Lecture 2: Poisson point processes: properties and statistical inference

Transcription

1 Lecture 2: Poisson point processes: properties and statistical inference Jean-François Coeurjolly 1 / 20

2 Definition, properties and simulation Statistical inference for the homogeneous Poisson point process Statistical inference for the inhomogeneous Poisson point process 2 / 20

3 Poisson point processes 2. Classical way of thinking of a Poisson point process m 1, bounded and disjoint B 1,..., B m S, the r.v. X B1,..., X Bm are independent. N (B) P ( B ρ(u)du) for any bounded B S. 1. Classical definition : X Poisson(S, ρ) X is a Poisson point process on S R d S <, with intensity function ρ (loc. integrable) if for any h : N lf R+, B S s.t. B ρ(u)du < E(h(X B )) = n 0 e B ρ(u)du n! B... h({x 1,..., x n }) B n ρ(x i )dx i. Theorem S R d, X Poisson(S, ρ) exists and is uniquely determined by v(b) = exp( B ρ(u)du), for any bounded B S. 3 / 20 i=1

4 A few properties of Poisson point processes Proposition : if X Poisson(S, ρ) 1. EN (B) = VarN (B) = ρ(u)du wich equals ρ B when ρ( ) = ρ B (homogeneouse case, i.e. stationary and istotropic case if S = R d ). 2. For any u, v S, ρ (2) (u, v) = ρ(u)ρ(v) (also valid for ρ (k), k 1) 3. and if S <, X admits a density w.r.t. Poisson(S, 1) given by f (x) = e S S ρ(u)du u x ρ(u). 4. Slivnyak-Mecke Theorem : for any non-negative function h : S N lf R +, then E h(u, X \ u) = E(h(u, X))ρ(u)du. u X S 4 / 20

5 Exercise 1 : Proof of 2-4 Exercises For property 2. Let A, B S and α (2) (A B) = µ(a)µ(b). For property 3. Let Y Poisson(S, 1) and prove that E(h(X)) = E(f (Y )h(y )) for any h : N lf R +. For property 4. Assume S < and the use the integral definition. Exercise 2. Let X be a homogeneous Poisson point process on S = [0, 1] 2. Compute the average number of R-closed neighbours of X in S, ie. average number of points which have at least a distinct neighbour in X at distance R. Extend Slivnyak-Mecke Formula to compute expectations of the form u,v h(u, v, X \ {u, v}). To what corresponds the average of u,v 1(d(u, X \ {u, v} R), d(v, X \ {u, v} R)). Compute this expectation using the extended SM formula. 5 / 20

6 Simulation of a Poisson p.p. on a bounded domain Let W be the window of simulation. Homogeneous case is straightforward : N (W ) P(ρ W ) ; let n the realization ; generate x 1,..., x n independent uniform points in W ; x = {x 1,..., x n }. Inhomogeneous case : a thinning procedure can be efficiently done if ρ(u) c : let y be a simulation of a Poisson(c,W) ; delete a point u y with prob. 1 ρ(u)/c. Find the (most probable) related realization in W = [ 1, 1] 2 for the intensities : ρ(u) = βe u 1 u 2 1.5u3 1 ; ρ = 200 ; ρ(u) = βe 2 sin(4πu 1 u 2 ) (β is adjusted s.t. the mean number of points in S, W ρ(u)du = 200.) 110 points / 20

7 Statistical inference for a Poisson point process Inference : consists in estimating ρ : homogeneous case. ρ( ; θ) : parametric inhomogeneous function. ρ(u) : non parametric estimation Note : All these estimates can be used even if the spatial point process is not Poisson (wait for a few 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,... 7 / 20

8 Homogeneous case We consider here the problem of estimating the parameter ρ of a homogeneous Poisson point process defined on S and observed on a window W S. Since N (W ) P(ρ W ), the natural estimator of ρ is ρ = N (W )/ W Properties 8 / 20

9 Homogeneous case We consider here the problem of estimating the parameter ρ of a homogeneous Poisson point process defined on S and observed on a window W S. Since N (W ) P(ρ W ), the natural estimator of ρ is ρ = N (W )/ W Properties (i) ρ corresponds to the maximum likelihood estimate. 8 / 20

10 Homogeneous case We consider here the problem of estimating the parameter ρ of a homogeneous Poisson point process defined on S and observed on a window W S. Since N (W ) P(ρ W ), the natural estimator of ρ is ρ = N (W )/ W Properties (i) ρ corresponds to the maximum likelihood estimate. (ii) ρ is unbiased. 8 / 20

