
06: if then
07:
08:
09: end if
10: end for
11:
12: end for
13: return
Algorithm 11.1 – Iterative scheduling procedure.
228 Chapter 11
time interval with a predefined minimum duration over which a specific facility is opened
(e.g. facility from 9:00 AM to 10:00 AM). The opening hours of a network of service
facilities can be represented through a set of MOIs, which is further referred to as a regime.
The procedure starts with a population of individuals ; a set of all MOIs to be
considered in the scheduling procedure, further referred to as the candidate set; a function
which returns the value of a person-based accessibility measure for an individual with
respect to a regime , such that higher values indicate higher accessibility; a function for
evaluating a regime on the basis of all accessibility values of all individuals in , such that
a more preferable regime yields a higher value; and the number of requested MOIs to be
included in the resulting regime, i.e. the output of the algorithm.
The output regime which consists of MOIs is built iteratively and in a bottom-up manner,
starting with zero MOIs. Each iteration runs through all MOIs in the candidate set which
are not yet included in so far. Each of these MOIs is a potential candidate to be added to
. The addition of each candidate is evaluated through the calculation of for
. The MOI which addition entails the highest value is then selected and added
permanently to , after which the algorithm takes the next iteration until contains
MOIs.
11.2.2 Equity approaches
In Neutens et al. (2011), the above procedure has been implemented to maximise the
accessibility to a set of public service facilities across a population. However, (public) service
providers are unlikely to maximise accessibility, independently from the distribution of
accessibility across individuals of the population. To account for this distribution,
appropriate evaluation functions ( ) will be specified. In the remainder of this section, we
will present three types of such functions, each of which applies to a different distribution
principle.
Utilitarian approach
In the approach of Neutens (2011), accessibility is maximised across the population,
whereas the equity of accessibility among individuals is disregarded. From an equity
perspective, this constitutes a utilitarian approach which means that utility is maximised
regardless of its distribution (Geurs & Ritsema van Eck 2001). The corresponding evaluation
function is given as the sum of individual accessibility values for a particular regime:
(11.1)
Egalitarian approach
While a utilitarian scheduling maximises the sum of individual accessibility across the
population, the approach will not necessarily lead to the most equitable regime as it
The impact of opening hours on the equity of individual space-time accessibility 229
implicitly favours those individuals who can obtain higher absolute accessibility levels.
Hence, we present an egalitarian scheduling which maximises the equity of individual
accessibility, as evaluated by a certain (in)equality metric. Various (in)equality metrics may
be adopted, ranging from simple statistical measures, such as quantile ratios, to more
complex and robust indicators such as the various Gini, Theil and Atkinson indices (Hao &
Naiman 2010). Each of these metrics, when expressed as a function in the appropriate
sense1, may be used as an evaluation function. Let denote such a function. Then,
an egalitarian evaluation function can be expressed as:
(11.2)
Distributive approach
Another alternative to the utilitarian approach, which treats all individuals equally
throughout the scheduling procedure, is a distributive scheduling which grants different
(groups of) individuals a different impact on the scheduling procedure. Distributive
evaluation functions can be expressed as a weighted sum of individual accessibility values.
Let denote the weight of an individual . Then, a distributive evaluation function can be
specified as:
(11.3)
Two cases can be distinguished: (i) a case where only positive or negative weights are
considered; and (ii) a case where both positive and negative weights occur. In case (i), the
scheduling procedure will give priority to the regime which maximises the overall
accessibility, allocating different individuals a different impact in the summation. In this way,
opening hours that are beneficial to individuals with larger weights are preferentially chosen
in the eventual regime. In case (ii), a subtraction is made such that the regime is selected
that maximises the difference in accessibility between a group of favoured and a group of
disfavoured individuals. In this way, the accessibility of one group tends to be increased or
decreased proportionally to the other.
11.3 Case study
An empirical case study is elaborated to illustrate and validate the approaches presented in
section 11.2. The study will explore their effects on the equity of individual accessibility
levels to public libraries in the city of Ghent (Belgium). To this end, a full week regime of
opening hours will be considered. The results will be validated against each other and
against the current regime of opening hours. The purpose of this exercise is to examine to
what extent policy makers can exert influence on the distribution of accessibility among the
1 Since evaluation functions are maximised, the functional form of inequality metrics will have to be rewritten
such that their value increases with increasing equality.
230 Chapter 11
population by rescheduling the opening hours of public services. The remainder of this
section will subsequently discuss input data, computation and results.
11.3.1 Input data
Public libraries, opening hours and candidate set
Information on the location and opening hours of Ghent’s municipal network of public
libraries2 is provided by the official city website (http://www.gent.be). The network consists
of one centrally located main library and 15 branch libraries dispersed across the city (Figure
11.1, Table 11.1). The libraries have a well-structured regime of weekly opening hours with
similar schedules for all branch libraries (Table 11.2). 50 (24%) of the total of 209 hours are
allocated to the main library, whereas most branch libraries individually account for merely
11 hours (5%). The common services delivered in each library include the lending of articles
(books, comic strips, DVDs, etc.), the consultation of reference works, magazines and
informative leaflets, and free surfing on the internet. The main library is by far the most
important in terms of service delivery, and it is the sole library with multiple subdivisions.
Figure 11.1 – Public libraries in Ghent.
2 Public archives and documentation centres will not be considered as they usually do not offer lending services
and attract a rather specific kind of visitors.
The impact of opening hours on the equity of individual space-time accessibility 231
No Name Collection Attractiveness
1 Zuid 368 907 12.82
2 Bloemekenswijk 7 387 8.91
3 Brugse Poort 7 669 8.94
4 Drongen-Baarle 7 314 8.90
5 Drongen-Centrum 16 543 9.71
6 Gentbrugge 13 791 9.53
7 Ledeberg 16 765 9.73
8 Mariakerke 15 330 9.64
9 Nieuw Gent 5 837 8.67
10 Oostakker 10 372 9.25
11 Sint-Amandsberg 19 228 9.86
12 St-Denijs-Westrem 9 723 9.18
13 Watersportbaan 4 900 8.50
14 Westveld 8 889 9.09
15 Wondelgem 8 057 8.99
16 Zwijnaarde 10 122 9.22
Table 11.1 – Library collection size (2010) and attractiveness estimate.
Library
Monday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
10:00 AM 11:00 AM 12:00 AM 1:00 PM 2:00 PM 3:00 PM 4:00 PM 5:00 PM 6:00 PM Tuesday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
10:00 AM 11:00 AM 12:00 AM 1:00 PM 2:00 PM 3:00 PM 4:00 PM 5:00 PM 6:00 PM Table 11.2 – Opening hours of public libraries in Ghent
232 Chapter 11
Wednesday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
10:00 AM 11:00 AM 12:00 AM 1:00 PM 2:00 PM 3:00 PM 4:00 PM 5:00 PM 6:00 PM Thursday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
10:00 AM 11:00 AM 12:00 AM 1:00 PM 2:00 PM 3:00 PM 4:00 PM 5:00 PM 6:00 PM
Library
Friday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
10:00 AM 11:00 AM 12:00 AM 1:00 PM 2:00 PM 3:00 PM 4:00 PM 5:00 PM
6:00 PM Saturday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
10:00 AM 11:00 AM 12:00 AM 1:00 PM 2:00 PM 3:00 PM 4:00 PM 5:00 PM
6:00 PM Table 11.2 – (continued)
To populate the candidate set , we have considered MOIs of one hour duration. As public
services are usually not offered on Sundays in Belgium (cf. Table 11.2), these will also not be
considered in the rescheduling. Furthermore, we limit our analysis per facility and per day
from Monday to Saturday, to the MOIs within the range from 8:00 AM to 8:00 PM which
The impact of opening hours on the equity of individual space-time accessibility 233
reflects the limits within which public facilities tend to offer services in Belgium. Thus, the
candidate set in this example consists of 1 152 MOIs (= 16 facilities x 6 days x 12 MOIs).
Sample population and accessibility measure
The population of library visitors is sampled through an activity/travel data set consisting of
two-day consecutive diaries of out-of-home activities of Ghent citizens aged five or more.
The sampled individuals are considered representative for the target constituency of Ghent’s
municipal libraries. The travel diaries have been collected in 2000 within the scope of the
SAMBA project (Spatial Analysis and Modeling Based on Activities) (Tindemans et al. 2005).
As households have been randomly sampled, the spatial distribution of home locations
reflects the actual population density with a sparsely populated industrial and port area in
the north of Ghent (Figure 11.2). Individuals sampled at the same day of the week have been
grouped under the assumption that their activities are representative for that weekday. In
total, 5 744 person-days have been used, ranging from Monday to Saturday.
The person-based accessibility measure in this case study relies on Burns’ (1979) utility-
theoretic framework for calculating individual accessibility. This framework has attracted
increased attention in recent years because it is theoretically appealing and can nowadays
straightforwardly be operationalized using geographical information systems (GIS) (see e.g.
Miller 1999, Neutens et al. 2008, 2010a, Hsu & Hsieh 2004, Ashiru, Polak & Noland 2003,
Ettema & Timmermans 2007). Also, prior research (Neutens et al. 2010a) has shown that
Burns-Miller measures are more successful in terms of signifying a state of equity than
traditional place-based measures, since they much better articulate inter-personal
differences for the various dimensions of accessibility they capture. Further, it is noted that
less variation in person-based accessibility would have been obtained, should we have used
a Lenntorp accessibility measure which expresses the cardinality of a choice set. This is
because the number of opportunities (i.e. libraries) in our case study is relatively small (see
Neutens et al., 2010a).
The accessibility for an individual to the MOI of service facility from to is specified
as:
(11.4)
With the set of non-overlapping time intervals within over which
can participate in an activity at ; the attractiveness of facility ; the travel
cost required for to participate in an activity at from to ; the calibration parameter
234 Chapter 11
Figure 11.2 – Sampled households and population density in Ghent.
of negative exponential travel cost decay.
