Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
An introduction to spatial point processes Jean-Fran¸cois Coeurjolly Laboratoire Jean Kuntzmann (LJK), Grenoble University
Spatial point pattern data
Definitions and Poisson case
1
Spatial point pattern data
2
Definitions and Poisson case
3
Summary statistics
4
Models for point processes
Summary statistics
Models for point processes
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
A very very brief introduction . . . The realization x , of a spatial point process defined on S and observed in a bounded domain is a finite set of objects xi ∈ S . x = {x1 , . . . , xn }
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Spatial point pattern (1) japanesepines ●
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Locations of 65 trees on a bounded domain.
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Japanesepines dataset (R package spatstat)
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S = R2 (equipped with k · k).
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Questions of interest : Can we estimate the number of trees per unit volume ? Homogeneous or inhomogeneous ? Is there any independence, attraction or repulsion between trees ?
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Spatial point pattern (2) longleaf
Longleaf dataset (R package spatstat)
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S= × (equipped with max(k · k, | · |)).
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R2
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Locations of 584 trees observed with their diameter at breast height.
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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 ?
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Spatial point pattern (3) ants
Ants dataset (R package spatstat)
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S= × {0, 1} (equipped with the metric max(k · k, dM ) for any distance dM on the mark space). R2
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Locations of 97 ants categorised into two species.
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Questions of interest : Competition inside one specie ? between the two species ?
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Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Spatial point pattern (4)
Questions of interest : Can the elevation field explain the arrangement of trees ? Among a large number of spatial covariates, which ones have the largest influence ?
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160
3604 locations of trees observed with spatial covariates (here the elevation field). S = R2 (equipped with the metric k · k), z (·) ∈ R2 .
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Spatial point pattern (6) Towards stochastic geometry . . . Voronoi tessellation generated from a repulsive point pattern ; used to model foams,. . . The centers form of the tessellation form a point process.
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Mathematical definition of a spatial point process ? S : Polish state space of the point process (equipped with the σ-algebra of Borel sets B). A configuration of points is denoted x = {x1 , . . . , xn , . . .}. For B ⊆ S : xB = x ∩ B . Nlf : space of locally finite configurations, i.e. {x , n(xB ) = |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 on some probability space (Ω, F , P ) with values on Nlf . Measurability of X ⇔ N (B ) = |XB | is a r.v. for any bounded B ∈ B.
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Theoretical characterization of the distribution of X Proposition The distribution of a point process X 1
is determined by the finite dimensional distributions of its counting function, i.e. the joint distribution of N (B1 ), . . . , N (Bm ) for any bounded B1 , . . . , Bm ∈ B and any m ≥ 1.
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Theoretical characterization of the distribution of X Proposition The distribution of a point process X 1
is determined by the finite dimensional distributions of its counting function, i.e. the joint distribution of N (B1 ), . . . , N (Bm ) for any bounded B1 , . . . , Bm ∈ B and any m ≥ 1.
2
is uniquely determined by its void probabilities, i.e. by P (N (B ) = 0),
for bounded B ∈ B.
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Theoretical characterization of the distribution of X Proposition The distribution of a point process X 1
is determined by the finite dimensional distributions of its counting function, i.e. the joint distribution of N (B1 ), . . . , N (Bm ) for any bounded B1 , . . . , Bm ∈ B and any m ≥ 1.
2
is uniquely determined by its void probabilities, i.e. by P (N (B ) = 0),
for bounded B ∈ B.
From now on, we assume that S = Rd (and even d = 2) or a bounded domain of R2 . Everything can de extended to marked spatial point processes and/or to more complex domains.
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Moment measures Moments play an important role in the modelling of classical inference. For point processes = moments of counting variables. Definition : for n ≥ 1 we define the n-th order moment measure (defined on S n ) by X µ(n) = E 1({u1 , . . . , un } ∈ D), D ⊆ S n . u1 ,...,un
the n-th order reduced moment measure (defined on S n ) by α(n) (D) = E
, X
1({u1 , . . . , un } ∈ D), D ⊆ S n .
u1 ,...,un
where the , sign means that the n points are pairwise distinct.
