Decomposition of variance for spatial Cox processes

Transcription

1 Decomposition of variance for spatial Cox processes Rasmus Waagepetersen Department of Mathematical Sciences Aalborg University Joint work with Abdollah Jalilian and Yongtao Guan November 8, /34

2 Background: Tropical rain forest ecology Fundamental questions: which factors influence the spatial distribution of rain forest trees and what is the reason for the high biodiversity of rain forests? Key factors: environment: topography, soil composition,... seed dispersal limitation: by wind, birds or mammals... 2/34

3 Example: Capparis Frondosa observation window W = 1000 m 500 m seed dispersal clustering environment inhomogeneity Elevation Potassium content in soil. Quantify dependence on environmental variables and seed dispersal using statistics for spatial point processes. 3/34

4 Example: modes of seed dispersal and clustering Three species with different modes of seed dispersal: Acalypha Diversifolia explosive capsules Loncocharpus Heptaphyllus wind Capparis Frondosa bird/mammal Is degree of clustering related to mode of seed dispersal? Quantify how much of the spatial variation is due to respectively environment and seed dispersal? Approach: Cox process model for joint effects of environment and seed dispersal. 4/34

5 Cox processes X is a Cox process driven by the non-negative random intensity function Λ if, conditional on Λ = λ, X is a Poisson process with intensity function λ. Intensity function Second-order product density ρ(u) = EΛ(u) ρ (2) (u,v) = EΛ(u)Λ(v) = Cov[Λ(u),Λ(v)]+ρ(u)ρ(v) Pair correlation function g(u,v) = ρ(2) (u,v) ρ(u)ρ(v) 5/34

6 Mean and covariances of counts A and B subsets of the plane EN(A) = A ρ(u)du A B Cov[N(A),N(B)] = = A B A B EΛ(u)du + A ρ(u)du + A B B Cov[Λ(u), Λ(v)]dudv ρ(u)ρ(v)[g(u,v) 1]dudv = Poisson variance + extra variance due to Λ 6/34

7 Decomposition of variance for a count Prediction of count N(B) given Λ : ˆN(B) = E[N(B) Λ] = B Λ(u)du VarN(B) = variation = 7/34

8 Decomposition of variance for a count Prediction of count N(B) given Λ : ˆN(B) = E[N(B) Λ] = B Λ(u)du VarN(B) = VarˆN(B) variation = structured variation 8/34

9 Decomposition of variance for a count Prediction of count N(B) given Λ : ˆN(B) = E[N(B) Λ] = B Λ(u)du VarN(B) = VarˆN(B)+Var [ N(B) ˆN(B) ] variation = structured variation + Poisson noise 9/34

10 Decomposition of variance for a count Prediction of count N(B) given Λ : ˆN(B) = E[N(B) Λ] = B Λ(u)du VarN(B) = VarˆN(B)+Var [ N(B) ˆN(B) ] variation = structured variation + Poisson noise Further, (Z =environmental variables) VarΛ(u) = variation of Λ = 10/34

11 Decomposition of variance for a count Prediction of count N(B) given Λ : ˆN(B) = E[N(B) Λ] = B Λ(u)du VarN(B) = VarˆN(B)+Var [ N(B) ˆN(B) ] variation = structured variation + Poisson noise Further, (Z =environmental variables) VarΛ(u) = VarˆΛ(u) variation of Λ = variation due to environment ˆΛ(u) = E[Λ(u) Z] 11/34

12 Decomposition of variance for a count Prediction of count N(B) given Λ : ˆN(B) = E[N(B) Λ] = B Λ(u)du VarN(B) = VarˆN(B)+Var [ N(B) ˆN(B) ] variation = structured variation + Poisson noise Further, (Z =environmental variables) VarΛ(u) = VarˆΛ(u)+Var [ Λ(u) ˆΛ(u) ] ˆΛ(u) = E[Λ(u) Z] variation of Λ = variation due to environment + other sources 12/34