The different components in (11.4) have been implemented as follows. The set
is composed of the positive time intervals within that start at the end
time of a fixed activity of plus the travel time to , and that end at the start time of ’s next
fixed activity minus the travel time from . The determination of thus
requires information on both individual fixed activities and travel times. As the fixity level of
activities has not been documented in the travel diaries, we had to extract fixed activities
manually from the diaries. To this end, the activities belonging to the purpose categories
“work”, “education”, “pick up/drop off” and the like have been considered fixed, given the
difficulty to conduct these activities at other places and times (Cullen & Godson 1975,
Schwanen, Kwan & Ren 2008).
The travel times, on the other hand, have been computed as shortest path times within
ESRI’s ArcGIS Network Analyst (9.3.1) based on TeleAtlas® MultiNet™ (2007.10) road network
data. To this end, we have geocoded the reported locations of fixed activities to the street
level. Shortest path times have been calculated according to the two predominant travel
modes in Ghent, i.e. car and bicycle. To account for individual differences in mobility
resources, it has been assumed that adult car owners with a driving license travel by car,
The impact of opening hours on the equity of individual space-time accessibility 235
whereas others travel by bicycle. In addition, car travel times have been corrected for
congestion by means of a factor based on road class, weekday, and time of day, following
Neutens et al. (2011). The congestion factor has been derived from average travel times
recently reported by Maerivoet and Yperman (2008) under the authority of the Federal
Government Service for Mobility and Transport. Each car travel time has been calculated as
the sum of the time shares spent along the different road classes within the shortest path,
weighted by their respective congestion factor. For bicycle travel times, a compromise
approach has been adopted due to the lack of information on dedicated bicycle facilities
(e.g. exclusive non-motorised paths) in Ghent. The approach consists of excluding highways
and other exclusive motorways from the road network and allowing travel directions for
non-motorised travellers3. The travel times have been estimated as the division of the
shortest path distance and an average cycling speed of 15 km/h (El-Geneidy, Krizek & Iacono
2007).
All travel costs in (11.4) have been computed as detour travel times for to
travel to in the time window delimited by the fixed activity immediately preceding and
the fixed activity immediately following , instead of travelling directly in between both fixed
activities. The travel times of the different elements in this detour calculation have been
assessed as described above. The decay parameter of the negative exponential deterrence
function in (11.4) has been estimated on the basis of the observed cumulative distribution of
reported travel times of travel diary trips of individuals visiting a service. Details of its
estimation are available in Neutens et al. (2011). Similar estimates for are obtained for car
(0.081) and bicycle travels (0.092).
For the attractiveness factor in (11.4), we have taken for each library the natural
logarithm of its collection size as a proxy (Table 11.1). The natural logarithm ensures that
attractiveness increases with collection size at a decreasing rate and that adheres to the
economic principle of declining marginal benefits. Ideally we would have liked to
operationalize attractiveness in a more holistic way, for instance by also considering the
variety of books/services on offer and the degree to which libraries are specialised in specific
genres. However, the relevant information for this broader operationalization was not
available to us and we leave this issue for future research. More generally, whilst the
measurements of the facilities’ attractiveness, the travel times and the activity participation
time may be refined in future research, we believe that the current procedures are adequate
for this illustrative case study.
To obtain the accessibility of an individual with respect to a complete regime , as required
in the scheduling procedure, we have applied (11.4) using the following maximative
function:
3 One-way streets for motorised vehicles passable in both directions for bicyclists are common in Ghent.
236 Chapter 11
(11.5)
Thus, in the case of concurrent MOIs of different facilities, only the MOI which offers the
highest accessibility to the individual at hand is accumulated. This is in line with the idea that
individuals may not benefit from having a larger choice set of facilities in case they deliver
very similar services, as is the case for the municipal libraries in Ghent. This reasoning also
relies on the potential of each individual to act as a rationale decision maker who is only
concerned with the most beneficial alternative. One advantage of this assumption and of
adopting a maximative formulation for the accessibility measure is that it becomes
consistent with rational utility theory (see Miller 1999 for more information on this), which is
extensively used for modelling choice behaviour in the field of activity-based travel demand
analysis. Note that, since our sample consists of a slightly different number of observations
per weekday, a weighting factor was added, such that each weekday has an equal weight.
The accessibility level for a regime to an individual obtained from (11.5) is a dimensionless
measure on a ratio scale. While the absolute value of this measure may be of limited value,
it is useful to compare the accessibility levels produced by different regimes. It is noted that
equation (11.5) is only one possible instance of an accessibility measure to be applied in the
iterative selection procedure. Future research may examine the effects of more complex
accessibility measures, such as those that account for a minimum activity duration or
interactions among different household members (Fan & Khattak 2009, Pendyala & Goulias
2002).
11.3.2 Evaluation functions and computation
Having specified the candidate set , the population and the accessibility function ,
we may now derive regimes on the basis of evaluation functions that correspond to the
different equity approaches given in section 11.2.2. In this empirical study, a utilitarian,
egalitarian, and distributive function will be illustrated. The utilitarian evaluation function
has been specified in (11.1). For the egalitarian evaluation function we have adopted a
negative form of Theil’s inequality index (Theil 1967):
with (11.6)
The Theil index is an inequality measure based on the concept of entropy, which turns to 0 in
the case of complete equality and to the natural logarithm of the sample size in the case of
complete inequality (i.e. = 8.66 in this case study). The Theil index was chosen in
this case study for various reasons. First, the Theil index is known to be anonymous and
scale-independent with respect to individual values. In addition, it implements the Pigou-
Dalton principle (Pigou 1912) which states that a transfer of value of higher ranked individual
to a lower ranked individual, as long as this does not inverse the ranking of the two, should
The impact of opening hours on the equity of individual space-time accessibility 237
result in greater equity. Also, the Theil index is decomposable, such that it can be obtained
from the weighted sum of Theil indices of different subgroups of the population. Finally,
from a computational point of view, the Theil index is preferable as it can be computed in
linear time with respect to the size of the population, whereas, e.g. for calculating a Gini
coefficient, quadratic time would be required. Since the scheduling procedure requires the
inequality measure to be calculated in each iteration for the addition of each remaining
candidate MOI (Algorithm 11.1), this is an important advantage of the Theil index. To obtain
a valid evaluation function that increases with the desirability of a regime, the negative Theil
index has been used for .
To explore the effects of a distributive scheduling approach, we will consider a progressive
evaluation function with balanced weights (i.e. positive and negative weights). A progressive
approach aims to favour disadvantaged individuals over others (Litman 2002). In the context
of our example, disadvantaged in terms of accessibility means having many space-time
constraints on visiting the municipal libraries. To assess the extent to which people
experience these space-time constraints, we will consider the level of accessibility they can
attain within their constraints in a regime consisting of all candidate MOIs. For individual ,
this level can be assessed as his/her total accessibility over all MOIs in the candidate set ,
i.e. . The evaluation function is specified as:
(11.7)
For we take the median value of over the population. Hence, the population is
split into two halves: a lower half consisting of individuals with more space-time constraints,
and an upper half comprised of individuals with fewer space-time constraints. It is important
to note that these halves represent a distinct social mix of persons in terms of socio-
demographics and residential neighbourhoods. One of the more salient differences between
both groups is the employment status of the individuals, since this characteristic determines
to a large degree the number of temporal constraints an individual faces (see also our earlier
findings in Neutens et al. 2010b). Figure 11.3 represents the composition of the upper and
lower half in terms of employment status. It is found that the lower half consists primarily of
full-time employed persons and students who typically experience many temporal
constraints, whereas the upper half includes more part-time employees and those who are
not gainfully employed (i.e. other) such as housewives, senior citizens and unemployed
persons who tend to have more hours per day available for conducting discretionary
activities such as library visits. Since intends to maximise the relative difference in
accessibility between the lower and upper half, it can be expected that the resulting regime
will alleviate the existing accessibility disparities between, among others, persons from
different employment categories.
238 Chapter 11
The ratio
in (11.7) has been introduced in order to express the accessibility of a regime
relative to (i.e. as a value in the range from 0 to 1). This has been done to ensure
that the impact of individuals on the scheduling procedure is independent of the absolute
value of accessibility (cf. utilitarian regime). In this way, we ensure that the lower and upper
halves have equal impact on the evaluation function.
Figure 11.3 – Composition of the lower and upper halves in terms of employment status.
Having specified , , and , we have automatically computed their corresponding
regimes through an implemented module of the iterative scheduling procedure (Algorithm
11.1). For each regime, we have set the requested number of MOIs to 209 in order to be
consistent with the current regime of opening hours (Table 11.2).
11.3.3 Results
The resulting utilitarian, egalitarian, and distributive regimes are presented in Tables 11.3-
11.5, respectively. For each MOI, the order of its allocation to the regime during the iterative
scheduling procedure has been indicated with a number. Additionally, the MOIs have been
gray-scaled into five equal interval classes of allocation order, with darker shading for earlier
allocated MOIs. The ranking provides insights into the relative importance of different
opening hours within each regime. The distributions of opening hours of different regimes
differ to a considerable degree from one another across both days of the week and libraries.
Utilitarian regime
The utilitarian regime (Table 11.3) clearly shows a hierarchy among the libraries. It allocates
opening hours to merely 7 of the 16 libraries. The first 72 hours have been assigned to the
main library covering the entire study period (Monday to Saturday from 8:00 AM to 8:00
PM). This can be explained by the central location of the main library in a well-populated
21%
42%
8%
29%
lower half
student
full-time employee
part-time employee
other
9%
25%
11%
56%
upper half
The impact of opening hours on the equity of individual space-time accessibility 239
area of the city and by its significantly higher attractiveness as a service facility relative to
the other libraries. Evening hours (6:00 PM to 8:00 PM) and hours on Saturday are selected
first by the algorithm and thus produce the highest accessibility over the entire population.
This is due to the fact that people have on average fewer space-time constraints resulting
from fixed activities during these periods. Next, branch libraries 2 and 3 are assigned
opening hours. While the collection size of these offices is rather modest, they are located
along the inner ring road around Ghent and are surrounded by major residential areas.
Hence, they can attract a large number of visitors for whom the main library is not the most
beneficial option in terms of accessibility. Again, opening hours outside the common
business hours in Belgium are assigned first. Finally, the algorithm allocates many opening
hours to branch library 11 as well as a few opening hours to branch libraries 5, 7 and 15.
