lecture 1: introduction to spatial point processes · 1.longleaf dataset (r package spatstat)....

Spatial datasets Definition and theoretical characterization Moment measures and intensity functions

Lecture 1: Introduction to spatial pointprocesses

Jean-François Coeurjollyhttp://www-ljk.imag.fr/membres/Jean-Francois.Coeurjolly/

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Description of the course

• This introduction is divided into 4 lectures :1. Introduction : examples, general definitions.2. Poisson point process.3. Summary statistics.4. Models for spatial point processes.

• Material : slides, datasets, R instructions used for thelectures can be found athttp://www-ljk.imag.fr/membres/Jean-

Francois.Coeurjolly/

• You are asked to prepare some of the theoretical exercisesand practical exercises (using the R software) at home.

• You are also asked to read a research paper (to be givennext week) and prepare a one-page summary of the paper(either in French or English).

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Special thanks to . . .

the Danish team : Jesper Mølle, Ege Rubak and RasmusWaagepetersen

and Adrian Baddeley (Australia), Rémy Drouilhet (Grenoble) andYongtao Guan (USA)

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Examples of spatial point pattern data

General definitions and characterization

Moment measures and intensity functions

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Spatial data . . .

. . . can be roughly and mainly classified into three categories :

1. Geostatistical data : modelling of a discrete/continuous(vector) real-valued random variable observed at fixedlocations of a continuous space.

2. Lattice data : modelling of a discrete/continuous (vector)real-valued random variable observed at fixed locations of adiscrete space.

3. Spatial point pattern : modelling of random locations ofpoints (or objects) observed on a continuous space.

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Geostatistical data• sic.100 dataset (R package geoR) : cumulative rainfall in

Switzerland the 8th May.

• The observation consists in a discretized sample path of arandom field, X = (Xu , u ∈ R2),Xu ∈ R.

• Scientific questions : spatial correlation modelling (covariance,variogram), trend and variogram estimation, spatial prediction =kriging.

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Lattice data

• Lightning density =number of lightning strikesper square kilometer peryear in 2013.

• The data are aggregated bydepartment ⇒ randomfield on a network,X = (Xu , u ∈ G),Xu ∈ Rwhere G is a graph.

• Scientific questions : spatial neighborhood correlation, spatialhomogeneity or inhomogeneity, discrete spatial modelling,. . .

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Spatial point pattern : forestry datasets• Observation of 65, 71 and 126 japanese, swedish and finnish pines

on W = [5.7, 5.7]2, [−5, 5] × [−8, 2] and [0, 9.6] × [0, 10] (in m).• The locations and the number of points are random !• The state space is denoted by S = R2 and is equipped with ‖ · ‖.

japanese pines

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Scientific questions :

• Understanding of the arrangement of the points : is thearrangement spatially homogeneous or inhomogeneous.

• Are the points independent one from each other (CSR) or dothey exhibit a particular structure (clustered or repulsion) ?

• Among the three species, which one has the highest intensity ? Isthis difference significant ? 8 / 19


Spatial point pattern : earth quakes dataset

• R package dataset.• The data set give the locations of 1000 seismic events of

MB > 4.0. The events occurred in a cube near Fiji since 1964.


• Can we estimate the spatialinhomogeneity of seismicevents ?

• Can we prove statistically thepresence of two main clusters ?

• Can we highlight theanisotropy ?

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New Zealand

New Caledonia

VanuatuFiji

Samoa

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Spatial point pattern : earth quakes dataset

• R package dataset.• The data set give the locations of 1000 seismic events of

MB > 4.0. The events occurred in a cube near Fiji since 1964.


• Can we estimate the spatialinhomogeneity of seismicevents ?

• Can we prove statistically thepresence of two main clusters ?

• Can we highlight theanisotropy ?

165 175 185

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New Zealand

New Caledonia

VanuatuFiji

Samoa

165 175 185

−35

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−15

9 / 19


Spatial point pattern : marked point process1. Longleaf dataset (R package spatstat). Locations of 584 trees

observed with their diameter at breast height. S = R2 × R+(equipped with max(‖ · ‖, | · |)).

2. Ants dataset (R package spatstat). Locations of 97 antscategorised into two species. S = R2 × {0, 1} (equipped with themetric max(‖ · ‖, dM ) for any distance dM on the mark space).

longleaf

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1. Are large trees independent ? Do large trees influence thepositions of smaller ones ?

2. Is there any competition inside each specie ? Are the two speciesindependent ?

10 / 19


Spatial point pattern with extra information

• chorley dataset (R packagespatstat)

• Cases of larynx and lungcancers and position of anindustrial incinerator.

• S = R2 × {0, 1} (equipped withthe metric max(‖ · ‖, dM ) forany distance dM on the markspace).

Chorley−Ribble Data

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• Can we input the spatial positions of cancer cases to thepresence of the industriel incinerator ? to any other extrainformation (population density,. . .) ?

11 / 19


Spatial point pattern with spatial covariates• Beischmedia dataset (R package spatstat, large tropical forest

dataset).

• 3604 locations of trees observed with two spatial covariates (herethe elevation field, gradient of the elevation).


• Can we explain the spatialinhomogeneity by the elevation,the slope ?

• Can we statistically prove thepositive influence of these twocovariates ?

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12 / 19


Spatio-temporal point process• Lightning strikes observed on the domain [−5, 9] × [42, 53]

(long,lat) observed temporally since the period 2010-2013.

• Both time and position are point processes ; S = R2 × R. Summer 2010

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lecture 1: introduction to spatial point processes · 1.longleaf dataset (r package spatstat)....

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