Second-Order Analysis of Spatial Point Processes

Transcription

1 Title Second-Order Analysis of Spatial Point Process Tonglin Zhang

2 Outline Outline Spatial Point Processes Intensity Functions Mean and Variance Pair Correlation Functions Stationarity K-functions Some Asymptotics Nonstationary Model Estimation

3 Spatial Point Processes Definition The definition of point processes is well-established and can be found in many textboods. Overall, a point process is defined on a complete separable metrix space S R d. Let N = N(A) be the number of points observed in A S, where S is the collection of Borel subsets of S. Then, N(A) is finite if A is bounded.

4 Intensity Functions Intensity Functions The k-th order intensity function of N (if it exists) as { } E[N(ds1 ) N(ds k )] λ k (s 1,, s k ) = lim, ds i 0,i=1,,k ds 1 ds k where s i are distinct points in S, ds i is an infinitesimal region containing s i S, and ds i is the Lebesgue measure of ds i.

5 Intensity Functions The first and second-order intensity functions It is of interest in the first and second-order intensity functions. The first-order intensity function is λ(s) = E[N(ds)] lim ds 0 ds and the second-order intensity function is where s 1 s 2. λ 2 (s 1, s 2 ) = E[N(ds 1 )N(ds 2 )] lim ds 1, ds 2 0 ds 1 ds 2

6 Mean and Variance Mean and Variance Functions The mean measure of N is µ(a) = A λ(s)ds, where λ(s) = λ ( s) is the first order intensity function. The second-order moment measure is µ (2) (A 1, A 2 ) = λ 2 (s 1, s 2 )ds 2 ds 1. A 2 A 1

7 Mean and Variance Then, the mean structure of N is E[N(A)] = µ(a) and the covariance structure of N is Cov[N(A 1 ), N(A 2 )] =µ (2) (A 1, A 2 ) + µ(a 1 A 2 ) µ(a 1 )µ(a 2 ) = [λ 2 (s 1, s 2 ) λ(s 1 )λ(s 2 )]ds 2 ds 1 + λ(s)ds. A 2 A 1 A 2 A 1

8 Pair Correlation Function Let g(s 1, s 2 ) = λ 2(s 1, s 2 ) λ(s 1 )λ(s 2 ). Then, g is called the pair correlation function and the covariance structure is Cov[N(A 1 ), N(A 2 )] = [g(s 1, s 2 ) 1]λ(s 1 )λ(s 2 )ds 2 ds 1 + A 2 λ(s)ds. A 1 A 2 A 1

9 Stationarity Strong Stationarity A spatial point process N is said strong stationary if for any measurable A 1,, A k B(R d ) the joint distribution of does not depend on s, where N(A 1 + s),, N(A k + s) A i + s = {s + s : s A i }.

10 Stationarity If λ(s) is a constant and Second-Order Stationarity λ 2 (s 1 s 2 ) = λ 2 (s 1 s 2 ), then N is called second-order stationary. In addition, if then N is called isotropic. If N is isotropic, then λ 2 (s 1 s 2 ) = λ 2 ( s 1 s 2 ), g(s 1, s) = g( s 1 s 2 ).

11 K-functions K-functions Suppose N is stationary. Let λ be the first-order intensity function. Then, the K-function is defined by K(t) = 1 E[number of extra events within λ distance of t of a randomly chosen event]. The L-function is L(t) = K(t) π t. In real application, K(t) is more often used.

12 K-functions Estimation Estimation of K-function, L-function and pair correlation can be found in R. The method is ˆK(t) = 1ˆλ I (d ij < t) w ij, n i j i where d ij is the distance, I is the indicator function, w ij is the edge correlation. In general, we choose w ij = 1 if the disc centered at s i with radius t is completely inside. Otherwise, w ij is just the proportion of the disc inside the domain.

13 Asymptotics Under Stationarity Strong mixing condition We say N satisfies the strong mixing condition if where lim α(ar, ad) = 0, (1) a α(r, d) = sup ρ(e 1, E 2 ) r d(e 1 ) d, d(e 2 ) d sup U 1 F(E 1 ) U 2 F(E 2 ) P(U 1 U 2 ) P(U 1 )P(U 2 ), d(e) = sup s,s E ρ(s, s ) is the diameter of E, d(e 1, E 2 ) = sup s E,s E ρ(s, s ) is the maximum distance between disjoint sets of points.

