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Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
An introduction to spatial point processes
Jean-Francois Coeurjolly
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
1 Examples of spatial data
2 Intensities and Poisson p.p.
3 Summary statistics
4 Models for point processes
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
A very very brief introduction . . .
The realization x , of a spatial point process defined on a spaceS is a (locally) finite set of objects xi ∈ S .
x = x1, . . . , xn , xi and n are random.
Thank you for your attention !
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
A very very brief introduction . . .
The realization x , of a spatial point process defined on a spaceS is a (locally) finite set of objects xi ∈ S .
x = x1, . . . , xn , xi and n are random.
Thank you for your attention !
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Spatial data . . .
. . . can be roughly and mainly classified into three categories :
1 Geostatistical data.
2 Lattice data.
3 Spatial point pattern
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Geostatistical data
observation of a (e.g.) continuous random variable at fixedlocations
meuse dataset (R package gstat) : topsoil heavy metalconcentrations, at the observation locations, collected in a floodplain of the river Meuse.
100
200
300
400
500
600
100
200
300
400
500
600
Main objective : interpolate the spatial data.
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Lattice data (1)
Eire dataset (R packagespdep)
% of people with groupA in eire, observed in 26regions.
The data are aggregatedon the region ⇒random field on anetwork.
Percentage with blood group A in Eire
under 27.9127.91 − 29.2629.26 − 31.02over 31.02
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Lattice data (2)
Lennon dataset (Rpackage fields)
Real-valued random field(gray scale image withvalues in [0, 1]).
Defined on the network1, . . . , 2562.
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Lattice data (3)
Over-interpretation : xkcd.
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Spatial point pattern (1)
Japanesepines dataset (Rpackage spatstat)
Locations of 65 trees on abounded domain.
S = R2 (equipped with ‖ · ‖).
japanesepines
Questions of interest :
Can we estimate the number of trees per unit volume ?Homogeneous or inhomogeneous ?
Is there any independence, attraction or repulsion betweentrees ?
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Spatial point pattern (2)
Longleaf dataset (R packagespatstat)
Locations of 584 treesobserved with theirdiameter at breast height.
S = R2 × R+ (equipped withmax(‖ · ‖, | · |)).
longleaf
Additional scientific questions :
Can the mark explain the intensity of the number of trees ?
Does a large tree tend to have smaller trees close to it ?
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Spatial point pattern (3)
Ants dataset (R packagespatstat)
Locations of 97 antscategorised into two species.
S = R2 × 0, 1 (equippedwith the metricmax(‖ · ‖, dM ) for anydistance dM on the markspace).
ants
Questions of interest :
Competition inside one specie ? between the two species ?
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Spatial point pattern (4)
3604 locations of trees observed with spatial covariates(here the elevation field).
S = R2 (equipped with the metric ‖ · ‖), z (·) ∈ R2.
Questions of interest :
Can the elevation field explain the arrangement of trees ?
Among a large number of spatial covariates, which oneshave the largest influence ?
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Spatial point pattern (4)
3604 locations of trees observed with spatial covariates(here the elevation field).
S = R2 (equipped with the metric ‖ · ‖), z (·) ∈ R2.
Questions of interest :
Can the elevation field explain the arrangement of trees ?
Among a large number of spatial covariates, which oneshave the largest influence ?
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Spatial point pattern (5)
Spatio-temporal point process on a complex spaceDaily observation of sunspots at the surface of the sun.can be viewed as the realization of a markedspatio-temporal point process on the sphere.S = S2 × R
+ × R+ (state, time, and mark).
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Spatial point pattern (6) : eye-movement data
Eye-movement (on an image or video) iscomposed of
sacades : exploratory step, local, veryquick 120ms.
fixations (< 1 of oscillation) ; analysingfixations allows to understand how asubject explores an image ; locations offixations as well as their number arerandom.
Oculo-nimbus project (Univ. Grenoble) : aim to understandmechanisms of newborns vision
Dozens of images
Newborns of 3-, 6-, 9- and12-month + adults controlgroup
' 40 subjects per age group
' 15 − 20 fixations bysubject
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Spatial point pattern (6) : eye-movement data
Eye-movement (on an image or video) iscomposed of
sacades : exploratory step, local, veryquick 120ms.
fixations (< 1 of oscillation) ; analysingfixations allows to understand how asubject explores an image ; locations offixations as well as their number arerandom.