Besides its high attractiveness and its proximity to densely populated areas, the importance
of library 11 can additionally be explained by its potential to attract visitors along their
commute between their home location and the major employment centre in the port area in
the north of Ghent. Furthermore, it is noted that, of all branches, library 11 currently has the
largest number of opening hours (Table 11.2Table 11.2).
In general, the utilitarian approach tends to cover each possible opening hour of the study
period for at least one of the libraries, whereas concurrent opening hours for two or more
libraries tend to be avoided. This is due to the competition effects that are accounted for by
the maximative form of (11.5), which limits the overall gain in accessibility that can be
obtained from the addition of a concurrent opening hour compared to the addition of a yet
uncovered hour of the study period. In other words, the best strategy to maximise the
overall library access in Ghent is to extend the current range of opening hours (cf. Table
11.2) and to reschedule concurrent hours to cover this extended range.
Egalitarian regime
The egalitarian regime (Table 11.4) is radically different from, and in many ways the
opposite, of the utilitarian regime. While the latter respects a strong hierarchy among
facilities, the egalitarian regime can be described as almost facility-independent: all 16
libraries have been allocated 12-17 opening hours. What is more, Table 11.4 shows that
equity of accessibility is almost entirely determined by the timing of opening hours. The
egalitarian regime clearly prioritizes the latest evening hour (7:00 PM – 8:00 PM) which is
allocated to all libraries on all days of the week. It also prioritizes the first morning hour (8:00
AM – 9:00 AM) which is selected in all cases, except on Monday and Saturday. On Saturday,
people appear to benefit in equal measure during noon (12:00 AM – 1:00 PM), since this
period tends to provide a high accessibility to most individuals. This is inter alia because only
few individuals in our sample reported fixed activities during this period (in-home activities
such as eating have not been reported in the travel diaries). The hour from 10:00 AM to
11:00 AM on Monday also enhances the equity of accessibility levels. This may be attributed
240 Chapter 11
Library
Monday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
8:00 AM 41
115
181
9:00 AM 67
129
192
10:00 AM 72
130
194
11:00 AM 70
124
191
12:00 AM 52
110
186
1:00 PM 57
116
185
2:00 PM 61
121
182
3:00 PM 54
119
175
4:00 PM 35
104
163
5:00 PM 15
84
155
6:00 PM 9
80
140
7:00 PM 3
78
136
Tuesday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
8:00 AM 38
137
199
9:00 AM 59
154
10:00 AM 64
159
11:00 AM 62
151
12:00 AM 47
138
1:00 PM 43
145
202
2:00 PM 42
153
197
3:00 PM 40
148
189
4:00 PM 26
123
177
5:00 PM 12
103
173
6:00 PM 7
86
205
167
7:00 PM 1
85
198
162
Wednesday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
8:00 AM 36
114
190
9:00 AM 50
132
10:00 AM 60
133
11:00 AM 55
131
12:00 AM 37
107
196
1:00 PM 34
106
187
2:00 PM 33
105
184
3:00 PM 30
102
183 4:00 PM 19
97
179
5:00 PM 11 83
178
6:00 PM 4 76
172
7:00 PM 2 75
169
Table 11.3 – Utilitarian regime of 209 opening hours, with indication of the allocation order of each
hour in the scheduling procedure. Allocated hours are gray-scaled according to an equal interval
classification into five classes of the allocation order.
The impact of opening hours on the equity of individual space-time accessibility 241
Library
Thursday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
8:00 AM 39
108
9:00 AM 63
125
10:00 AM 68
120
11:00 AM 71
122
12:00 AM 53
109
1:00 PM 56
111
2:00 PM 58
117
3:00 PM 45
112
4:00 PM 32
96
188
5:00 PM 13
79
174
6:00 PM 8
74
170
7:00 PM 5
73
168
207
Friday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
8:00 AM 44 113
193
9:00 AM 65 134
200
10:00 AM 69 135
204
11:00 AM 66 139
208
12:00 AM 49 127
201
1:00 PM 48 126
2:00 PM 51 128
3:00 PM 46 118
195
203
4:00 PM 31 95
176
180
5:00 PM 14 82
165
171
6:00 PM 10 81
161
166
7:00 PM 6 77
160
164
Saturday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
8:00 AM 18 88
144
9:00 AM 27 98
158
10:00 AM 28 100
156
11:00 AM 29 101
152
12:00 AM 25 99
141
1:00 PM 22 91
143
2:00 PM 24 93
149
3:00 PM 23 94
150
4:00 PM 21 89
157
5:00 PM 20 90
206
147 6:00 PM 17 92
146
7:00 PM 16 87
209
142
Table 11.3 – (continued)
242 Chapter 11
Library
Monday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
8:00 AM
9:00 AM
10:00 AM 188 167 129 185 182 156 187 159 138 140 179 181 100 112 139 173 11:00 AM
12:00 AM
1:00 PM
2:00 PM
3:00 PM
4:00 PM
2 198
196 201
194 197 199
195
5:00 PM
6:00 PM
7:00 PM 91 22 64 146 86 92 28 40 46 38 16 80 9 48 50 97
Tuesday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
8:00 AM 190 142 172 164 135 153 178 180 117 141 143 166 89 5 116 183 9:00 AM
10:00 AM
11:00 AM
12:00 AM
1:00 PM
2:00 PM
3:00 PM
4:00 PM
5:00 PM
205
207
6:00 PM
7:00 PM 101 21 76 93 45 24 61 56 31 43 54 55 10 95 57 94
Wednesday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
8:00 AM 192 174 130 4 148 128 177 162 118 104 151 165 87 122 115 158 9:00 AM
10:00 AM
11:00 AM
12:00 AM
1:00 PM
2:00 PM
3:00 PM
4:00 PM
5:00 PM
206
209
6:00 PM
7:00 PM 110 18 59 136 65 25 71 74 33 58 49 82 14 8 39 96
Table 11.4 – Egalitarian regime of 209 opening hours, with indication of the allocation order of
each hour in the scheduling procedure. Allocated hours are gray-scaled according to an equal
interval classification into five classes of the allocation order.
The impact of opening hours on the equity of individual space-time accessibility 243
Library
Thursday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
8:00 AM 186 169 119 3 161 127 184 107 114 102 176 150 85 149 133 157 9:00 AM
10:00 AM
11:00 AM
12:00 AM
1:00 PM
2:00 PM
3:00 PM
4:00 PM
5:00 PM
200
203
204
6:00 PM
7:00 PM 103 19 44 121 72 20 73 66 34 35 63 78 13 7 41 83
Friday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
8:00 AM 191 171 125 134 170 126 189 105 123 168 154 160 98 23 131 145 9:00 AM
10:00 AM
11:00 AM
12:00 AM
1:00 PM
2:00 PM
3:00 PM
4:00 PM
202
5:00 PM
6:00 PM
7:00 PM 90 29 60 84 32 12 75 6 30 42 68 70 17 120 51 53
Saturday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
8:00 AM
208
9:00 AM
10:00 AM
11:00 AM
12:00 AM 193 99 132 1 62 113 175 144 111 147 152 52 88 106 137 163 1:00 PM
2:00 PM
3:00 PM
4:00 PM
5:00 PM
6:00 PM
7:00 PM 108 11 79 109 155 15 67 47 36 77 27 124 26 69 37 81
Table 11.4 (continued)
244 Chapter 11
Library
Monday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
8:00 AM 65
156
144
155 173 202
9:00 AM
10:00 AM
11:00 AM
12:00 AM
1:00 PM
2:00 PM
3:00 PM
4:00 PM
5:00 PM 12
126 178
88 56 153 76 120 177 104 39 158
6:00 PM 5
179 64 204 53 37 121 34 146 147 93 62 116 194 7:00 PM 2
151 57 201 42 27 160 19 59 166 103 81 91 115
Tuesday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
8:00 AM 18
189
142
186 129
9:00 AM
10:00 AM
11:00 AM
12:00 AM
1:00 PM
2:00 PM
3:00 PM
4:00 PM 167 171
169
172
170 174 168 205
5:00 PM 11 50
54
163
98
152 71 79 52 159
6:00 PM 8 28
165 36
141 122 63
92 72 89 44 143
7:00 PM 1 26
176 21
83 124 67 123 30 60 84 68 85 188
Wednesday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
8:00 AM 17
139
108
130
9:00 AM
10:00 AM
11:00 AM
12:00 AM
1:00 PM
2:00 PM
3:00 PM 4:00 PM
5:00 PM 15
95
80
110 181 164
6:00 PM 10 61
41
55
99 97
7:00 PM 6 48
31
25 208
105 109
Table 11.5 – Distributive regime of 209 opening hours, with indication of the allocation order of
each hour in the scheduling procedure. Allocated hours are gray-scaled according to an equal
interval classification into five classes of the allocation order.
The impact of opening hours on the equity of individual space-time accessibility 245
Library
Thursday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
8:00 AM 16
182
149
9:00 AM
10:00 AM
11:00 AM
12:00 AM
1:00 PM
2:00 PM
3:00 PM
4:00 PM 197
199 207
200 198
5:00 PM 13 100
154 69 145 196 150 90 46 157
114
6:00 PM 7 45 180 187 40 138 193 106 87 24 161
96 190
7:00 PM 3 35 162 70 23 119 183 94 102 20 75 195
136 140
Friday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
8:00 AM 32
184 74
9:00 AM
10:00 AM
11:00 AM
12:00 AM
1:00 PM
2:00 PM
3:00 PM
4:00 PM
132
206
134
131
133 135
5:00 PM 14
175
49 117
112 86 38
137
66 111
6:00 PM 9
82
43 113 203 148 77 29
125
51 101
7:00 PM 4 58 185 127 33 107 191 118 73 22 209 128
47 78 192
Saturday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
8:00 AM
9:00 AM
10:00 AM
11:00 AM
12:00 AM
1:00 PM
2:00 PM
3:00 PM 4:00 PM
5:00 PM
6:00 PM
7:00 PM
Table 11.5 – (continued)
246 Chapter 11
to the fact that many people are prohibited from visiting a library within this hour, since they
are constrained by fixed activities reported in their diary. In short, it seems that the best
strategy to improve equity of accessibility among the population is to select simultaneous
opening hours for different facilities at strategic times of the day.