11 Homogeneous case We consider here the problem of estimating the parameter ρ of a homogeneous Poisson point process defined on S and observed on a window W S. Since N (W ) P(ρ W ), the natural estimator of ρ is ρ = N (W )/ W Properties (i) ρ corresponds to the maximum likelihood estimate. (ii) ρ is unbiased. (iii)var ρ = ρ W. Proof : (i) follows from the definition of the density (ii-iii) can be checked using the Campbell formulae. 8 / 20

12 Asymptotic results Homogeneous case (2) 9 / 20

13 Homogeneous case (2) Asymptotic results For large N (W ), ρ W N(ρ W, ρ W ) and so W 1/2 ( ρ ρ) N(0, ρ). (the approximation is actually a convergence as W R d ) 9 / 20

14 Homogeneous case (2) Asymptotic results For large N (W ), ρ W N(ρ W, ρ W ) and so W 1/2 ( ρ ρ) N(0, ρ). (the approximation is actually a convergence as W R d ) Variance stabilizing transform : 2 W 1/2 ( ρ ρ) N(0, 1) 9 / 20

15 Homogeneous case (2) Asymptotic results For large N (W ), ρ W N(ρ W, ρ W ) and so W 1/2 ( ρ ρ) N(0, ρ). (the approximation is actually a convergence as W R d ) Variance stabilizing transform : 2 W 1/2 ( ρ ρ) N(0, 1) We deduce a 1 α (α (0, 1)) confidence interval for ρ IC 1 α (ρ) = ( ρ ± z α/2 ) 2. 2 W 1/2 9 / 20

16 A simulation example We generated m = replications of homogeneous Poisson point processes with intensity ρ = 100 on [0, 1] 2 (blcak plots) and on [0, 2] 2 (red plots). Histograms of ρ Histograms of 2 W 1/2 ( ρ ρ) Density [0,1]^2 [0,2]^2 N(100,100) N(100,100/4) Density [0,1]^2 [0,2]^2 N(0,1) / 20

17 A simulation example We generated m = replications of homogeneous Poisson point processes with intensity ρ = 100 on [0, 1] 2 (black plots) and on [0, 2] 2 (red plots). W = [0, 1] 2 W = [0, 2] 2 Emp. Mean of ρ Emp. Var. of ρ Emp. Coverage rate of 95% confidence intervals 95.31% 94.78% 11 / 20

18 Quadrat counting based test Performed as follows : 1. divide W into quadrats B 1,..., B m. 2. count n j = n(x Bj ) for j = 1,..., m. Under the null hypothesis, then n j are realizations of P(ρ B j ). 3. Construct the χ 2 statistic (n j e j ) 2 (n j ρ B j ) 2 T = = e j ρ B j where ρ = n/ W 4. Under the null hypothesis if T χ 2 (m 1) Example for the swedishpines dataset : j j / 20

19 Applications in R Pines datasets Consider the three unmarked datasets : japanesepines, swedishpines, finpines. Plot the data, estimate the intensity parameter. Construct a confidence interval for each of them. Which one is significantly the most abundant? Judge the assumption of the Poisson model using a GoF test based on quadrats. bei dataset Show that the bei (tropical forest dataset), the qk (earth quakes), the lightning strikes (for each summer) datasets cannot be modelled by a homogeneous Poisson point process. 13 / 20

20 Inhomogeneous case : parametric estimation Assume that ρ is parametrized by a vector θ R p (p 1). The most well-known model is the log-linear one : ρ(u) = ρ(u; θ) = exp(θ z(u)) where z(u) = (z 1 (u), z 2 (u),..., z p (u)) corresponds to known spatial functions or spatial covariates. θ can be estimated by maximizing the log-likelihood on W l W (X, θ) = log ρ(u; θ) + (1 ρ(u; θ))du u X W W = W + θ z(u) exp(θ z(u))du. u X W } W {{ } :=l W (X,θ) In other words { } d θ = Argmax θ l W (X, θ) = Argmin θ dθ l W (X; θ). 14 / 20

