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Regionalization of Information Space with Adaptive Voronoi
Diagrams
René F. ReitsmaDept. of Accounting, Finance & Inf. Mgt.
Oregon State University
Stanislaw TrubinDept. of Electrical Engineering and Computer Science
Oregon State University
Saurabh SethiaDept. of Electrical Engineering and Computer Science
Oregon State University
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Regionalization of Information Space with Adaptive Voronoi Diagrams
Information space: contents & usage. Pick or infer a spatialization? Loglinear/multidimensional scaling approach. Regionalization based on distance: Voronoi Diagram. Regionalization based on area: Inverse/Adaptive Voronoi
Diagram. Conclusion and discussion.
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Information Space
Dodge & Kitchin (2001) Mapping Cyberspace. Dodge & Kitchin (2001) Atlas of Cyberspace. Chen (1999) Information Visualization and Virtual
Environments. J. of the Am. Soc. for Inf. Sc. & Techn. (JASIST). ACM Transactions/Communications. Annals AAG: Couclelis, Buttenfield & Fabrikant, etc. IEEE INTERNET COMPUTING. INFOVIS Conferences.
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Information Space - Analog Approaches
Cox & Patterson (National Center for Supercomputing Applications - Cox & Patterson (National Center for Supercomputing Applications - NCSA) (1991) Visualization of NSFNET trafficNCSA) (1991) Visualization of NSFNET traffic
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Information Space - Analog Approaches
Card, Robertson & York (Xerox) (1996) WebBookCard, Robertson & York (Xerox) (1996) WebBook
ContentUsage
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Information Space - Other A Priori Approaches
WebMap Technologies WebMap Technologies
ContentUsage
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Information Space - Other A Priori Approaches
SOM: Kohonen, Chen, et al.
ContentUsage
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Information Space - Other A Priori Approaches
Inxight hyberbolic web site map viewer
ContentUsage
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Information Space - Other A Priori Approaches
Chi (2002)Chi (2002)
ContentUsage
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Information Space - A Posteriori Approaches
Infer or resolve geometry (dimension & metric) from secondary data using ordination techniques:
Factorial techniques. Vector space models. Multidimensional scaling. Spring models.
Sources of secondary data: Content. Relationships (structure). Navigational records.
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Buttenfield/Reitsma Proposal Distance is inversely proportional to traffic volume. Observed data are noisy manifestation of a stable process.
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Building as a Learning Tool (BLT)
Can this space be regionalized? If so, how?
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Criteria for Regionalization
Define our points as 'generators.'
Distance point of view:
Nongenerator points get allocated to the closest generator --> Voronoi Diagram.
Area point of view:
Generators have claims on the surrounding space --> Inverse Voronoi Diagram.
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Voronoi Diagram Regionalization Based on Distance
Okabe A., Boots, B., Sugihara, K., Chiu,S.N. (2000) Spatial Tesselations; Wiley Series in Probability and Statistics.
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Voronoi Diagrams
Honeycombs are regionalizations. Regularly spaced 'generators.' Coverage is inclusive. Mimimum material, maximum
area. Minimum generator distance.
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Ordinary Voronoi Diagrams
Vi = {x | d(x, i) d(x, j) , i j}
Thiessen Polygons. Bisectors are lines of
equilibrium. Bisectors are straight lines. Bisectors are perpendicular to
the lines connecting the generators.
Bisectors intersect the lines connecting the generators exactly half-way.
Three bisectors meet in a point.
Exterior regions go to infinity.
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Ordinary Voronoi Diagrams
Vi = {x | d(x, i) d(x, j) , i j} is a special case:
Assignment (static) view: Distance (friction) is uniform in all directions for all
generators.
Growth (dynamic) view: All generators grow their regions at the same rate. All generators start growing at the same time. Growth is uniform in all directions.