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Intensity functions Assume µ(1) and α(2) are absolutely continuous w.r.t. Lebesgue measure, and denote by ρ and ρ(2) the densities.
Campbell Theorems 1
For any measurable function h : S → R Z X E h(u) = h(u)ρ(u)du. S
u∈X 2
For any measurable function h : S × S → R E
, X u,v ∈X
h(u, v ) =
Z Z S
h(u, v )ρ(2) (u, v )dudv . S
ρ(u)du ' Probability of the occurence of u in B (u, du) ρ (u, v ) ' Probability of the occurence of u in B (u, du) and v in B (v , dv ). (2)
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Poisson point processes Classical definition : X ∼Poisson(S , ρ) ∀m ≥ 1, ∀ bounded and disjoint B1 , . . . , Bm ⊂ S , the r.v. XB1 , . . . , XBm are independent. R N (B ) ∼ P B ρ(u)du for any bounded A ⊂ S . ∀B ⊂ S , ∀F ∈ Nlf P (XB ∈ F ) =
X e− n≥0
R B
ρ(u)du
Z
n!
B
...
Z
1({x1 , . . . , xn })
B
n Y i=1
If ρ(·) = ρ, X is said to be homogeneous which implies EN (B ) = ρ|B |,
VarN (B ) = ρ|B |.
and if S = Rd , X is stationary and isotropic.
ρ(xi )dxi .
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
A few realizations on S = [−1, 1]2 3
2
ρ(u) = βe −u1 −u1 −.5u1 . ρ = 200. ρ(u) = βe 2 sin(4πu1 u2 ) . ρ(u)du = 200.)
R
(β is adjusted s.t. the mean number of points in S ,
S
110 points
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Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
A few properties of Poisson point processes Proposition : if X ∼Poisson(S , ρ) Void probabilities : v (B ) = P (N (B ) = 0) = e −
R B
(ρ(u)du)
.
For any u, v ∈ S , ρ (u, v ) = ρ(u)ρ(v ) (also valid for ρ , k ≥ 1) (2)
(k )
and if |S | < ∞, X admits a density w.r.t. Poisson(S , 1) given by R Y f (x ) = e |S |− S ρ(u)du ρ(u). u∈x
Slivnyak-Mecke Theorem : for any non-negative function h : S × Nlf → R+ , then Z X E h(u, X \ u) = Eh(u, X )ρ(u)du. u∈X
Example : if ρ(·) = ρ, E
P
u∈X ∩[0,1]2
S
1(d(u, X \ u) ≤ R) = ρ 1 − e −ρπR
2
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Statistical inference for a Poisson point process Simulation : homogeneous case : very simple non-homogeneous case : a thinning procedure can be efficiently done.
Inference : consists in estimating ρ, ρ(·; θ) or ρ(u) depending on the context. All these estimates can be used even if the spatial point process is not Poisson (wait for 2 slides) Asymptotic properties very simple to derive under the Poisson assumption.
Goodness-of-fit tests : tests based on quadrats counting, based on the void probability,. . .
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Objective and classification Objective : Define some descriptive statistics for s.p.p. (independently on any model so). Measure the abundance of points, the clustering or the repulsiveness of a spatial point pattern w.r.t. the Poisson point process. Classification : First-order type based on the intensity function. Second-order type statistics : pair correlation function, Ripley’s K 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)
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
First order type statistics Essentially using Campbell formulae, we can estimate the intensity of a s.p.p as follows : 1 2
if X is stationary b ρ = N (W )/|W | is an estimate of ρ. Non-stationary, parametric estimation of the intensity : if ρ(u) = ρ(u; θ) can be used using the “Poisson likelihood”, i.e. Z X lW (X , θ) = log ρ(u; θ) − ρ(u; θ)du. W
u∈XW 3
Non stationary, non-parametric estimation of the intensity (see previous chapter for notation) : ! X 1 kv − uk −1 b ρh (v ) = Kh (η) k . h hd u∈X W
(asymptotic properties are more awkward to derive).