Distributive regime
The distributive regime (Table 11.5) returns another distinct pattern. Whereas in the
utilitarian and egalitarian case, opening hours are well distributed among all days, the
distributive regime has no opening hours on Saturday. This is an artefact of the non-
longitudinal travel diary data consisting of person-days: since most individuals have much
discretionary time on Saturday, it appears to be difficult to advantage individuals with fewer
space-time constraints (lower half) over individuals with more constraints (upper half) on
Saturdays. The same reasoning, albeit to a lesser extent, applies to Wednesdays when most
part-time workers do not work or only work half a day in Belgium. The evening, late
afternoon and early morning hours on other weekdays, on the other hand, are abundantly
covered by all libraries. This pattern mirrors the common timing of discretionary time
budgets of the persons who experience many space-time constraints, including in particular
full-time workers and students (see Figure 11.3).
Accessibility and equity
Beyond the variations in opening hours, we can also evaluate the regimes in terms of their
distribution of accessibility levels4 ( ) across individuals and the equity thereof. To this end,
Figure 11.4 displays a Box-and-Whisker diagram which depicts the spread of accessibility
levels for each regime, and Figure 11.5 presents the inequality of this distribution, measured
by the Theil index. The current regime has been included in these figures as a reference.
Figure 11.4 and Figure 11.5 demonstrate that the distribution of accessibility among
individuals strongly depends on the scheduling approach. The largest increase in accessibility
relative to the current regime can be realised with the utilitarian approach (Figure 11.4),
which confirms its objective. However, the accessibility levels in this regime have a large
spread and are relatively unequally distributed among the population, as is reflected by the
high Theil index (Figure 11.5). The egalitarian approach, on the other hand, offers the lowest
average accessibility level, but produces by far the smallest spread in accessibility levels
(Figure 11.4) and the most equity (i.e. the lowest Theil index) (Figure 11.5). In other words, in
our case study striving for equity among the entire population comes at the expense of the
absolute level of accessibility. This is because, while it is feasible to offer every individual a
comparably low level of accessibility, high accessibility levels cannot be allocated equally
given the substantial differences in the extent of space-time constraints across the
respondents in the sample. Finally, the distributive regime lies somewhere in between the
4 The accessibility levels (A) are calculated for each regime using equation (11.5).
The impact of opening hours on the equity of individual space-time accessibility 247
current and egalitarian regime, in terms of the average, the spread and the equity of
accessibility levels. This can be explained by the progressive configuration of the distributive
evaluation function (section 11.3.2): it intends to favour the individuals with more space-
time constraints (lower half) relative to the individuals with fewer space-time constraints
(upper half).
To further validate the distributive approach, we will consider the Box-and-Whisker diagram
of individual accessibility for the lower and upper half of the population separately, as
displayed in Figure 11.6. While for all regimes, the upper half has higher accessibility levels
than the lower half, this difference is smaller for the egalitarian and distributive regimes. In
terms of average accessibility, the distributive regime yields the smallest absolute and
relative difference among both halves (14.9 for the lower half and 19.7 for the upper half).
Also, this regime entails the second highest average accessibility level for the lower half. In
other words, changing the opening hours of public service facilities using a distributive
approach can be a successful strategy to alter the existing disparities in accessibility between
people with different space-time constraints. However, the average accessibility level over
the entire population (17.3) is lower compared to the current (24.2) and utilitarian regime
(38.6). Thus, as for the egalitarian regime, a progressive distribution of accessibility comes at
the expense of the absolute level of accessibility. Nonetheless, given that the decrease in
average accessibility relative to the current regime is smaller than in the egalitarian case, a
progressive distributive approach may be more appropriate if the aim is to improve the
accessibility of those who experience most space-time constraints in their daily life.
Figure 11.4 – Box-and-Whisker diagrams of the accessibility level per regime.
0
20
40
60
80
100
120
Current Utilitarian Egalitarian Distributive
Acc
essi
bili
ty le
vel (
A)
Regime
248 Chapter 11
Figure 11.5 – Theil index of the accessibility level per regime.
Figure 11.6 – Box-and-Whisker diagrams per regime for the lower (left) and upper (right) halves of
the population.
0.33
0.27
0.10
0.30
0,0
0,1
0,2
0,3
0,4
Current Utilitarian Egalitarian Distributive
Thei
l In
dex
Regime
0
20
40
60
80
100
120
140
Current Utilitarian Egalitarian Distributive
Acc
essi
bili
ty le
vel (
A)
Regime
The impact of opening hours on the equity of individual space-time accessibility 249
11.4 Conclusion
In contrast to prior accessibility studies that have focused on the spatial organisation of
public service delivery, this chapter has explored the ways in which equity of individual
accessibility to services within the population can be influenced by adapting the opening
hours of service facilities. To this end, three different scheduling approaches – utilitarian,
egalitarian and distributive – have been elaborated within a generalised iterative scheduling
procedure. While the utilitarian approach aims to compose a regime that offers the highest
overall accessibility, the egalitarian approach seeks to find a regime that maximises the
equity of accessibility levels across individuals. The distributive approach, on the other hand,
uses different weights for different individuals in the scheduling procedure to favour certain
(groups of) individuals and/or disfavour others.
The three scheduling approaches have been implemented and applied in a detailed
empirical case study focusing on the rescheduling of a standard week regime of opening
hours for the municipal libraries in Ghent (Belgium). The resulting time schedules showed
significant differences in terms of the distribution of opening hours among facilities, as well
as across the days of the week and times of the day. We have also demonstrated that
rescheduling according to the various approaches strongly affects the distribution of
individual accessibility and the equity of accessibility. All scheduling approaches have
thereby clearly shown to validate their purpose. Of all scheduling regimes, the utilitarian
regime caused the largest increase in average accessibility, while the egalitarian regime
produced the most equitable regime. However, the improvement in equity realised by the
egalitarian regime was offset by a decrease in the absolute level of accessibility. Finally, we
were able to demonstrate that the distributive approach is effective in redistributing
accessibility among different groups of individuals. The distributive regime combines to a
certain extent the merits of the other two approaches: it offers a higher level of accessibility
than the egalitarian approach and a more compact and equitable distribution of individual
accessibility compared to the utilitarian approach.
This chapter extends the existing literatures about space-time accessibility analysis, urban
service delivery and social exclusion by showing to what extent equity in individual space-
time accessibility can be influenced by changing the opening hours of service delivery. From
a policy point of view, this is an important achievement because it enables ex-ante and ex-
post evaluations of different configurations of opening hours of services and their
consequences in terms of equity. Understanding the relationship between opening hours
and equity of accessibility is also important in view of the growing awareness of the impact
of urban time policies on people’s quality of life (Neutens et al. 2011). In Ghent as well as in
many other European cities, local authorities are currently re-examining the historically
emerged opening hours of their municipal services in order to better attune these to the
temporal needs and desires of the citizens, especially those who have multiple competing
250 Chapter 11
claims on their time (Mareggi 2002, Boulin 2006). This chapter contributes to these lines of
inquiry by providing additional insights into how (equity of) individual accessibility can be
improved by amendments to the temporal structure of urban systems.
The generality of the proposed iterative scheduling procedure and its arbitrary evaluation
function enables to apply the methodology to various aspects linked to the space-time
accessibility of opportunities to individuals. For instance, beyond the equity of accessibility,
the effects of opening hour scheduling on other variables connected to accessibility within
social welfare research and policies (cf. Rouwendal & Rietveld 1999), including indicators of
urban liveability (Pacione 1990) and social capital (Adler & Kwon 2002), can be assessed.
Other possible extensions can be found in urban and transport planning and policy making.
For example, it would be useful to consider evaluation functions which incorporate the
effects of opening hours and space-time accessibility on congestion times in order to control
the flows of traffic by means of rescheduling business hours. The results of such studies
could be compared to the predictions of alternative planning initiatives such as investments
in the spatial transport infrastructure or road pricing systems (Gutiérrez, Condeço-
Melhorado & Martín 2010). It is our aim to continue this line of research about the effects of
opening hour policies in future studies.
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Discussion and conclusions 253
12 Discussion and conclusions
This concluding chapter discusses the main contributions, results, and findings of this
dissertation, as well as the limitations of these achievements and opportunities for further
research. Given that specific contributions and conclusions have already been stated within
each chapter, this section in addition intends to disclose the links that interconnect these
chapters as well as their situation within the broader literature. In support of the discussion
and to offer feedback on the earlier stated objectives and research questions (section 1.2),
an overview table of main and application-oriented contributions is presented in Table 12.1.
Part I – Moving objects
To begin with, a number of important contributions are made to the Qualitative Trajectory
Calculus (QTC). Given that QTC is unique in being a qualitative spatiotemporal calculus
dedicated to handling interactions among moving objects, these contributions add to a
notably exclusive framework in the research field of moving objects. The theoretical
overview presented in Chapter 2 may in the first place be contemplated an umbrella
contribution vis-à-vis QTC in general, and the related chapters of this work in particular
(Chapters 3-5). However, the chapter also makes some prominent theoretical contributions,
including the introduction of a conceptual neighbourhood diagram (CND) and composition-
rule tables (CRTs) for QTCC. These additions supply to the missing links in earlier works and
augment the reasoning power of QTCC significantly. Furthermore, Chapter 2 discusses some
vital issues regarding QTC-based information systems, such as potential extensions of QTC
and how the calculus can deal with incomplete information (e.g. tracking data).