Oculo-nimbus project (Univ. Grenoble) : aim to understandmechanisms of newborns vision
Dozens of images
Newborns of 3-, 6-, 9- and12-month + adults controlgroup
' 40 subjects per age group
' 15 − 20 fixations bysubject
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Mathematical definition of a spatial point process
Do you really want to look at this ?
S : Polish state space of the point process (equipped with theσ-algebra of Borel sets B).
A configuration of points is denoted x = x1, . . . , xn , . . .. ForB ⊆ S : xB = x ∩ B .
Nlf : space of locally finite configurations, i.e.
x ,n(xB ) < ∞,∀B bounded ⊆ S
equipped with Nlf = σ(x ∈ Nlf ,n(xB ) = m,B ∈ B,B bounded,m ≥ 1
).
Definition
A point process X defined on S is a measurable application defined onsome probability space (Ω,F ,P ) with values on Nlf .
Measurability of X⇔ N (B ) is a r.v. for any bounded B ∈ B.
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Theoretical characterization of the distribution of X
Proposition
The distribution of a point process X is determined
1 by the joint distribution of N (B1), . . . ,N (Bm ) for any boundedB1, . . . ,Bm ∈ B and any m ≥ 1.
2 or by its void probabilities, i.e. by
P (N (B ) = 0), for bounded B ∈ B.
For the rest of the talk :
let S = R2 or a bounded domain of R2.
everything can ± be extended to marked spatial point processes,spatio-temporal point processes, manifold-values point processes.
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Theoretical characterization of the distribution of X
Proposition
The distribution of a point process X is determined
1 by the joint distribution of N (B1), . . . ,N (Bm ) for any boundedB1, . . . ,Bm ∈ B and any m ≥ 1.
2 or by its void probabilities, i.e. by
P (N (B ) = 0), for bounded B ∈ B.
For the rest of the talk :
let S = R2 or a bounded domain of R2.
everything can ± be extended to marked spatial point processes,spatio-temporal point processes, manifold-values point processes.
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Theoretical characterization of the distribution of X
Proposition
The distribution of a point process X is determined
1 by the joint distribution of N (B1), . . . ,N (Bm ) for any boundedB1, . . . ,Bm ∈ B and any m ≥ 1.
2 or by its void probabilities, i.e. by
P (N (B ) = 0), for bounded B ∈ B.
For the rest of the talk :
let S = R2 or a bounded domain of R2.
everything can ± be extended to marked spatial point processes,spatio-temporal point processes, manifold-values point processes.
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Moment measures
Moments (mean, variance, covariance,. . .) play an important rolein the characterization of a random variable (or a time series,random field) ;
For point processes : moment measures which are related tomoments of counting variables ;
Definition : for n ≥ 1 we define
the n-th order (reduced) moment measure (defined on Sn) by
α(n)(D) = E,∑
u1,...,un∈X
1(u1, . . . , un ∈ D), D ⊆ Sn .
where the , sign means that the n points are pairwise distinct.
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Intensity functions
Often (always) assumed that moment measures areabsolutely continuous w.r.t. Lebesgue measures, so insteadof the moment measures, the quantitites of interest (i.e.the ones you should keep in mind !) are :
1 Intensity function : ρ(·) : Rd → R+
ρ(u) = lim|du |→0
P(one event in B (u ,du))|du |
2 Second-order intensity function : ρ(2)(·, ·) : Rd × Rd → R+
ρ(2)(u , v ) = lim|du |,|dv |→0
P(2 distinct events in B (u , du) and B (v ,dv ))|du ||dv |
3 k -th order intensity function . . .
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Campbell formula (1)
Valid for any point process (having an intensity function)
Campbell Theorem
For any measurable function h : Rd → R (such that . . .. . .)
E∑u∈X
h(u) =
∫h(u)ρ(u)du .
Examples :
h(u) = 1(u ∈W ), for W ⊂ Rd
EN (W ) =
∫Wρ(u)du
(= ρ|W | if ρ(·) = ρ, homogeneous case
)h(u) = 1(u ∈W )ρ(u)−1, for W ⊂ Rd
E∑
u∈X∩W
ρ(u)−1 = |W |.
