Estimating functions for inhomogeneous spatial point processes with incomplete covariate data

Transcription

1 Estimating functions for inhomogeneous spatial point processes with incomplete covariate data Rasmus aagepetersen Department of Mathematics Aalborg University Denmark August 15, / 23

2 Data (Barro Colorado Island Forest Dynamics Plot) Observation window: S = [0, 1000] [0, 500]m 2 Beilschmiedia Ocotea Elevation Gradient norm (steepness) Question: tree intensities related to elevation and gradient? 2 / 23

3 Log-linear models for intensity function z(u) = ( z 1 (u),...,z p (u) ) vector of covariates for each location u in observation window. E.g. z(u) = (1, z elev (u), z grad (u)) for rain forest example. Log-linear model for intensity function: Interpretation: λ(u; β) = exp(z(u)β T ) λ(u; β)dn u = P(point occurs in neighbourhood N u around u) 3 / 23

4 Estimation of β X: point pattern observed in Poisson process log likelihood function: l(β) = z(u)β T u X Poisson score estimating function: u(β) = z(u) u X λ(u; β)du z(u)λ(u; β)du Also applicable for non-poisson point processes with intensity function λ( ; β) (Schoenberg, 2005, aagepetersen, 2007) 4 / 23

5 Missing covariate data Elevation covariate interpolated from elevation observations on grid. However, evaluating u(β) = z(u) z(u)λ(u; β)du u X requires z(u) observed for any u! 5 / 23

6 Approximations of log likelihood I Suppose z(u) observed at finite set of locations Q. Rathbun (1996) approximate z(u)λ(u; β)du z(u)λ(u; β)du where z(u)λ(u; β) unbiased prediction of z(u)λ(u; β), u given z(u), u Q. 6 / 23

7 Approximations of log likelihood I Suppose z(u) observed at finite set of locations Q. Rathbun (1996) approximate z(u)λ(u; β)du z(u)λ(u; β)du where z(u)λ(u; β) unbiased prediction of z(u)λ(u; β), u given z(u), u Q. Riemann approximation: z(u)λ(u; β)du w(u)z(u)λ(u; β) u Q where w(u) quadrature weight for u Q. 7 / 23

8 Approximations of log likelihood II: spatstat Approximation of score function used in R package spatstat (Baddeley and Turner) u(β) u spat (β) = u X z(u) u Q w(u)z(u)λ(u; β) but now Q = X D includes observed points in addition to dummy points D. Note: X events of point pattern (supplied by nature ) D dummy points controlled by scientist. Two types of weights: grid or dirichlet 8 / 23

9 Quadrature schemes in spatstat Grid Dirichlet * * * * * * * * * * * * * w(u) = C v #(X C v )1, u C v where = v D C v w(u) area of Dirichlet cell for u in Dirichlet tesselation generated by Q. 9 / 23

10 spatstat: relation to generalized linear models and iterative weighted least squares Estimating function u spat (β) = u X z(u) u X D w(u)z(u)λ(u; β) formally equivalent to score function of weighted Poisson regression (log link): u spat (β) = u X D w(u)z(u) [ y u λ(u; β) ] with weights w(u) and observations y u = 1[u X]/w(u). Hence implementation straightforward using e.g. glm() in R (Poisson family, log link). 10 / 23

11 Distribution of parameter estimates from approximate score functions? Problem: hard to obtain distribution of parameter estimates from approximate score functions 11 / 23

12 Monte Carlo approximation of integral Consider M random uniform dummy points D on (simple random dummy points). Assume wlog = 1. Rathbun et al. (2006): Monte Carlo approx. of integral: u rath (β) = z(u) 1 z(u)λ(u; β) M u X u D = u(β) [ 1 z(u)λ(u; β) M u D z(u)λ(u; β)du ] 12 / 23

13 Monte Carlo approximation of integral Consider M random uniform dummy points D on (simple random dummy points). Assume wlog = 1. Rathbun et al. (2006): Monte Carlo approx. of integral: u rath (β) = z(u) 1 z(u)λ(u; β) M u X u D = u(β) [ 1 z(u)λ(u; β) M u D z(u)λ(u; β)du ] CLT for Monte Carlo approximation: M 1/2[ 1 z(u)λ(u; β) z(u)λ(u; β)du] d N(0, G) M u D where G asymptotic covariance matrix. 13 / 23

14 Distribution of parameter estimate (Poisson process case) Poisson score asymptotically normal where i(β) Fisher information. u(β) N ( 0, i(β) ) Suppose ˆβ solution of u rath (β) = 0 where u rath (β) = u(β) [ 1 z(u)λ(u; β) M u D z(u)λ(u; β)du ] 14 / 23