Chapter 3 contributes to the formal definition of QTCN and explores its reasoning power
through elaborating the composition of QTCN relations and examining the ability to use QTCN
for answering qualitative questions. QTCN is a key type of QTC due to its consideration of
network-constrained movement, since networks are by far the most commonly modelled
and studied travel environment for conducting mobile objects in geographical space. Its
application field therefore primarily lies in transportation, and particularly in geographical
information systems for transportation (GIS-T) (Shaw 2010). The restriction of QTCN to static
networks may however be regarded as a harmful limitation, given that in many cases
transportation networks can be more realistically modelled as dynamic environments with
time-varying properties. Chapter 4 tackles this issue through investigating how dynamic
networks would affect QTCN relations as well as their transitions, conceptual
neighbourhoods and compositions. These findings are embraced in the definition of a novel
calculus (QTCDN’) which differs from QTCN in its underpinning of dynamic networks able to
undergo topological changes. Although other types of changes may influence QTCN relations,
topological changes merit particular attention since they are the most fundamental changes
in a network, impinging on its structure of connected nodes and edges. It is worthwhile to
254 Chapter 12
Cap Main contributions
(objective 1)
Application-oriented contributions (objective 2)
Pa
rt
I
– M
ov
ing
ob
je
ct
s 2
Overview of QTC
CND and CRTs for QTCC
Extensions of QTC
Representing incomplete information
Traffic scene example
3
Formal definition of QTCN
Canonical cases for QTCN
Composition of QTCN relations
Transformation to RTCN relations
Police/gangster example
Collision avoidance applications
4 Definition of QTCDN’
CND for QTCDN’ Extension of QTCN to dynamic
networks
5
Conceptual model for QTC-based information systems
Implementation methodology
Implementation prototype
QTC-based information system
Traffic scene case study
Squash rally case study
6 Extension of sketch map ontology
Typology of MPO trajectory representations in sketch maps
Theoretical support for sketch-based information systems
Pa
rt
II
–
Tr
av
ell
ing
su
bj
ec
ts
7 Application of sequence alignment
methods to analyse patterns in Bluetooth tracking data
Empirical case study on visitors at a trade fair
8 Incorporation of uncertainty in STPs
Formal definition of rough obstacle-constrained STPs
Implementation of rough obstacle-constrained STPs
Example case study
9
Formal definition of reverse STPs
Formal definition and implementation of place-based accessibility measures derived from reverse STPs
User-friendly designed end-user GIS toolkit for measuring and mapping accessibility
Example case study
10 Procedure for scheduling service
opening hours to optimise individual accessibility
Empirical case study on accessibility of government offices
11
Generalisation of scheduling procedure of Chapter 10
Utilitarian, egalitarian and distributive scheduling approaches
Empirical case study on accessibility of municipal libraries
Table 12.1 – Main and application-oriented contributions.
remark that the topological changes modelled in QTCDN’ may not only represent physical
topological changes, such as those controlled by switches in railway networks. They may also
model virtual changes such as temporary disconnections of roads at a red traffic light (Ding
& Güting 2004), or temporary connections among different transport modes in multi-modal
transportation networks (Lysgaard 1992). The definition of QTCDN’ is a major step forward in
view of potential network analysis applications for QTC, also because nowadays information
systems such as routing services and monitoring systems increasingly adopt dynamic
networks (Jahn et al. 2005, Zografos & Androutsopoulos 2008, Köhler, Möhring & Skutella
Discussion and conclusions 255
2009). Although discrete topological changes support the modelling of many different types
of changes in dynamic networks, they are inappropriate to represent changes that affect
travel costs continuously. The effect on travel time of a growing traffic jam in a road
network, for example, may be more adequately modelled as a continuous function rather
than a discontinuous shift. This particularly concerns to QTC because of its consideration of
continuous space and time. Although this issue has already been briefly studied in earlier
work (Delafontaine et al. 2006, Bogaert & Delafontaine 2006), a dedicated extension of QTCN
to formally incorporate continuous network changes would be a significant complementary
step.
Although Chapters 3 and 4 have not explicitly addressed the implementation of network-
based QTC calculi in information systems, they at least provide the necessary theoretical
basis to support such operationalizations. The issue of implementing QTC is thoroughly
addressed in Chapter 5, presenting a conceptual data model for QTC-based information
systems, an application prototype (QTCAnalyst), and two illustrative case studies. Although
QTCAnalyst is confined to the basic (QTCB) and double-cross (QTCC) calculi, the proposed
conceptual model is generic for QTC, thus including the network-based calculi. Therefore,
the contributions of Chapter 5 have taken QTC from a purely theoretical formalism to the
point where it can be picked up by computer scientists, software engineers and information
system developers.
The use of QTC in applications has been highlighted in several discussions and case studies
throughout Chapters 2-5. Especially applications related to traffic scene analysis and
monitoring have been put forward (e.g. sections 2.8, 3.7.2, and 5.4.1). Importantly, even
though most traffic scenes are embedded in networks, the traffic scene examples in
Chapters 2 and 5 have been elaborated using non-network-constrained calculi. This reveals
how the complementarities among different QTC types may support a certain scaling ability
in the analysis. For example, while road networks may be modelled at the overall spatial
level by the graph-based sets of interconnected linear features adopted in QTCN and QTCDN’,
a two-dimensional constrained open space will be more appropriate for an analysis of car
interactions and driving behaviour at the level of individual roads and lanes. The scaling
ability of QTC could be extended further by defining even more types of QTC, thereby
considering increasingly finer categorisations of the quantity spaces that underlie the
qualitative relations, ultimately arriving at quantitative information in case of an infinite
discretisation. This idea, however, conflicts with the basic tenet of qualitative information
which intends to distinguish categories that are relevant and essential to the behaviour
being modelled (Cohn 1996). In QTC, simple {-,0,+} quantity spaces are adopted, and
therefore, QTC relations involve a fair degree of abstraction. Whether or not, and to what
extent this level of abstraction restricts the applicability of the calculus remains a pertinent
question which has only been answered partly in this dissertation.
256 Chapter 12
When reconsidering the first and second research questions of section 1.2, it is noted that,
while the adequacy of QTC to represent and reason about moving objects in realistic
scenarios has been discussed and illustrated in several applications, a profound study of the
added value of the calculus versus other (prevailing) formalisms, e.g. alternative qualitative
approaches such as those of Dylla et al. (2007), Wolter et al. (2007), Hornsby & King (2008),
and Pommerening, Wölfl & Westphal (2009), has been lacking in each of these cases. Given
the yet mentioned abstraction of QTC relations, one of the calculus’ plus-points may be its
robustness to handle large sets of moving objects, e.g. moving objects databases (MODs)
(Güting & Schneider 2005, Wolfson & Mena 2005, Revesz 2010). Yet, since each of the
examples in Part I has involved but a limited number of objects, an empirical assessment of
this robustness is still to be done. Also, and despite this potential robustness, it is
questionable whether a calculus of binary relations is suitable for reasoning about multiple
objects. Although the matrix representation proposed in section 2.7.1 offers an elegant
solution, it bypasses the question of whether reasoning about more than two moving
objects also requires relations that associate more than two objects.
Against the above implied opportunities for further elaboration, it is observed that some
theoretical results of Part I warrant weaknesses of the QTC calculus that should not be
denied. Among these are the weak results obtained from the composition of QTCN1 relations
(section 3.5), and the granularity dependence of stable relations (section 5.5(e)). An
important lesson to be learned from these deficiencies is that implementations such as
QTCAnalyst should not be regarded as stand-alone solutions. Rather they require to be
embedded as assistant tools which confront and integrate QTC results with information
obtained from other sources or formalisms. In this respect, an important opportunity to
underpin the further development of QTCAnalyst to one or more genuine and application-
specific end-user information systems consists of equipping the prototype with quantitative
information and operations. Quantitative information and methods are known to
complement their qualitative counterparts, and – given the limitations of QTCAnalyst listed
in section 5.5 – their integration may hence be inevitable in order to maximise the
application capabilities. For instance, when applying QTC for reasoning about large
databases of moving objects, such as cars in a city or visitors at a mass event, a function for a
priory selecting those objects that are possibly interacting with each other would be
recommended (cf. section 5.5(g)). Such a selection function cannot be obtained via QTC
alone. Instead, a quantitative distance measure could be introduced to implement such a
function. Quantitative methods would also be useful in support of more detailed posterior
analyses of relevant events, situations, or behaviours that have been a priory detected
through the analysis of QTC relations.
1 Even weaker composition results may be expected for QTCDN’ due to the relaxation of spatiotemporal
constraints in dynamic networks.
Discussion and conclusions 257
In addition and in contrast to the integration of quantitative methods, another worthwhile
opportunity to extent the QTCAnalyst prototype relates to the competence of qualitative
formalisms to handle incomplete or imperfect information (Kuipers 1994), in particular
information originating from human cognition or communication such as natural language
expressions and freehand drawings (Forbus 1997, Kuehne & Forbus 2002, Van de Weghe et
al. 2007). To this end, appropriate modalities to input and extract moving objects data
stemming from such sources are to be developed. In this respect, Chapter 6 contributes to
the support of sketch-based input, by extending the concept and ontology of freehand
sketch maps in order to represent moving point objects and their trajectories. A typology of
glyphs representing trajectories in sketch maps is introduced and some important links
between spatiotemporal characteristics of trajectories and such glyphs are revealed. These
are relevant in order to automate the interpretation of trajectories from sketches.
Information systems to reason about moving objects interpreted from sketch maps could be
useful to assist empirical evaluations of human sketching behaviour observed when
sketching about dynamic phenomena (cf. Blaser 2000), e.g. to substantiate an objective
categorisation of sketching behaviour, or to appraise human adequacy and skills in sketching
about moving objects and/or in understanding such sketches. Such systems are, among
others, relevant to the field of cognitive mapping and wayfinding (Golledge 1999), which
often relies on sketch maps in empirical studies, in order to enhance the interpretation of
human travelling behaviour (cf. Part II). However, the typology presented in Chapter 6 also
heralds some pitfalls which hamper a straightforward operationalization of information
systems involving sketch-based moving objects. Perhaps the most stringent of these is the
difficulty to represent objects that evolve continuously and simultaneously in time through
conventional freehand sketching. This shortcoming is also relevant in view of the possible
development of sketch-based input modalities for QTCAnalyst given the continuous
interactions which underlie QTC relations. One solution to this problem, left for future
research, may be the consideration of additional sketch content obtained from other
communication means apart from sketching s.s., such as speech and annotations.