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Campbell formula (2)
Valid for any point process (having a 2-nd . . .)
Campbell Theorem
For any measurable function h : Rd × Rd → R (such that . . .. . .)
E,∑
u ,v∈X
h(u , v ) =
∫ ∫h(u , v )ρ(2)(u , v )dudv .
Examples :
h(u) = 1(u ∈ A, v ∈ B ) for A,B ⊂ Rd s.t. A ∩ B = ∅
E,∑
u ,v∈X
1(u ∈ A)1(v ∈ B ) = E (N (A)N (B )) =
∫A
∫Bρ(2)(u , v )dudv .
⇒ Modelling/estimating ρ(2) allows to understand/modelcovariances of counting variables.
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Covariances of counting variables
For A,B ⊂ Rd s.t. A ∩ B = ∅
Cov (N (A),N (B )) =
∫A
∫B
ρ(2)(u , v ) − ρ(u)ρ(v )
dudv
=
∫A
∫B
ρ(u)ρ(v ) g(u , v ) − 1dudv .
where the function g given by
g(u , v ) =ρ(2)(u , v )ρ(u)ρ(v )
is called the pair correlation function.
The departure of g to 1 will measure some kind of independencefor a point process X (wait for a few slides).
If X is isotropic (i.e the distribution of X in invariant underrotation), then g(u , v ) = g(‖v − u‖).
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Poisson point processes
Intuitive definition : X ∼Poisson(S , ρ)
∀m ≥ 1, ∀ bounded and disjoint B1, . . . ,Bm ⊂ S , the r.v.XB1
, . . . ,XBmare independent.
N (B ) ∼ P(∫
Bρ(u)du
)for any bounded A ⊂ S .
Poisson process is the reference model for point processes.
PPP model points without any interaction !
If ρ(·) = ρ, X is said to be homogeneous which implies
EN (B ) = ρ|B |, VarN (B ) = ρ|B |.
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
A few realizations on W
u = (u1, u2) ∈Wρ(u) = βe−u1−u
21−.5u
31 , W = [0, 1]2.
ρ = 100, W = [0, 1]2.
ρ(u) = βe2 sin(4πu1u2), W = [−1, 1]2.
(β is adjusted s.t. the mean number of points in W ,∫Wρ(u)du = 200.)
110 points
050
100
150
2040
6080
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
A few properties of Poisson point processes
Proposition : if X ∼Poisson(W , ρ)
Void probabilities : v (B ) = P (N (B ) = 0) = e−∫B
(ρ(u)du).
For any u , v ∈ Rd ,
ρ(2)(u , v ) = ρ(u)ρ(v )⇒ g(u , v ) =ρ2(u , v )ρ(u)ρ(v )
= 1
(also valid for ρ(k ), k ≥ 1)
Hence, for a general point process X
g(u , v ) < 1 means that two points are less likely to appear at u , vthan for the Poisson model.⇒ characteristic for repulsive patterns
g(u , v ) > 1 means that two points are more likely to appear atu , v than for the Poisson model.⇒ characteristic for clustered patterns
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Statistical inference for a Poisson point process
Simulation :
homogeneous case : very simplenon-homogeneous case : a thinning procedure can beefficiently done.
Inference :
consists in estimating ρ, ρ(·; β) or ρ(u) depending on thecontext.All these estimates can be used even if the spatial pointprocess is not Poisson (wait for 2 slides)Asymptotic properties very simple to derive under thePoisson assumption.
Goodness-of-fit tests : tests based on quadrats counting,based on the void probability,. . .
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Parametric intensity estimation
Problem : model ρ(u) = ρ(u; β) for β ∈ Rp , p ≥ 1 and estimate β.
Example : forestry dataset
ρ(u; β) = exp (β1 + β2Alt(u) + β3Slope(u))
where Alt(u) and Slope(u) are spatial covariates corresponding tomaps of altitude and slope of elevation.
Assume we have a Poisson point process with intensity ρ(u; β)observed in W .
Poisson likelihood
It can be shown that the log-likelihood for this model writes (up to anormalizing constant)
`W (X; β) =∑
u∈X∩W
log ρ(u; β) −∫W
ρ(u; β)du
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Towards estimating equations
Rathburn and Cressie (98) : MLE is consistent and
asymptotically normal when X=Poisson ;
. . .. . .but the procedure is acutally valid for much more generalpoint processes
Evaluate the score function (gradient vector with length p)
`′(X, β) =∑
u∈X∩W
ρ′(u; β)ρ(u; β)
−
∫W
ρ′(u; β)du .