14 Asymptotics Under Stationarity Some Results (Theorems 4.1 and 4.3 in Ivanoff (1982)). Let N be a strong stationary spatial point process and also satisfies the strong mixing condition. If 0 < C 2 = Γ(0, s)ds <, R d where Γ(0, s) is the covariance function of N, then for any A R d and N(ηA) E[N(ηA)] M η (A) = η d 2 D N(0, A ) (M η (A 1 ),, M η (A k )) D (M(A 1 ),, M(A k )) for any disjoint Borel A 1,, A k R d, where M(A 1 ),, M(A k ) are independent normal random variables.

15 Asymptotics Under Stationarity (Corollary 7.2 in Ivanoff (1982)). If the kth-order factorial cumulant density function Q k of N satisfies s 1,,s k R d Q k (s 1,, s k ) ds 1 ds k < C 1, k = 2, 3, 4, and s 1,,s k R d Q k (s 1,, s k ) ds 1 ds k 1 < C 2, k = 2, 3, 4 for some constants C 1, C 2 R, then M η ( d i=1 (0, t i]) with t i > 0 for i = 1,, d weakly converges to a Gaussian process which can be determined by the previous conclusion.

16 Nonstationary Model The Second-Order Intensity-Reweighted Stationary Model Baddeley, Moller and Waagepetersen (2000) propose the model. It requires that the pair correlation function has g(s 1, s 2 ) = λ 2(s 1, s 2 ) λ(s 1 )λ(s 2 ) = g(s 1 s 2 ) some times, it even requires that g(s 1, s 2 ) = g( s 1 s 2 ).

17 Nonstationary Model If the process has an isotropic pair correlation function, then K(s) = 2π s 0 ug(u)du, s > 0. For an inhomogeneous Poisson process, K(s) = πs 2. For a cluster process, K(s) > πs 2. For an inhibition process, K(s) < πs 2.

18 Estimation Stochastic Integral Let S be an set. Then, N(S) = and E[N(S)] = S A N(ds) E[N(ds)] = A λ(s)ds.

19 Estimation We consider T = = i i, s i s j ϵ S 1 S si s j s s ϵ,s s 1 S si s j N(ds )N(ds).

20 Estimation Then, E(T ) = = = S S s s ϵ,s s 1 s s ϵ t S s, s t ϵ =πϵ 2. S s s E[N(ds )N(ds)] λ 2 (s s 1 ) S s s ds ds λ 2 (t) S s s dsdt Therefore, if we assume N is strong stationary, then Ĉ 2 = 1ˆλ 1 S si s j πϵ2ˆλ + 1 where ˆλ is N(S)/ S. i i, s i s j ϵ

21 Estimation Using the same idea, we can show that the estimate of K function under the inhomogeneous model is ˆK(t) = 1 S I ( s s t) ˆλ(s)ˆλ(s s s S,s s ) This is a moment estimator of the K-function..

22 Estimation Likelihood function for the first-order intensity function If N is Poisson process, then the loglikelihood function is l(β) = n i=1 log[λ(s i, β)] λ(s, β)ds. S

23 Future Research Second-Order Parametric and Nonparametric Inference There is no second-order parametric model. Even there is a parametric model, it is still unknown how to write down the likelihood function. Although there are a few method to estimate the second-order parameter, the behavior is still unknown.

24 Reference Reference Baddeley, A.J., Møller, J., and Waagepetersen, R. (2000). Non- and semi-parametric estimation of interaction in inhomogeneous point patterns. Statistica Neerlandica, 54, Diggle, P.J., Gómez-Rubio, V., Brown, P.E., Chetwynd, A.G., and Gooding, S. (2007). Second-order analysis of inhomogeneous spatial point processes using case-control data. Biometrics, 63, Guan, Y. (2008). A KPSS test for stationarity for spatial point processes. Biometrics, 64, Ivanoff, G. (1982). Central limit theorems for point processes. Stochastic Processes and Their Applications, 12,