Yet another avenue for further development lies in the extraction of information through
the analysis of conceptual animations or patterns of QTC relations, such as the overtake
events or squash rally patterns extracted in section 5.4. Conceptual animations may offer
insights into movement behaviour over a longer time window. For instance, drawing on the
cognitive aspects of the QTCC reference frame, which is based on Freksa’s (1992) double
cross, the observed relation patterns in the squash rally example (section 5.4.2) may actually
capture essential distinctions about the movement interaction perceived by both players
and by consequence reflect their perception and cognition during the course of the rally. If
the existence of such correlations can be empirically assessed, this would allow comparative
analyses of QTC animations across both games and players to be linked to the results of the
games and to the performance of the players, and in this respect for instance assist
258 Chapter 12
applications in sports performance analysis and management (Gratton & Jones 2009,
O’Donoghue 2009, Horne et al. 2011, Skinner & Edwards 2011). One notable technique to
consider in order to further elaborate the analysis of QTC animations is sequence alignment
(Morrison 2010). Alike qualitative calculi, sequence alignment methods (SAM) rely on a
discrete categorisation of quantitative information and they employ an explicit capability to
handle incomplete knowledge. Moreover, SAM may offer an interesting opportunity to
extract information across multiple objects in order to make inferences about individual QTC
animations.
Part II – Travelling subjects
The issue of SAM leads to the second part of this dissertation. In Chapter 7, SAM have been
applied as a data mining technique to detect behavioural patterns from Bluetooth tracking
data of visitors at a big indoor trade fair. The empirical results suggest that the approach is
suitable to discover knowledge about revealed space-time behaviour, for instance, to
underpin a typology of behavioural patterns. Furthermore, the case study demonstrates that
Bluetooth tracking is a powerful data acquisition method in this context which allows for the
anonymous and unannounced tracking of a mass of individuals. Although the positional and
temporal accuracy of Bluetooth sensing is limited, it is to some degree controllable and
perhaps preferable to other techniques, thereby considering that GPS signals are usually
obstructed in indoor environments. Chapter 7 shows that the limitations of Bluetooth
tracking data are partially offset by the theoretical ingenuity of SAM, which underlines the
usefulness of the approach. For example, to fill the gaps in the observation sequences of one
individual, SAM exploit the entire dataset of Bluetooth observations. A further validation of
the empirical results would however benefit from ongoing research regarding the
calibration, reliability, and scalability of SAM when applied to tracking data (Wilson 2006,
Shoval & Isaacson 2007).
Apart from its yet described strengths, some significant weaknesses of Bluetooth tracking
techniques are worthwhile discussing in the light of the third research question (section 1.2).
To begin with, apart from increasing the data acquisition accuracy and reducing data noise,
an important sampling problem has to be tackled. The uncertainty about the ratio of tracked
individuals to the total population considerably limits the interpretation and extrapolation of
results of analyses of Bluetooth observations. Given that the actual ratio of tracked devices
may well fluctuate in space and time as well as across different socio-economic groups of
individuals, the applicability of Bluetooth tracking experiments that do not include an
alternative reference data collection method is questionable. According to Girardin et al.
(2008), Bluetooth tracking experiments also suffer from a low scalability in terms of the
ability to deploy these across different spatial contexts. Another of their concerns relates to
the privacy and ethical issues of Bluetooth tracking experiments without individual’s consent
(see also Wong & Stajano 2005, Gutman & Stern 2007, Hay & Harle 2009). The risk of
Discussion and conclusions 259
identifying individuals or organisations is especially pertinent in view of the improvement of
the sensing accuracy. Furthermore, unannounced tracking experiments generally go hand in
hand with a lack of additional attributes such as socio-economical variables, which severely
reduces the potential for conducting empirical studies beyond investigations of strictly
spatiotemporal behaviour. Therefore, important research challenges lie ahead with respect
to combining and enriching Bluetooth tracking with other data collection techniques.
Bluetooth tracking data usually consists of discrete time-stamped observations of individuals
– or rather their devices – in the neighbourhood of Bluetooth sensors. In many cases, the
chronological path of Bluetooth observations sampled from one individual cannot be
interpolated to a representative continuous trajectory, which precludes the derivation of
higher motion attributes such as motion azimuth and velocity, and by consequence
conducting analyses building on these attributes, such as reasoning with QTC. One way to
cope with this shortcoming is to consider potential paths. In other words, while failing to
delineate an individual’s precise trajectory (i.e. revealed behaviour), we may focus on where
(s)he could (not) have been given the information at hand (i.e. potential behaviour).
Chapter 8 offers an original approach to enable the analysis of potential behaviour starting
from raw tracking data. To this end, the classical time geographical concept of a space-time
prism (STP) is enriched in two ways. First, since each data acquisition method involves a
certain amount of uncertainty stemming from spatial and temporal inaccuracies, errors and
failures, noise, etc., the traditional STP is extended to account for the uncertainty of sampled
anchor points. Drawing on the principles of rough set theory, this uncertainty is accounted
for through the definition of rough STPs. Second, to overcome the restriction of
conventional STPs to model travel in isotropic unconstrained spaces, these are embedded in
a travel environment constrained by discrete impassable obstacles.
The consideration of obstacle-constrained environments goes against the tendency to model
STPs in network-constrained environments (e.g. Neutens et al. 2008b, Miller & Bridwell
2009, Kuijpers & Othman 2009, Kuijpers et al. 2010). Actually, it complements this tendency
because network-based models are inappropriate to handle constrained open spaces such as
public parks and gardens, golf courses, or beaches. Also, more than network-based
frameworks, the obstacle-constrained worldview supports analyses at a micro level spatial
scale. Especially when zooming in on dense urban and built areas, many environments,
including indoor locations (e.g. the trade fair in Chapter 7), may be perceived as accessible
spaces populated with inaccessible obstacles. The combination of both rough and obstacle-
constrained STPs affords analysts the valuable potential to study potential behaviour starting
from tracking data collected within obstacle-constrained environments. Not only does
Chapter 8 provide a formal definition for rough obstacle-constrained STPs, an algorithm to
automatically generate these prisms given a set of individual sample points, uncertainty
parameters, and obstacles is developed and illustrated as well. Hence, beyond theoretically
260 Chapter 12
contributing to time geography, the chapter offers a ready-made procedure which opens the
door for empirical investigations of potential behaviour.
One theme of empirical studies where the analysis of potential behaviour by definition plays
a key role is accessibility. Accessibility relates to the ease with which people can conduct
activities in time and across space. Notwithstanding the central role of persons, accessibility
is usually measured as an attribute of locations (place-based perspective) in lieu of
individuals (person-based perspective) (Neutens et al. 2010). Chapter 9 adds to the
incorporation of potential behaviour in place-based assessments of accessibility to services.
Through the introduction of location-centred STPs – in lieu of conventional STPs which join
individual anchor points – a theoretical basis is formed for measuring and mapping place-
based accessibility while reckoning with person-based constraints, albeit universally
postulated constraints rather than genuine individualised constraints. More than that,
founded on this framework, a toolkit PrismMapper, which implements some complementary
network-based accessibility measures in a GIS, is developed and released in order to offer
scientists, planners and decision makers the ability to employ these novel place-based
measures for a GIS-based evaluation of accessibility.
Given that, on the one hand, place-based approaches have hitherto been by far the most
frequently applied in accessibility evaluations, and that, on the other hand, this trend is
being heavily criticised in scientific literature from a person-based perspective (Miller 2007,
Kwan 2009), the contributions of Chapter 9 – the toolkit in particular – are unique in bridging
both perspectives. Also, PrismMapper has been designed with specific attention to
transparency, user-friendliness, employability, and comprehensibility with respect to non-
technically oriented end-users who may not be acquainted with the theoretical background
of accessibility research. The toolkit thereby copes with the disadvantages of related tools
which implement procedures that are often obscure to the user, and/or which settings and
results are too complex to handle by non-specialists, such as the benefit values produced by
the tools of Miller & Wu (2000) or Neutens, Versichele & Schwanen (2010).
Chapter 9 tackles another persistent shortcoming of place-based evaluations of accessibility:
the negligence of temporal constraints on the opportunities to be accessed. Services, for
example, are only delivered within a regime of well-chosen opening hours. This is illustrated
in a case study where it is shown how the possibilities to pay an evening library visit in Ghent
(Belgium) differ strongly between different days of the week. Chapter 10 enables a more
systematic investigation of the gravity of overlooking the effects of service opening hours on
accessibility. It is examined and demonstrated how the accessibility of services to individuals
largely depends on their opening hour schedules and how these timetables can be
manipulated in order to optimise the absolute level of accessibility. In an empirical case
study on the accessibility of government offices to citizens in Ghent, the approach is shown
to be successful in increasing the average level of individual accessibility drastically.
Discussion and conclusions 261
In Chapter 11, the optimisation procedure to distribute opening hours among service
facilities in function of individual accessibility has been generalised in support of any
function of accessibility, rather than solely the absolute level of accessibility cumulated
across the population. Thus, a generic approach is obtained for allocating opening hours in
function of other aspects related to individual accessibility. In the case of Chapter 11, the
aspect under scrutiny is the equity of accessibility across individuals, which is an important
issue within the evaluation of social exclusion (Cass, Shove & Urry 2005). Three different
scheduling approaches are put forward according to three different equity principles and
these are implemented in an empirical case study concerning the accessibility of public
libraries in Ghent. The results illustrate how the rescheduling of library opening hours can be
an instrument to effectively control the (un)equity of accessibility among individuals. These
outcomes may assist library policies that focus on attracting specific social groups or on
increasing the average number of book issues per capita. A striking outcome of the study is
that, through rescheduling, both the absolute level of individual accessibility as well as the
equity of its distribution across individuals can be increased considerably whilst preserving
or even reducing the total amount of allocated service opening hours. The accomplishment
of this threefold condition is in particular appealing to public authorities, planners and
decision makers who are expected to fulfil this triple aim, particularly in the context of
austerity measures including budget cuts and service declination (e.g. Muir & Douglas 2001).