15 Distribution of parameter estimate (Poisson process case) Poisson score asymptotically normal where i(β) Fisher information. u(β) N ( 0, i(β) ) Suppose ˆβ solution of u rath (β) = 0 where u rath (β) = u(β) [ 1 z(u)λ(u; β) M u D z(u)λ(u; β)du ] Then ˆβ N ( β, i(β) 1 i(β) 1 G i(β) 1) NB: increasing intensity asymptotics: N = E#X! (fixed observation window ) 15 / 23

16 Stratified dummy points Alternative: one uniformly sampled dummy point in each of M cells (stratified dummy points) Suppose f continuously differentiable. Then CLT [ 1 ] d M f (u) f (u)du N(0, G) M u D (faster rate of convergence). Asymptotic covariance matrix G depends on partial derivatives f i u j of f. 16 / 23

17 Monte Carlo versions of spatstat (aagepetersen, 2007) D point process of dummy points of intensity M. Monte Carlo version of dirichlet u dir (β) = u X z(u) u X D (either simple or stratified dummy points) λ(u; β) z(u) λ(u; β) M 17 / 23

18 Monte Carlo versions of spatstat (aagepetersen, 2007) D point process of dummy points of intensity M. Monte Carlo version of dirichlet u dir (β) = u X z(u) u X D (either simple or stratified dummy points) λ(u; β) z(u) λ(u; β) M Monte Carlo version of grid: (stratified dummy points) u grid (β) = u X z(u) 1 M u X D 1 z(u)λ(u; β) #(X C u ) 1 Advantage: implementation using glm() just as for usual spatstat. 18 / 23

19 Distribution of parameter estimates (Poisson process case) Grid: u grid (β) and u rath (β) with stratified dummy points same asymptotic distribution - hence same asymptotic distribution for parameter estimates. Dirichlet: more subtle behaviour stratified dummy points: u dir (β) less efficient than u grid (β)/u rath (β) simple dummy points: asymptotic covariance matrix tends to that of MLE when q = M/N increases (N = E#X). 19 / 23

20 Numerical example: Poisson process Case of Poisson process with covariate vector (1, z elev ) β elev = 0.1. Ratios of asymptotic standard errors for estimating function estimate ˆβ elev relative to asymptotic standard error for MLE. simple stratified q u rath u grid u dir u dir q = M/N proportion of dummy points. u dir better than Rathbuns Monte Carlo approximation u rath but not useful in case of stratified dummy points. 20 / 23

21 Clustered point processes Consider cluster process X = c Y X c where Y stationary Poisson point process of intensity κ > 0. Given Y, clusters X c are independent Poisson processes with intensity functions λ c (u) = α exp(z 2:p (u)β T 2:p)h(u c; ω) Intensity function of X is then of log-linear form exp(z(u)β T ) where β 1 = log(κα) and z 1 (u) = 1. Estimate ˆβ using u(β), u dir (β) or u grid (β) still asymptotically normal but covariance matrix now depends on ψ = (κ, ω). 21 / 23

22 Minimum contrast estimation of ψ Closed form expression K(t; ψ) and estimate ˆK(t) = u,v X 1[ v u t] λ(u; ˆβ)λ(v; ˆβ) v u available for K-function (extension to inhomogeneous case). 22 / 23

23 Minimum contrast estimation of ψ Closed form expression K(t; ψ) and estimate ˆK(t) = u,v X 1[ v u t] λ(u; ˆβ)λ(v; ˆβ) v u available for K-function (extension to inhomogeneous case). Second step for estimation of ψ: ˆψ = arg min ψ r 0 [ˆK(t) c K(t; ψ) c ] 2 dt NB: only requires z(u) for u X. 23 / 23

24 Inference regarding clustering So far focused on intensity parameter β. Sometimes clustering of main interest but still need to estimate β to adjust for aggregation due to covariates. Joint asymptotic normality of (ˆβ, ˆψ) established in & Guan (2007) for wide class of mixing point processes including cluster processes log Gaussian Cox processes (ψ covariance parameter of Gaussian field). 24 / 23

25 References Baddeley, A., Møller, J., and aagepetersen, R. (2000). Non- and semi-parametric estimation of interaction in inhomogeneous point processes, Statistica Neerlandica, 54, aagepetersen, R. (2007). An estimating function approach to inference for inhomogeneous Neyman-Scott processes, Biometrics, 63, aagepetersen, R. (2007) Estimating functions for inhomogeneous spatial point processes with incomplete covariate data, submitted. aagepetersen, R. and Guan, Y. (2007). Two-step estimation for inhomogeneous spatial point processes, in preparation. 25 / 23