⇒ Campbell form. (valid for any X) : E `′(X, β) = 0
So, `′(X, β) is a nice estimating equation (see RW’s talk).
(Properties for the resulting estimator require more techniques).
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Back to Eye-movement data
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Example of modelling
log ρ(u ,m; β) = β1Saliency(u) +
4∑m ′=1
(βm
′
0 + βm′
1 Ad(u))1(m′ = m)
where
Saliency(u) : Deterministic model of intensity map.
Ad(u) : binary map built from the top 5% of ρAd(u).
βm′
0 : can be ,, the different number of points per agegroup.
βm′
1 : parameters of interest ; the values are not interesting
but β11 < · · · < β
41 is the cognitive hypothesis to test.
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Adjusted contrasts tests
⇒ Significant differences for 5 out of the 6 images.
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Objective and classification
Objective :
Define some descriptive statistics for s.p.p. (independently onany model so).
Measure the abundance of points, the clustering or therepulsiveness of a spatial point pattern w.r.t. the Poisson pointprocess.
Classification :
First-order type based on the intensity function.
Second-order type statistics : pair correlation function, Ripley’sK function.
Statistics based on distances : empy space function F ,nearest-neigbour G , J function.
(We assume that ρ and ρ(2) exist in the rest of the talk)
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Ripley’s K function (isotropic and planar case)
Definition : let r ≥ 0
K (r ) =ρ−1 E(number of extra events within distance r of a randomly chosen event
)=ρ−1E
(N (B (0, r ) \ 0)|0 ∈ X )
L(r ) =√K (r )/π
Properties :
Under the Poisson assumption, K (r ) = πr2 ; L(r ) = r .
If K (r ) > πr2 or L(r ) > r (resp. K (r ) < πr2 or L(r ) < r) wesuspect clustering (regularity) at distances lower than r .
Application in practice :
define a grid r values : r1, . . . , rI ;
find an estimator of K (ri ) or L(ri ), say K (ri ) and L(ri ) ;
Plot e.g. (ri , L(ri )) and compare with the Poisson case.
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Edge corrected estimation of the K function
Definition
We define
the border-corrected estimate as
KBC (r ) =1
ρ
1
N (Wr )
∑u∈Wr
N (B (u , r )) − 1
where Wr = u ∈W : B (u , r ) ⊆W is the erosion of W by r .
the translation-corrected estimate as
KTC (r ) =1
ρ2
,∑u ,v∈XW
1(v − u ∈ B )|W ∩Wv−u |
where Wu = W + u = u + v : v ∈W .
Remark : everything extends to 2nd-order reweighted stationary point processes ;
asymptotic properties depend on mixing conditions,. . .
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Example of L function for a Poisson point pattern
106
0.00 0.05 0.10 0.15 0.20 0.25
0.00
0.05
0.10
0.15
0.20
0.25
r
Linh
om(r)
Linhomobs(r)Linhom(r)Linhomhi(r)Linhomlo(r)
The enveloppes are constructed using a Monte-Carloapproach under the Poisson assumption.
⇒ we don’t reject the Poisson assumption.
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Example of L function for a repulsive point pattern
113
0.00 0.05 0.10 0.15 0.20 0.25
0.00
0.05
0.10
0.15
0.20
0.25
r
Linh
om(r)
Linhomobs(r)Linhom(r)Linhomhi(r)Linhomlo(r)
⇒ the point pattern does not come from the realization ofa homogeneous Poisson point process.
exhibits repulsion at short distances (r ≤ .05)
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Example of L function for a clustered point pattern
Xth
0.00 0.05 0.10 0.15 0.20 0.25
0.00
0.05
0.10
0.15
0.20
0.25
r
Linh
om(r)
Linhomobs(r)Linhom(r)Linhomhi(r)Linhomlo(r)
⇒ the point pattern does not come from the realization ofa homogeneous Poisson point process.
exhibits attraction at short distances (r ≤ .08).