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Ripley’s K function We assume (for simplicity) the stationarity and isotropy of X . Definition The Ripley’s K function is literally defined for r ≥ 0 by 1 E number of extra events within distance r of a randomly chosen event ρ =E N (B (0, r ) \ 0)|0 ∈ X )
K (r ) =
We define the L function as L(r ) = (K (r )/π)1/2 . Properties : Under the Poisson case, K (r ) = πr 2 ; L(r ) = r . If K (r ) > πr 2 or L(r ) > r (resp. K (r ) < πr 2 or L(r ) < r ) we suspect clustering (regularity) at distances lower than r .
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Edge corrected estimation of the K function Definition We define the border-corrected estimate as X 1 bBC (r ) = 1 K {N (B (u, r )) − 1} b ρ N (W r ) u∈W r
where W r = {u ∈ W : B (u, r ) ⊆ W } is the erosion of W by r . the translation-corrected estimate as bTC (r ) = 1 K b ρ2
, X 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,. . .
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Example of L function for a Poisson point pattern ●●
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The enveloppes are constructed using a Monte-Carlo approach under the Poisson assumption. ⇒ we don’t reject the Poisson assumption.
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Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Example of L function for a repulsive point pattern
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⇒ the point pattern does not come from the realization of a homogeneous Poisson point process. exhibits repulsion at short distances (r ≤ .05)
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Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Example of L function for a clustered point pattern 0.25
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Spatial point pattern data
Definitions and Poisson case
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. 2
Poisson case : ∀r > 0, F (r ) = G(r ) = 1 − e −πr , 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 .
Spatial point pattern data
Definitions and Poisson case
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 ) its cardinality. Then, the (border corrected) estimator of F is b (r ) = F
1 X 1(d (u, X ) ≤ r ) n(Ir ) u∈I r
where Ir = I ∩ W r . The (border corrected) estimator of G is b )= G(r
X 1 1(d (u, X \ u) ≤ r ) N (W r ) u∈X ∩W r
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
1.0
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Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
More realistic models than the Poisson point process We can distinguish several classes of models for spatial point processes 1
point processes based on the thinning of a Poisson point processes, on the superimposition of Poisson point processes. [sometimes hard to relate the stochastic process producing the realization and the physical phenomenon producing the data]
2
Cox point processes (which include Poisson Cluster point processes,. . . ).
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Gibbs point processes. Strong links with statistical physics
4
Determinental point processes. Links with random matrices
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
An attempt to classify these models . . . Model
Allows to model
Are moments explicit ? yes
Density w.r.t. Poisson ? no
Cox
attraction
Gibbs
repulsion but also attraction
no
yes
Determinental
repulsion
yes
yes
This classification is really important since the methodologies to infer these models will be based either on moment methods or on conditional densities w.r.t. Poisson point process. asymptotic results require different tools : e.g. CLT based on mixing conditions (for Cox, determinental point process) or on a “martingale-type” condition for Gibbs point process.
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Cox point processes (1) Definition Suppose that Z = {Z (u) : u ∈ S } is a nonnegative random field so that with probability one, u → Z (u) is a locally integrable function. If the conditional distribution of X given Z is a Poisson process on S with intensity 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) =
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The void probabilities are given by ! Z v (B ) = E exp − Z (u)du B
for bounded B ⊆ S .
E[Z (u)Z (η)] . ρ(u)ρ(η)
Spatial point pattern data
Definitions and Poisson case
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 ) be independent 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∈C Xc is a Neymann-Scott process with cluster centres C and clusters Xc , c ∈ C .
Spatial point pattern data
Definitions and Poisson case
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 ) be independent 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∈C Xc is a Neymann-Scott process with cluster centres C and clusters Xc , c ∈ C . X is a Cox process on Rd driven by Z (u) =
P
c∈C
αk (u − c).
When k is the Gaussian kernel, X is called the Thomas process.
Spatial point pattern data
Definitions and Poisson case
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
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Definitions and Poisson case
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 = exp Y where Y is a Gaussian random field. Then, X is said to be a log Gaussian Cox process (LGCP). Basic properties : let m and c denote the mean function and the covariance 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)).