Beyond the application potential of the procedures developed in Chapters 10 and 11, these
chapters pioneer an innovative research direction, considering that comparable detailed
investigations of the effects of opening hours on person-based accessibility are – to our
knowledge – non-existing. Obviously, this novel line of research may be deepened further in
many respects. To this end, in order to better attune the scheduling procedure and in line
with the generalisation made in Chapter 11, many opportunities consist of incorporating
other aspects that relate to accessibility on the one hand or by including additional effects of
changes in opening hours on the other hand. For instance, one could explore the
consequences of rescheduling on the uncoupling and temporal fragmentation of everyday
human activities (Schwanen, Dijst & Kwan 2008, Hubers, Schwanen & Dijst 2008), intra-
household relations (Schwanen, Ettema & Timmermans 2007, Schwanen, Kwan & Ren 2008,
Schwanen & de Jong 2008), joint accessibility (Neutens et al. 2008a), etc. To reinforce the
practicability of the scheduling procedures elaborated in Chapters 10 and 11, a pertinent
challenge lies in obtaining an objective estimation of the feasibility of modifications to
opening hour regimes from the viewpoint of service suppliers, as well as service users. From
the supply side, the scheduling algorithms should account for the costs and benefits related
to each specific candidate regime, including a trade-off with individual accessibility. From the
demand side, on the other hand, challenges lie in estimating a population’s adaptability to
opening hour modifications as well as in integrating individual desirability for certain
(combinations of) opening hours. The latter aspect would benefit from the consideration of
262 Chapter 12
probabilistic models of human behaviour such as random utility models (Manski 1977,
Cascetta 2009), Markov chains (Brémaud 1999), and stochastic frontiers (Kumbhakar & Knox
Lovell 2000).
In view of the common focus of Chapters 9-11, a more general discussion on accessibility
measures is apposite here. Although Chapter 9 seeks to combine the best of both place- and
person-based accessibility measures, sides have to be taken in the end, given the
fundamental discordance between both approaches. In this case, place-based measures
have been obtained. Yet, this outcome is not indiscriminate given PrismMapper’s target
audience of planners, decision makers and authorities, who employ practically exclusively
place-based approaches to date (e.g. see the indicators included in the Stadsmonitor 2008
report, available on the Flemish urban policy website http://www.thuisindestad.be). Yet, the
lack of person-based accessibility measures in applied science provokes thought, since these
are believed to complement place-based measures and to articulate personal and social
differences in accessibility much more nuanced (Kwan 1998, Neutens et al. 2010). However,
little discussion has addressed the reasons for this hiatus. As already argued in Chapter 9,
these may be found in two interrelated aspects: (i) the absence of appropriate data sources
or the difficulty to acquire these, and (ii) their reliability and representativeness. The
importance of the latter aspect may not be underestimated, in particular vis-à-vis decision
makers or authorities responsible for entire spatial districts and their inhabitants. While
Chapters 10 and 11 provide substantial insights on person-based accessibility, it remains
undetermined whether the behaviour extracted from the pooled activity-travel dairy
samples underlying their analyses are representative for the behaviour of Ghent’s citizens,
and by consequence which lessons the city council of Ghent should draw from such analyses.
A straightforward answer to the question of representativeness is hampered by the fact that
individual behaviour is highly complex and in many cases unpredictable, if not indeterminate
(Cziko 1989) on the one hand, and by its unstableness and changeability in time on the
other. This variability is inconvenient in empirical studies with eye on long-term planning and
policies, which explains their appeal for place-based approaches which outcomes can be
considered far less a product of time. From the viewpoint of the individual, this
inconvenience may however be to a certain extent unjust, in the sense that person-based
approaches call for a change of mentality towards a dynamic and individualized planning of
service delivery, rather than a static permanent operating regime at fixed locations and
regular times, as is and will be increasingly supported by location-aware technologies and
ICTs in nowadays and future information societies.
Conclusion
In conclusion, it is noticed that the modelling and analysis of moving objects and travelling
subjects has resulted in a broad and fairly diverse, yet fascinating assembly of original
contributions with respect to both theoretical and application-oriented research. The
Discussion and conclusions 263
contribution of this dissertation cuts across fundamental models as well as empirical
findings, including results ranging from theorem proofs and formal axiomatisations to clear-
cut end-user deliverables such as PrismMapper. Existing frameworks and common research
practices have been adapted or extended, novel methods have been proposed and
implemented, and these efforts have been illustrated in a wide variety of conceptual
examples and empirical case studies offering important insights and lessons to be learned.
The merit of this dissertation is therefore believed to be in its application-oriented
achievements which open up the further continuation and application of its outcomes.
A final remark concerns the observation that each of the research questions addressed in
this work has been answered only partially. In part, this is due to the fact that alternative
answers to these comprehensive questions already exist in science. In that sense, this
dissertation has contributed by closing some well-chosen gaps in GIScience and related
fields. On the other hand, numerous unresolved issues leave room for improvement and
open avenues for further investigation. Since much of the research on analysing moving
entities is driven by technological progress (e.g. positioning systems), new answers may be
searched for in that direction as well. One remarkable challenge in this respect – perhaps the
most exciting one – lies in a profound integration of information stemming from different
data collection technologies. In the context of travelling individuals such an integration may
encompass technologies such as, among others, satellite positioning systems (e.g. GPS),
wireless communication technologies (e.g. Wi-Fi, Bluetooth, IrDA, NFC, RFID, GPRS, UMTS),
nanotechnology, microchip implants, camera surveillance and monitoring systems, and
automated ticketing, counting, and money tracking systems (e.g. see Shlesinger 2006). Such
multi-technological approaches have the potential to revolutionize both the spatial and
temporal scalability of tracking experiments and consequently increase the abilities to
capture the longitudinal travelling behaviour of a large number of individuals across diverse
environments (e.g. indoor and outdoor) and geographical scales. On the other side of the
medal, a far-reaching integration and scalability of tracking data may come at the prize of
location privacy (Duckham 2009) and will therefore require sufficient legislative adaptations
as well. Yet, this evolution is already palpable through the recent trends of neogeography
(Turner 2006, Graham 2010) and volunteered geographic information (Goodchild 2007,
Goodchild 2010), and ambient intelligence (Weber, Rabaey & Aarts 2005, Mikulecký et al.
2009, Augusto et al. 2010).
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268
Samenvatting (Dutch summary)
Bewegingen maken inherent deel uit van het gedrag van mensen, dieren, goederen en
gegevens. Bewegende entiteiten vormen daarom een fundamentele onderzoekseenheid in
heel wat wetenschappelijke disciplines, zoals artificiële intelligentie, gedragswetenschappen,
ethologie, geografische informatiewetenschap, robotica, sportwetenschappen en verkeer-
en vervoerslogistiek. De wetenschappelijke interesse in onderzoek over bewegende
objecten is de laatste decennia sterk aangezwengeld, voornamelijk omwille van
technologische vooruitgang. Enerzijds hebben steeds geavanceerdere transportmiddelen
ertoe geleid dat een groeiend volume aan entiteiten met toenemende snelheden over
grotere afstanden kan worden verplaatst. Anderzijds heeft de belangrijke progressie in de
ontwikkeling van positionering- en trackingsystemen zoals GPS de mogelijkheden tot het
verzamelen van gedetailleerde gegevens omtrent bewegende objecten in aanzienlijke mate
vergroot. Verder biedt de evolutie in informatie- en communicatietechnologie (ICT)
onderzoekers in ijl tempo de nodige ondersteuning aangaande het opslaan, beheren,
bevragen, verwerken en communiceren van alsmaar grotere datavolumes.
Dit proefschrift tracht een bijdrage te leveren aan het wetenschappelijk onderzoek dat
verband houdt met het modelleren en analyseren van bewegende objecten in het algemeen
en specifiek binnen het kader van de geografische informatiewetenschap. Geografische
informatiewetenschap (GIW) vormt de wetenschappelijke ruggengraat van geografische
informatiesystemen (GIS) en richt zich op het ontwikkelen en toepassen van theorieën,
methodes, technologieën en gegevens om geografische processen, relaties en patronen te
begrijpen (Mark 2003). Aan het onderzoek in GIW met betrekking tot bewegende objecten
zijn twee algemenere onderzoekstendensen voorafgegaan. Aangezien deze tendensen voor
een stuk tot op heden doorlopen, doorspekken zij in verschillende opzichten de inhoud van
dit werk. Een eerste trend houdt verband met het tijdruimtelijke fenomeen dat beweging is.
Het modelleren en analyseren van bewegende objecten omvat namelijk naast ruimtelijke
ook belangrijke temporele aspecten. Daarom volgt zij de strekking binnen GIW die zich heeft
gericht op de theoretische omkadering van de temporele component in geografische
informatie en de bijhorende ondersteuning van tijdruimtelijk GIS. De tweede
vermeldenswaardige verschuiving is ontsproten aan de vaststelling dat modellen, methodes
en technieken voor het bestuderen van individuen en hun gedrag binnen GIW doorgaans
een plaatsgebaseerde aanpak hanteren en zo voorbijgaan aan essentiële kenmerken van
personen (Miller 2007). Men gaat bijvoorbeeld onrechtstreeks het individu herleiden tot een
bepaalde – zogenaamd representatieve – locatie zoals zijn/haar woonplaats en maakt op die
manier een voor vele toepassingen onaanvaardbare abstractie van zijn/haar tijdruimtelijk
gedrag. Deze gedachte heeft in GIW een mentaliteitswijziging teweeggebracht van
plaatsgebaseerde naar persoonsgebaseerde benaderingen waarin individuen centraal staan.
Diezelfde gedachtegang is onder andere weerspiegeld in de onderverdeling van dit werk.
Samenvatting (Dutch Summary) 269
Deze doctoraatsverhandeling bestaat uit een compilatie van tien manuscripten
(Hoofdstukken 2-11) voorafgegaan door een algemene inleiding (Hoofdstuk 1) en afgesloten
met een concluderende discussie (Hoofdstuk 12). Elk manuscript is aan een internationaal
peer review proces onderworpen en in een internationaal wetenschappelijk tijdschrift of
boek verschenen, of ter publicatie voorgelegd. De manuscripten zijn ontwikkeld met het oog
op twee algemene objectieven:
Objectief 1 Een originele wetenschappelijke bijdrage leveren en in het bijzonder een
bijdrage tot de mogelijkheden die GIW biedt tot het modelleren en
analyseren van bewegende objecten en personen die zich verplaatsen.