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Statistics based on distances : F , G and J functions
Assume X is stationary (definitions can be extended in the general case)
Definition
The empty space function is defined by
F (r ) = P (d (0,X ) ≤ r ) = P (N (B (0, r )) > 0), r > 0.
The nearest-neighbour distribution function is
G(r ) = P (d (0,X \ 0) ≤ r |0 ∈ X )
J -function : J (r ) = (1 −G(r ))/(1 − F (r )), r > 0.
Poisson case : ∀r > 0, F (r ) = G(r ) = 1 − e−πr2, J (r ) = 1.
F (r ) < Fpois (r ), G(r ) > Gpois (r ), J (r ) < 1 : attraction at dist. < r .
F (r ) > Fpois (r ), G(r ) < Gpois (r ), J (r ) > 1 : repulsion at dist. < r .
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Non-parametric estimation of F , G and J
As for the K and L functions, several edge corrections exist. We focus here only on
the border correction. We assume that X is observed on a bounded window W
with positive volume.
Definition
Let I ⊆W be a finite regular grid of points and n(I ) itscardinality. Then, the (border corrected) estimator of F is
F (r ) =1
n(Ir )
∑u∈Ir
1(d (u ,X ) ≤ r )
where Ir = I ∩Wr .
The (border corrected) estimator of G is
G(r ) =1
N (Wr )
∑u∈X∩Wr
1(d (u ,X \ u) ≤ r )
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Application to a clustered point pattern dataXth
0.00 0.02 0.04 0.06 0.08 0.10
0.0
0.2
0.4
0.6
0.8
1.0
r
F(r)
Fobs(r)F(r)Fhi(r)Flo(r)
0.000 0.005 0.010 0.015 0.020
0.0
0.2
0.4
0.6
0.8
r
G(r)
Gobs(r)G(r)Ghi(r)Glo(r)
0.00 0.02 0.04 0.06 0.08 0.10
05
1015
2025
r
J(r)
Jobs(r)J(r)Jhi(r)Jlo(r)
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
More realistic models than the Poisson point process
We can distinguish several classes of models for spatial pointprocesses. Among them :
1 Cox point processes (which include Poisson Cluster pointprocesses,. . . ).
2 Gibbs point processes.Strong links with statistical physics
3 Determinantal point processes.Links with random matrices
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
An attempt to classify these models . . .
Model Allows to model Are moments Density w.r.t.explicit ? Poisson ?
Cox attraction yes no
Gibbs repulsion no yesbut also attraction
Determinantal repulsion yes yes
This classification is really important since the
methodologies to infer these models will be based either onmoment methods or on conditional densities w.r.t. Poisson pointprocess.
asymptotic results require different tools : e.g. CLT based onmixing conditions (for Cox, determinantal point process) or on a“martingale-type” condition for Gibbs point process.
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Cox point processes (1)
Definition
Suppose that Z = Z (u) : u ∈ S is a nonnegative random field so thatwith probability one, u → Z (u) is a locally integrable function. If theconditional distribution of X given Z is a Poisson process on S withintensity function Z , then X is said to be a Cox process driven by Z .
It is straightforwardly seen that
1 Provided Z (u) has finite expectation and variance for any u ∈ S
ρ(u) = EZ (u), ρ(2)(u , η) = E[Z (u)Z (η)], g(u , η) =E[Z (u)Z (η)]ρ(u)ρ(η)
.
2 The void probabilities are given by
v (B ) = E exp
(−
∫B
Z (u)du)
for bounded B ⊆ S .
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Cox point processes (2) : Neymann-Scott process
Definition
Let C ∼Poisson(Rd , κ). Conditional on C , let Xc ∼Poisson(Rd , ρc) beindependent Poisson processes for any c ∈ C where
ρc(u) = αk (u − c)
where α > 0 is a parameter and k is a kernel (i.e. for all c ∈ Rd ,u → k (u − c) is a density function). Then X = ∪c∈CXc is aNeymann-Scott process with cluster centres C and clustersXc , c ∈ C .
X is a Cox process on Rd driven by Z (u) =∑
c∈C αk (u − c).
When k is the Gaussian kernel, X is called the Thomas process.
