Point Processes. Tonglin Zhang. Spatial Statistics for Point and Lattice Data (Part II)

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1 Title: patial tatistics for Point Processes and Lattice Data (Part II) Point Processes Tonglin Zhang

2 Outline Outline imulated Examples Interesting Problems Analysis under tationarity Analysis under Nonstationarity Likelihood Analysis tochastic Integral Asymtotic Frameworks

3 imulated Examples Poisson Processes Let be the study area. A poisson process is derived if N(A 1 ),, N(A k ) are iid Poisson random variable with mean µ(a 1 ),, µ(a k ) if A 1,, A k are disjoint subsets of. Extensions: Cox process: µ is a random measure (e.g. log µ is given by a Gaussian random field). Mixed poisson process (mean measure is given by Y µ, where Y is a nongenative random variable).

4 imulated Examples y Homogeneous Poisson Process x Figure : A Homogeneous Poisson Point Process on [0, 1] 2.

5 imulated Examples Neyman cott Cluster Process A Neyman cott cluster process has: A parent process and an off-spring process. The parent process is Poisson. The off-spring process is derived from the parent process: each parent point generate a Poisson number of off-spring points around it. Example: Let s 1,, s m be parent point. Then, There are m clusters. Cluster 1 is around s1 ; cluster 2 is around s 2 ; and so on.

6 imulated Examples y Homogeneous Poisson Cluster Process x Figure : A Homogeneous Neyman cott Cluster Process on [0, 1] 2.

7 imulated Examples How to generate an inhomogeneous Poisson point process Let f be a PDF on s. Then, we can Generate n Poisson(b). Generate n observations independently from a PDF f. Let them be s 1,, s n. Then, {s 1,, s m } is a sample from Poisson point process with intensity λ(s) = bf (s).

8 imulated Examples Other Processes Compound Point Process Generate n from a nonnegative discreate random varible. Generate n observations independent from f.

9 Interesting Research Problems tationarity: λ(s) is a constant; λ 2 (s 1, s 2 ) = λ 2 (s 1 s 2 ); g(s 1, s 2 ) = g(s 1 s 2 ); and etc. Isotropic: tationarity, and λ 2 (s 1, s 2 ) = λ 2 ( s 1 s 2 ). Nonstationary process: estimation of intensity functions. tationary temporal point process: return (or recurrence) intervals. econd-order analysis: K-function, L-function, Pair correlation function. Return Intervals. Marked point processes: relationship between points and marks.

10 Analysis under tationarity The K-Function (Ripley 1976) Let {s 1,, s n } be observed points from a stationary point process with intensity λ. K-function describes the second-order properties, which is t K(t) = 2π ug(u)du, 0 where u is the pair correlation function.

11 Analysis under tationarity We can estimate ˆK(t) = 1 A λ 2 n w ij I (d ij t), where A is a disc, w ij is the weight for edge correction, and d ij = s i s j (λ may be replaced as ˆλ if λ is unknown). i=1 There is a local version, which is called the local K-function, which is ˆK local (t; s i ) = 1 w ij I (d ij t). A λ j i j i

12 Analysis under tationarity If N is a homogeneous Poisson point process, then For any s in, let K(t) = 2π t 0 udu = πt 2. U s,t = {s : 0 < s s } t}. If there is a point at s, then E[N(U s,t ) s] = πt 2. Therefore, the L-function K(t) L(t) = π t is useful.

13 Analysis under tationarity If N is a homogeneous cluster point process with cluster size k, then g(u) k. Therefore, Then, There is For any A, there is K(t) kπt 2. E[N(U s,t ) s] kπt 2. L(t) (k 1)t > 0. E[N(A)] = λ A and V [N(A)] ke[n(a)] = kλ A

14 Analysis under tationarity Pair Correlation Function If N is a Neyman cluster process, then we roughly have g(u) k < 1 for some k. Then, L(t) < 0.

15 Analysis under tationarity Pair Correlation Function If N is strong stationary, then (under for conditions for weak dependence) for any A, we roughtly have V [N(A)] = ϕe[n(a)] where ϕ = λ [g(u) 1]du + 1. R d

16 Analysis under tationarity A imulated Example We evaluate the value ϕ = V [N(A)]/E[N(A)] based on the following example. N is a cluster process on [0, 1] 2 with cluster size k. The cluster is determined by a bivariate normal with center at its parent point and standard deviance σ. The parameters are (λ, σ, k). A is a rectangle at (0.2, 0.2), (0.2, 0.8), (0.8, 0.2), and (0.8, 0.8). I simulated 1000 times.