21 A very short background on estimating equations θ = ArgminU n (θ), θ 0 : true parameter. U n (θ) is said to define an unbiased estimating equation if EU n (θ 0 ) = 0. We usually define (a) Σ = Var(U n (θ 0 )) the variance-covariance matrix of U n (θ 0 ). (b) S = S(θ 0 ) where S(θ) = E ( d dθ U n(θ) ) the sensitivity matrix. If U n (θ 0 ) approx. N(0, Σ), then under some technical and regularity assumptions θ θ 0 approx. N(0, S 1 ΣS 1 ) Sketch of the proof : there exists θ s.t. θ θ 0 θ θ 0 U n (θ 0 ) U n ( θ) = U n (θ 0 ) = ( θ θ 0 )S( θ) ( θ θ 0 )S so θ θ 0 S 1 U n (θ 0 ). 15 / 20

22 Inhomogeneous case : parametric estimation (2) The score function s W (X, θ) = d dθ l W (X, θ) is an unbiased estimating equation! Indeed, 16 / 20

23 Inhomogeneous case : parametric estimation (2) The score function s W (X, θ) = d dθ l W (X, θ) is an unbiased estimating equation! Indeed, s W (X, θ) = z(u) u X W W z(u) exp(θ z(u)) du } {{ } :=ρ(u) 16 / 20

24 Inhomogeneous case : parametric estimation (2) The score function s W (X, θ) = d dθ l W (X, θ) is an unbiased estimating equation! Indeed, s W (X, θ) = z(u) u X W and from Campbell Theorem W z(u) exp(θ z(u)) du } {{ } :=ρ(u) 16 / 20

25 Inhomogeneous case : parametric estimation (2) The score function s W (X, θ) = d dθ l W (X, θ) is an unbiased estimating equation! Indeed, s W (X, θ) = z(u) u X W W z(u) exp(θ z(u)) du } {{ } :=ρ(u) and from Campbell Theorem Es W (X, θ) = z(u) ( exp(θ0 z(u)) exp(θ z(u)) ) du = 0 when θ = θ 0. W Exercise : verify that Σ = S = W z(u)z(u) exp(θ z(u))du. Hence, θ θ 0 N(0, S 1 ), i.e. S( θ) 1/2 ( θ θ 0 ) N(0, I p ). 16 / 20

26 Inhomogeneous case : parametric estimation (2) The score function s W (X, θ) = d dθ l W (X, θ) is an unbiased estimating equation! Indeed, s W (X, θ) = z(u) u X W W z(u) exp(θ z(u)) du } {{ } :=ρ(u) and from Campbell Theorem Es W (X, θ) = z(u) ( exp(θ0 z(u)) exp(θ z(u)) ) du = 0 when θ = θ 0. W Exercise : verify that Σ = S = W z(u)z(u) exp(θ z(u))du. Hence, θ θ 0 N(0, S 1 ), i.e. S( θ) 1/2 ( θ θ 0 ) N(0, I p ). Asymptotic results formalized by Rathbun and Cressie (1994). 16 / 20

27 Data example : dataset bei A point pattern giving the locations of 3605 trees in a tropical rain forest. Accompanied by covariate data giving the elevation (altitude) (z 1 ) and slope of elevation (z 2 ) in the study region. elevation, z1 elevation gradient, z Assume an inhomogeneous Poisson point process (which is not true, see the next chapter) with intensity log ρ(u) = β + θ 1 z 1 (u) + θ 2 z 2 (u). Question : how can we prove that each covariate has a significant and positive influence? / 20

28 Nonparametric estimation (see e.g. Diggle 2003) Idea is to mimic the kernel density estimation to define a nonparametric estimator of the spatial function ρ. Let k : R d R + a symmetric kernel with intensity one. Examples of kernels Gaussian kernel : (2π) d/2 exp( y 2 /2). Cylindric kernel : 1 π 1( y 1). Epanecnikov kernel : 3 4 1( y < 1)(1 y 2 ). Let h be a positive real number (which will play the role of a bandwidth window), then the nonparametric estimate (with border correction) at the location v is defined as ( ) ρ h (v) = K h (v) 1 1 v u h k, K d h (v) = h d k h h u X W Analogy with heatmaps! ( v u ) du 18 / 20

29 Intuitively, this works... Indeed, using the Campbell formula and a change of variables we can obtain ( ) E ρ h (v) = K h (v) 1 1 v u E h k d h u X W ( ) = K h (v) 1 1 v u W h k ρ(u)du d h = K h (v) 1 k ( ω ) ρ(ωh + v)dω h small ρ(v). W v h K h (v) 1 W v h k ( ω ) ρ(v)dω More theoretical justifications and properties and a discussion on the bandwidth parameter and edge corrections can be found in Diggle (2003). 19 / 20

30 Application to the qk dataset (quakes) sigma=.5 sigma=1 sigma= sigma= / 20