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Cox point processes (2) : Neymann-Scott process
Definition
Let C ∼Poisson(Rd , κ). Conditional on C , let Xc ∼Poisson(Rd , ρc) beindependent Poisson processes for any c ∈ C where
ρc(u) = αk (u − c)
where α > 0 is a parameter and k is a kernel (i.e. for all c ∈ Rd ,u → k (u − c) is a density function). Then X = ∪c∈CXc is aNeymann-Scott process with cluster centres C and clustersXc , c ∈ C .
X is a Cox process on Rd driven by Z (u) =∑
c∈C αk (u − c).
When k is the Gaussian kernel, X is called the Thomas process.
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Four realizations of Thomas point processes
κ = 50, σ = 0.03, α = 5
κ = 100, σ = 0.03, α = 5
κ = 50, σ = 0.01, α = 5
κ = 100, σ = 0.01, α = 5
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Cox point processes (4) : Log-Gaussian Cox processes
Definition
Let X be a Cox process on Rd driven by Z = expY where Y is aGaussian random field. Then, X is said to be a log Gaussian Coxprocess (LGCP).
Basic properties : let m and c denote the mean function and thecovariance function of Y
1 the intensition function of X is
ρ(u) = exp (m(u) + c(u , u)/2) .
2 The pair correlation function g of X is
g(u , η) = exp(c(u , u)).
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Four realizations of (stationary) LGCP point processes
with exponentialcorrelation function (δ = 1).
The mean m of theGaussian process is suchthat ρ = exp(m + σ2/2).
σ = 2.5, α = 0.01, ρ = 100
σ = 2.5, α = 0.005, ρ = 100
σ = 2.5, α = 0.01, ρ = 200
σ = 2.5, α = 0.005, ρ = 200
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Cox point processes (5) : parametric estimation method
For most of the models, the likelihood is not available butmoments are accessible.
Then the idea is then to estimate θ using a minimum contrastapproach : i.e. define θ as the minimizer of∫ r2
r1
∣∣∣∣K (r )q −Kθ(r )q∣∣∣∣2 dr or
∫ r2
r1
∣∣∣g(r )q − gθ(r )q∣∣∣2 dr
where
K (r ) and g(r ) are the nonparametric estimates of K (r ) andg(r ).where [r1, r2] is a set of r fixed values.q is a power parameter (adviced in the literature to be setto q = 1/4 or 1/2).
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Gibbs point process (1)
We focus on the case S bounded.
Definition
A finite point process X on a bounded domain S is said to be a Gibbspoint process if it admits a density f w.r.t. a Poisson point processwith unit rate, i.e. for any F ⊆ Nf
P (X ∈ F ) =∑n≥0
e−|S |
n!
∫S
. . .
∫S
1(x1, . . . , xn ∈ F )f (x1, . . . , xn )dx1 . . . dxn
where the term n = 0 is read as exp(−|S |)1(∅ ∈ F )f (∅).
Gpp can be viewed as a perturbation of a point process.
f is easily interpretable ' weight w.r.t. a Poisson process.
f specified up to an unknown constant f = c−1h with
c =∑n≥0
exp(−|S |)n!
∫S
. . .
∫S
h(x1, . . . , xn )dx1 . . . dxn = E[h(Y )]
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Gibbs point process (2) : the most well-known classDefinition
An istotropic and homogeneous parwise interaction point process hasa density of the form (for any x ∈ Nf )
f (x ) ∝ βn(x )∏u ,v ⊆x
φ2(‖v − u‖)
where φ2 : R+∗ → R
+ is called the interaction function.
The main example is the Strauss point process defined by
f (x ) ∝ βn(x )γsR(x ) where sR(x ) =∑u ,v ∈x
1(‖v − u‖ ≤ R)
where β > 0,R < ∞, γ is called the interaction parameter :
γ = 1 : homogeneous Poisson point process with intensity β.
0 < γ < 1 : repulsive point process.
γ = 0 : hard-core process with hard-core R.
γ > 1 : the model is not well-defined.
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Realizations of a Strauss point process
(simulation of spatial Gibbspoint processes can be doneusing spatial birth-and-deathprocess or using MCMC withreversible jumps, see Møllerand Waagepetersen fordetails)
β = 100, γ = 0, R = 0.075
β = 100, γ = 0.3, R = 0.075
β = 100, γ = 0.6, R = 0.075
β = 100, γ = 1, R = 0.075
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Gibbs point processes (3) : inference
Likelihood unavailable : normalizing constant unknown, momentsnot expressible (e.g. in the stationary case ρ = Eλ(0,X )).