17 Analysis under tationarity y Example for Dispersion Effect x Figure : An Example for Dispersion Effect under tationarity

18 Analysis under tationarity Table : Averages under tationarity in 10, 000-run simulations λ σ k E[N(A)] V [N(A)] ϕ

19 Annalysis under tationarity Return Intervals. Let N be a stationary point process on R. Let τ be the time for the next occurrence. If N is Poisson, then the average return time of a point is E(τ) = 1 λ. If N is a cluster process, then we should consider the return of parent points. It is about E(τ) = 1 kλ. If there is a return, then it is a return of cluster.

20 Annalysis under Nonstationarity Assume N is not stationary with first-order intensity λ(s) and second-order intensity λ 2 (s 1, s 2 ). Recently, most research focuses on the second-order intensity-reweighted stationary or isotropic procoess, where the first is defined by and the second is defined by g(s 1, s 2 ) = λ 2(s 1, s 2 ) λ(s 1 )λ(s 2 ) = g(s 1 s 2 ) g(s 2, s 2 ) = g( s 1 s 2 ).

21 Analysis under Nonstationarity For any A, there is V [N(A)] (ϕ 1)E 2 [N(A)] + E[N(A)] for a centain ϕ. If N is an inhomogeneous cluster process, then we roughly have V [N(A)] ke[n(a)], where k is the cluster size. Therefore, an inhomogeneous cluster process is not a second-order intensity-reweighted stationary process.

22 Analysis under Nonstationarity If λ(s) is given, then the K-function is estimated by ˆK(t, λ) = 1 A n i=1 j i where A is a disc and w ij is edge-correction. w ij I (d ij t) λ(s i )λ(s j ),

23 Analysis under Nonstationarity One should Estimate λ(s); Estimate λ 2 (s 1, s 2 ) by estimating g(u) or K(u); It is hard to provide nonparametric estimation for both the first-order and the second-order intensity functions. Therefore, one often (e.g. Guan (2009) JAA, ) considers: A parametric analysis for λ(s); and a nonparametric analysis for λ 2 (s 1, s 2 ).

24 Analysis under Nonstationarity A imulated Example We generate data from cluster process with λ(s) = bβ(2, 2) y Inhomogeneous Poisson Cluster Process x

25 Analysis under tationarity Table : Averages under Nonstationarity in 10, 000 imulations b σ k E[N(A)] V [N(A)] ϕ

26 Likelihood Analysis If N is an inhomogeneous Poisson process with intensity function λ θ (s), then the loglikelihood function is l(θ) = n i=1 log λ θ (s i ) λ θ (u)du. If N is not Poisson, the above is the composite loglikelihood function, which can also provide a consistent estimator.

27 tochastic Integral Let f (s) be a function. We can use tochastic Integral method as Then, and E(U 2 ) = Therefore, V (U) = U = n f (s i ) = i=1 E(U) = f (s)n(ds). f (s)λ(s)ds f (s)f (s )λ 2 (s, s )ds ds + f 2 (s)λ(s)ds. f (s)f (s )[g(s, s ) 1]λ(s)λ(s )ds ds+ f 2 (s)λ(s)ds.

28 tochastic Integral For a bivariate function f (s, s ), there is U = f (s, s )N(ds)N(s ). Then, E(U) = f (s, s )λ 2 (s, s )dsds + f (s, s)λ(s)ds.

29 tochastic Integral In addition, there is E(U 2 ) = f (s, s )f (s, s )λ 4 (s, s, s, s )dsds ds ds + [2f (s, s )f (s, s ) + f (s, s )f (s, s ) + f (s, s )f (s, s ) + 2f (s, s )f (s, s)]λ 3 (s, s, s )dsds ds + [f (s, s)f (s, s ) + f (s, s )f (s, s ) + f (s, s )f (s, s)]λ 2 (s, s )dsds + f 2 (s, s)λ(s)ds.

30 tochastic Integral For l(θ), there is l(θ) = Then, we have l(θ) θ j = log λ θ (s)n(ds) λ θ (u)du. 1 λ θ (s) λ θ (s) N(ds) θ j λ θ (u) du. θ j Thus, E[ l(θ) θ j ] = 0.

31 Asymptotic Frameworks There are two different types of asymtotic frameworks: Increasing domain: increases to R d but intensity functions do not vary (focused by most research). Fixed domain: does not vary but intensity functions goes to infnity (e.g. λ(s) = bλ 0 (s) as b, rarely discussed).