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Four realizations of (stationary) LGCP point processes σ = 2.5, α = 0.01, ρ = 100
σ = 2.5, α = 0.005, ρ = 100
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σ = 2.5, α = 0.005, ρ = 200
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σ = 2.5, α = 0.01, ρ = 200 ●
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Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Cox point processes (5) : parametric estimation method For most of the models, the likelihood is not available but moments are accessible. Then the idea is then to estimate θ using a minimum contrast approach : i.e. define θˆ as the minimizer of Z r2 Z r2 2 2 b q b q g (r )q − gθ (r )q dr K (r ) − Kθ (r ) dr or r1
r1
where b (r ) and b K g (r ) are the nonparametric estimates of K (r ) and g(r ). where [r1 , r2 ] is a set of r fixed values. q is a power parameter (adviced in the literature to be set to q = 1/4 or 1/2).
Spatial point pattern data
Definitions and Poisson case
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 Gibbs point process if it admits a density f w.r.t. a Poisson point process with unit rate, i.e. for any F ⊆ Nf P (X ∈ F ) =
X e −|S | Z n≥0
n!
Z . . . 1({x1 , . . . , xn } ∈ F )f ({x1 , . . . , xn })dx1 . . . dxn
S
S
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 −1 h with Z X exp(−|S |) Z c= ... h({x1 , . . . , xn })dx1 . . . dxn = E[h(Y )] n! S S n≥0
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Gibbs point process (2) : the most well-known class Definition An istotropic and homogeneous parwise interaction point process has a density of the form (for any x ∈ Nf ) Y φ2 (kv − uk) f (x ) ∝ βn(x ) {u,v }⊆x
where φ2 : R+∗ → R+ is called the interaction function. The main example is the Strauss point process defined by X f (x ) ∝ βn(x ) γsR (x ) where sR (x ) = 1(kv − uk ≤ R) {u,v }∈x
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.
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Realizations of a Strauss point process β = 100, γ = 0, R = 0.075 ●
β = 100, γ = 0.3, R = 0.075 ●
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β = 100, γ = 0.6, R = 0.075 ● ● ●
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β = 100, γ = 1, R = 0.075
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(simulation of spatial Gibbs point processes can be done using spatial birth-and-death process or using MCMC with reversible jumps, see Møller and Waagepetersen for details)
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Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Gibbs point processes (3) : inference Likelihood unavailable : normalizing constant unknown, moments not expressible (e.g. in the stationary case ρ = Eλ(0, X )).
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Gibbs point processes (3) : inference Likelihood unavailable : normalizing constant unknown, moments not expressible (e.g. in the stationary case ρ = Eλ(0, X )). Models (even when S = Rd ) can be defined through the Papangelou conditional intensity f (x ∪ u) λ(u, x ) = , x ∈ Nlf , u ∈ S . f (x ) Key-concept since several alternatives methods exist based on λ (and not on f ) including the pseudo-likelihood Z X LPLW (x ; θ) = λ(x , x \ u; θ) − λ(u, x ; θ)du. u∈xW
W
Spatial point pattern data
Definitions and Poisson case
Summary statistics
Models for point processes
Gibbs point processes (3) : inference Likelihood unavailable : normalizing constant unknown, moments not expressible (e.g. in the stationary case ρ = Eλ(0, X )). Models (even when S = Rd ) can be defined through the Papangelou conditional intensity f (x ∪ u) λ(u, x ) = , x ∈ Nlf , u ∈ S . f (x ) Key-concept since several alternatives methods exist based on λ (and not on f ) including the pseudo-likelihood Z X LPLW (x ; θ) = λ(x , x \ u; θ) − λ(u, x ; θ)du. u∈xW
W
Approaches and diagnostic tools use Georgii-Nguyen-Zessin formula Z X E h(u, X \ u) = E(h(u, X )λ(u, X ))du. u∈X
Spatial point pattern data
Definitions and Poisson case
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,. . .
Spatial point pattern data
Definitions and Poisson case
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, phase transition. Many classical statistical methodologies need to be adapted (and proved) to s.p.p. : robust methods, resampling techniques, multiple hypothesis testing. High-dimensional problems : S = Rd with d large, selection of variables, regularization methods,. . . Space-time point processes.
Spatial point pattern data
Definitions and Poisson case
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.