Models (even when S = Rd) can be defined through thePapangelou conditional intensity
λ(u , x ) =f (x ∪ u)f (x )
, x ∈ Nlf , u ∈ S .
Key-concept since several alternatives methods exist based onλ (and not on f ) including the pseudo-likelihood
LPLW (x ; θ) =∑u∈xW
λ(x , x \ u; θ) −∫W
λ(u , x ; θ)du .
Approaches and diagnostic tools use Georgii-Nguyen-Zessinformula
E∑u∈X
h(u ,X \ u) =
∫E(h(u ,X )λ(u ,X ))du .
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Gibbs point processes (3) : inference
Likelihood unavailable : normalizing constant unknown, momentsnot expressible (e.g. in the stationary case ρ = Eλ(0,X )).
Models (even when S = Rd) can be defined through thePapangelou conditional intensity
λ(u , x ) =f (x ∪ u)f (x )
, x ∈ Nlf , u ∈ S .
Key-concept since several alternatives methods exist based onλ (and not on f ) including the pseudo-likelihood
LPLW (x ; θ) =∑u∈xW
λ(x , x \ u; θ) −∫W
λ(u , x ; θ)du .
Approaches and diagnostic tools use Georgii-Nguyen-Zessinformula
E∑u∈X
h(u ,X \ u) =
∫E(h(u ,X )λ(u ,X ))du .
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Gibbs point processes (3) : inference
Likelihood unavailable : normalizing constant unknown, momentsnot expressible (e.g. in the stationary case ρ = Eλ(0,X )).
Models (even when S = Rd) can be defined through thePapangelou conditional intensity
λ(u , x ) =f (x ∪ u)f (x )
, x ∈ Nlf , u ∈ S .
Key-concept since several alternatives methods exist based onλ (and not on f ) including the pseudo-likelihood
LPLW (x ; θ) =∑u∈xW
λ(x , x \ u; θ) −∫W
λ(u , x ; θ)du .
Approaches and diagnostic tools use Georgii-Nguyen-Zessinformula
E∑u∈X
h(u ,X \ u) =
∫E(h(u ,X )λ(u ,X ))du .
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Conclusion
The anaysis of spatial point pattern
very large domain of research including probability,mathematical statistics, applied statistics
own specific models, methodologies and software(s) to deal with.
is involved in more and more applied fields : economy, biology,physics, hydrology, environmentrics,. . .
Still a lot of challenges
Modelling : the “true model”, problems of existence, phasetransition.
Many classical statistical methodologies need to be adapted (andproved) to s.p.p. : robust methods, resampling techniques,multiple hypothesis testing.
High-dimensional problems : S = Rd with d large, selection ofvariables, regularization methods,. . .
Space-time point processes.
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
Conclusion
The anaysis of spatial point pattern
very large domain of research including probability,mathematical statistics, applied statistics
own specific models, methodologies and software(s) to deal with.
is involved in more and more applied fields : economy, biology,physics, hydrology, environmentrics,. . .
Still a lot of challenges
Modelling : the “true model”, problems of existence, phasetransition.
Many classical statistical methodologies need to be adapted (andproved) to s.p.p. : robust methods, resampling techniques,multiple hypothesis testing.
High-dimensional problems : S = Rd with d large, selection ofvariables, regularization methods,. . .
Space-time point processes.
Examples of spatial data Intensities and Poisson p.p. Summary statistics Models for point processes
References
A. Baddeley and R. Turner.Spatstat : an R package for analyzing spatial point patterns.Journal of Statistical Software, 12 :1–42, 2005.
N. Cressie.Statistics for spatial data.John Wiley and Sons, Inc, 1993.
P. J. Diggle.Statistical Analysis of Spatial Point Patterns.Arnold, London, second edition, 2003.
X. Guyon.Random Fields on a Network.Springer-Verlag, New York, 1991.
J. Illian, A. Penttinen, H. Stoyan, and D. Stoyan.Statistical Analysis and Modelling of Spatial Point Patterns.Statistics in Practice. Wiley, Chichester, 2008.
J. Møller and R. P. Waagepetersen.Statistical Inference and Simulation for Spatial Point Processes.Chapman and Hall/CRC, Boca Raton, 2004.