Objectief 2 Bijdragen tot of verbetering brengen in het in praktijk brengen of inzetbaar
maken van bestaand theoretisch werk zodoende de kloof tussen wetenschap
en technologie en tussen theorie en praktijk te helpen dichten.
Het eerste objectief omschrijft de focus van dit werk en impliceert tevens de vereisten die de
redacteurs en uitgevers van de opgenomen manuscripten vooropstellen met betrekking tot
de originaliteit en toegevoegde waarde daarvan. Het tweede objectief vloeit voort uit het
gevaar dat elke wetenschap loopt om losgekoppeld te raken van de toepassingen en
technologieën die ze hoort te ondersteunen en zich te verliezen in een louter theoretisch
bestaan. Deze verhandeling poogt de kloof die aldus kan ontstaan, in dit geval specifiek die
tussen GIW en GIS, te overbruggen en schenkt daarom bijzondere aandacht aan het
implementeren en ontwikkelen van applicaties.
Een onderscheid werd gemaakt tussen de hoofdstukken die handelen over bewegende
objecten in het algemeen (Deel I, Hoofdstukken 2-6) en de hoofdstukken die betrekking
hebben op personen die zich verplaatsen (Deel II, Hoofdstukken, 7-11). Deel I beschouwt
bewegende objecten in hun meest elementaire vorm: entiteiten waarvan de positie en/of
geometrische kenmerken doorheen de tijd veranderen. In dit opzicht worden zij in Deel I,
zoals gebruikelijk in GIW, gemodelleerd als bewegende puntenobjecten. Deze kernachtige
conceptualisatie laat toe om abstractie te maken van complexere geometrieën die in
analyses vaak irrelevant zijn en bovendien vanuit algoritmisch oogpunt aanzienlijk minder
performant.
Het eerste deel richt zich tot het kwalitatief voorstellen van en redeneren over bewegende
objecten. Kwalitatief redeneren is een traditioneel onderzoeksdomein in artificiële
intelligentie dat later in GIW zijn intrede heeft gemaakt en zich richt op het ontwikkelen van
informatiesystemen die in staat zijn te redeneren over fysieke systemen zonder daarbij op
precieze kwantitatieve informatie te berusten (Weld & de Kleer 1989). Een van de
technieken die daartoe kan leiden is het opstellen van een kwalitatieve calculus, hetgeen
omschreven kan worden als een verzameling van paarsgewijs verschillende en onderling
exclusieve relaties en een verzameling bewerkingen om over deze relaties te redeneren
270
(Ligozat & Renz 2004). Hoofdstukken 2 tot 5 zijn toegewijd aan een specifieke kwalitatieve
calculus, namelijk de Qualitative Trajectory Calculus (QTC) (Van de Weghe 2004) die relaties
tussen twee bewegende puntobjecten definieert. Hoofdstuk 2 (Delafontaine et al. 2011b)
biedt een theoretisch overzicht van alle fundamentele types van QTC en bespreekt hoe de
calculus kan worden uitgebreid en hoe ze kan omgaan met onvolledige informatie en met
ruwe trajectgegevens. Hoofdstukken 3 en 4 werken een bepaald type van QTC, namelijk de
Qualitative Trajectory Calculus on Networks (QTCN) verder uit. QTCN beschouwt relaties
tussen objecten die bewegen in netwerken, zoals auto’s in wegennetwerken. In Hoofdstuk 3
(Delafontaine et al. 2011a) wordt QTCN formeel gedefinieerd en wordt nagegaan in welke
mate de calculus in staat is kwalitatieve vragen te beantwoorden. Hoofdstuk 4 (Delafontaine
et al. 2008) breidt QTCN uit naar dynamische netwerken, meer bepaald naar netwerken die
topologische veranderingen kunnen ondergaan. Deze uitbreiding is zinvol om het maken of
verbreken van fysieke en/of virtuele connecties te modelleren, zoals het tijdelijk blokkeren
van een weg aan een slagboom. Hoofdstuk 5 behandelt de implementatie van QTC in een
informatiesysteem. Daarbij wordt QTCAnalyst, een prototype van een op QTC gebaseerd
informatiesysteem, ontwikkeld en geïllustreerd aan de hand van twee gevalstudies.
Hoofdstuk 6 gaat dieper in op één van mogelijkheden om QTCAnalyst verder uit te breiden.
Aangezien het algemeen aanvaard is dat kwalitatieve meer dan kwantitatieve methodieken
overeenstemmen met de manier waarop mensen redeneren en communiceren, loont het de
moeite QTCAnalyst uit te breiden naar gegevens die direct uit menselijke communicatie
voortkomen en/of de menselijke perceptie weerspiegelen. Het ondersteunen van manueel
geschetste trajecten is één van die mogelijkheden. In Hoofdstuk 6 (Delafontaine & Van de
Weghe 2009) wordt onderzocht hoe bewegende objecten op basis van ingeschetste
trajecten in informatiesystemen kunnen worden gemodelleerd.
In het tweede deel wordt dieper ingegaan op het verplaatsingsgedrag van individuen.
Daarbij worden twee perspectieven gehanteerd. Enerzijds wordt het geobserveerd gedrag
van personen geanalyseerd. Daartoe worden in Hoofdstuk 7 twee onderzoekslijnen met
succes gecombineerd: op tracking gegevens verzameld door middel van Bluetooth sensoren
wordt sequence alignment – een techniek overgenomen uit de bio-informatica (Morrison
2010) – toegepast, zodoende gedragspatronen in kaart te brengen. Een van de nadelen van
Bluetooth gegevens is echter dat zij in de meeste gevallen niet toelaten een nauwkeurig
continu tijdruimtelijk traject van een individu af te leiden. Mogelijke individuele trajecten
kunnen daarentegen wel worden geëxtraheerd. Daarom is verder aandacht besteed aan het
modelleren en analyseren van potentieel gedrag. In Hoofdstuk 8 wordt het klassieke concept
van een tijd-ruimte prisma uit de tijdgeografie (Hägerstrand 1970) uitgebreid, zodoende een
analytisch kader te creëren dat toelaat op basis van discrete observaties het potentieel
verplaatsingsgedrag van individuen binnen open, door obstakels begrensde ruimtes te
analyseren. Deze uitbreiding is complementair ten opzichte van eerdere implementaties van
tijd-ruimte prisma’s die voornamelijk op netwerk omgevingen waren geënt. Ook Hoofdstuk 9
Samenvatting (Dutch Summary) 271
gaat dieper in op potentieel gedrag, namelijk op het aspect bereikbaarheid dat daarmee
verband houdt, maar dat desalniettemin het vaakst op een plaatsgebaseerde manier wordt
gemeten (bv. hoeveel winkels liggen er in een straal van 1 km van plaats x?). Tegenover
persoonsgebaseerde bereikbaarheidsindicatoren hebben plaatsgebaseerde indicatoren wel
het belangrijke voordeel dat ze in een gebiedsdekkende kaart kunnen worden voorgesteld.
Daarom wordt in Hoofdstuk 9 getracht, op basis van alweer een aangepast tijd-ruimte
prisma, plaatsgebaseerde indicatoren te verrijken met essentiële persoonsgebaseerde
kenmerken. Deze innovatieve bereikbaarheidsindicatoren worden dan geïmplementeerd in
een GIS-applicatie PrismMapper zodoende hun berekening en kartering te automatiseren,
hetgeen vervolgens in een gevalsstudie wordt geïllustreerd. PrismMapper is in de eerste
plaats bedoeld voor eindgebruikers zoals planologen, beleidsmakers, enz. en is daartoe heel
bewust op een gebruiksvriendelijke en bevattelijke manier ontworpen. Een andere
meerwaarde tegenover gerelateerde plaatsgebaseerde applicaties is dat PrismMapper in het
evalueren van de bereikbaarheid van voorzieningen rekening houdt met hun openingsuren.
Hoofdstukken 10 en 11 werken dit vernieuwende aspect verder uit in een systematische
studie over de effecten van verschillende openingsuren op individuele bereikbaarheid. In
Hoofdstuk 10 wordt een algoritme opgesteld dat toelaat een bepaalde hoeveelheid
openingsuren te verdelen over een gegeven aantal voorzieningen om zo hun bereikbaarheid
ten aanzien van een gegeven populatie te maximaliseren. De procedure wordt daarna
toegepast in een uitgebreide gevalsstudie betreffende de bereikbaarheid van
gemeentekantoren in Gent (België). In Hoofdstuk 11 ten slotte, wordt die procedure
gegeneraliseerd naar de maximalisatie van eender welke functie van individuele
bereikbaarheid en aansluitend toegepast op drie functies die overeenstemmen met
verschillende gelijkheidsprincipes ten aanzien van de verdeling van individuele
bereikbaarheid binnen de populatie. Deze drie benaderingen worden achtereenvolgens
geïllustreerd in een empirische studie over de bereikbaarheid van bibliotheken in Gent.
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Biographical sketch 273
Biographical sketch
Matthias Delafontaine was born on April 20, 1984 in Bruges
(Belgium). After graduating from high school at Sint-Jozefsinstituut-
College in Torhout (Belgium) in 2002, he started his academic
education in geography and geomatics at Ghent University where
he obtained his Master’s degree magna cum laude in 2006. In the
same year he joined the Department of Geography (Ghent
University) and started working on various scientific projects in GIS,
cartography, and spatial planning. In 2007, he received a doctoral
grant of the Research Foundation – Flanders and became a PhD
student. As an assistant teacher, he was involved in the courses of
geographical information science, GIS programming, spatial
analysis, map algebra and geostatistics. In 2011, he obtained a
postgraduate degree from the Doctoral School of Natural Sciences
(Ghent University). Matthias has participated in many major
international conferences and symposia and he is the author of
several publications in leading international journals in geography,
geographical information science and GIS.