Boots (1980) Economic Geography: Weighted versions “produce patterns which are free of the
peculiar and, in an empirical sense, unrealistic characteristics of patterns created by the Thiessen polygon model.”
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Weighted Voronoi Diagrams
Multiplicatively Weighted Voronoi Diagram:
Vi = {x | d(x, i)/wi d(x, j)/wj , i j}
wi = wj ==> Ordinary Voronoi Diagram.
wi wj:
Static View: distance friction i distance friction j.
Dynamic View: generators start growing at the same time, but grow at different rates.
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WeightedVoronoi Diagrams Cont.'d
Multiplicatively Weighted Voronoi Diagram: Vi = {x | d(x, i)/wi d(x, j)/wj , i j}
Bisectors are lines of equilibrium.
Bisectors become curved when wi wj.
Bisectors divide the lines connecting generators i and j in portions wi/(wi + wj) and wj/(wi + wj).
Low weight regions get surrounded by high weight regions.
Highest weight region goes to infinity (surrounds all others).
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Weighted Voronoi Diagrams Cont.'d
Bisectors are Appolonius Circles: “Set of all points whose distances from two fixed points are in a constant ratio” (Durell, 1928).
(j – q) / (i – q) = (j – p) / (p - i) = wj / wi = 5
q cannot be -p = -1 as (j – q) / (i – q) = (6 - -1) / (0 - -1) = 7 5
(6 – q) / –q = 5 ==> q = -1.5
As wj increases, p decreases, q increases ==> hence, i's (circular) region gets smaller.
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Weighted Voronoi Diagrams Cont.'d
Other weighting schemes:
Additively Weighted: Vi = {x | d(x, i) - wi d(x, j) - wj , i j}
Generators grow at identical rates but start growing at different times.
Bisectors are hyperboles.
Compoundly Weighted: Vi = {x | d(x, i)/wi1 - wi2 d(x, j)/wj1 - wj2 , i j}
Power Diagram: Vi = {x | d(x, i)p- wi d(x, j)p - wj , i j}
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Weighted Voronoi Diagrams Cont.'d
Applications in Geography: Huff, D. (1973) Delineation of a National System of Planning
Regions on the Basis of Urban Spheres of Influence; Regional Studies; 7; 323-329.
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Inverse Voronoi Diagrams
Voronoi Diagrams:
Based on distance: Area = f(position, weight).
Peripheral generators claim peripheral space. Landlocking.
Based on area: Generator regions have areas proportional to a(ny) given
variable. Space is uniform; i.e., distance friction is uniform in all
directions. Weight = f(position, area). Inverse Voronoi diagram.
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Inverse Voronoi Diagrams Cont.'d
MWVD is a nice starting point:
Multiplicity reflects multiplicity in area.
Distance friction is uniform in all directions ==> concentric allocation.
By increasing weights landlocked generators can 'escape.'
However:
Weights represent distance rather than area.
Area proportionality requires bounding polygon.
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Adaptive MW Voronoi Diagram
Weight = f(position, area)
Let Ai = target area of generator i (prop.).
Let ai,j = allocated area of generator i (prop.) after iteration j.
Objective function: minimize Ai - ai,j
Let wi,j = weight of generator i at iteration j.
wi,0 = Ai
wi,j+1 = wi,j + w
wi,j+1 = wi,j (1 + k(Ai - ai,j))
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Adaptive MW Voronoi Diagram
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Adaptive MW Voronoi Diagram
Summary:
Interest in information space visualization.
LLM/MDS method provides dimensionality, location and a measure of size or 'force' (od).
MW Voronoi diagrams provide a good 'multiplicative' starting point but area = f(position, distance).
AMW Voronoi diagrams can solve for weights = f(position, area).
Applies to dimensionalities > 2.
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Some issues...
• How to select k in wi,j+1
= wi,j (1 + k(A
i - a
i,j))?
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Any Applicability to the World?
Search-and-rescue? Crop dusting and harvesting? Others?