13 Measure of influence of environmental covariates Z: R 2 = VarˆΛ(u) VarΛ(u) = VarE[Λ(u) Z] VarΛ(u) (right hand side does not depend on u in case of stationary environment) Analogy to linear regression: SSR = E [ˆΛ(u) ρ(u) ] 2 du = W VarˆΛ(u) W [ SST = E Λ(u) ρ(u)] 2 du = W VarΛ(u) Then W R 2 = SSR SST 13/34

14 Additive model for Λ Additive model: Λ(u) = Z(u)+Λ 0 (u) Z contribution from environment: Z(u) = βz(u) T Z(u) = (Z 1 (u),...,z p (u)) Λ 0 : structured variation (e.g. seed dispersal) independent of environment. Cox process X union of independent Cox processes with random intensity functions Z(u) and Λ 0 (u). Assume Z and Λ 0 non-negative and stationary. 14/34

15 R 2 for additive model R 2 = σ2 Z σ 2 Z +σ 2 0 σ 2 Z = Var Z(u) and σ 2 0 = VarΛ 0 (u) 15/34

16 Log-linear model Λ(u) = Λ 0 (u)exp[ Z(u)] Interpretation in terms of survival of seedlings: X 0 seedlings: stationary Cox process with random intensity function Λ 0. X thinning of X 0 with survival depending on environment Z. Z does not need to be non-negative. 16/34

17 Example: Cox/cluster process: Inhomogeneous Thomas process X 0 Λ0 shot-noise process X0 cluster process: Offspring distributed according to Gaussian density around Poisson parents 17/34

18 Example: Cox/cluster process: Inhomogeneous Thomas process X 0 Λ0 shot-noise process X0 cluster process: Offspring distributed according to Gaussian density around Poisson parents Inhomogeneity: offspring in X 0 survive according to probability p(u) exp(z(u)β T ) depending on covariates (independent thinning). X /34

19 R 2 for log-linear model R 2 = σ 2 exp Z ] σ 2 exp 0[ Z +σ2 σ 2 exp Z +ρ2 exp Z σ 2 = Varexp[ Z(u)] exp Z ρ exp Z = Eexp[ Z(u)] σ 2 0 = VarΛ 0 (u) 19/34

20 Models for c 0 (u v) = Cov[Λ 0 (u),λ 0 (v)] NB: any positive definite function is a covariance function but not necessarily for a non-negative random process Λ 0. Use covariance functions from explicit constructions of Λ 0. Log-Gaussian: Λ 0 (u) = exp[y(u)] c 0 (u v) = ρ 2 0exp[Cov(Y(u),Y(v))] ρ 2 0 where Y Gaussian field and ρ 0 = EΛ 0 (u). 20/34

21 Models for c 0 (u v) = Cov[Λ 0 (u),λ 0 (v)] NB: any positive definite function is a covariance function but not necessarily for a non-negative random process Λ 0. Use covariance functions from explicit constructions of Λ 0. Log-Gaussian: Λ 0 (u) = exp[y(u)] c 0 (u v) = ρ 2 0exp[Cov(Y(u),Y(v))] ρ 2 0 where Y Gaussian field and ρ 0 = EΛ 0 (u). Shot-noise: Λ 0 (u) = αk(u v) c 0 (u v) = κα 2 k(w)k(w +v u)dw u C R 2 where C homogeneous Poisson with intensity κ and k( ) probability density. 21/34

22 Matérn class Λ 0 shot-noise process: sum of K (ν 1)/2 Bessel densities k( ) centered around points of homogeneous Poisson process. c 0 (h) = κα R 2 k(w)k(w +v u)dw = σ 2 ( h /η) ν K ν ( h /η) ν 1 Γ(ν) With ν = 1/2 (exponential covariance function) c 0 (h) = σ0 2 exp( h /η) With ν = ( Gaussian covariance function) c 0 (h) = σ0 2 exp[ ( h /η)2 ] 22/34

23 Parameter estimation (β) X observed within W R 2 Estimate β and ψ = (σ0 2,η,ν) using X Z. First-order composite likelihood: CL 1 (β) = logρ(u Z;β) u X W ρ( Z,β) intensity function for X Z. For additive model: W ρ(u Z;β)du ρ(u Z,β) = ρ 0 +βz(u) T ρ 0 = EΛ 0 (u) 23/34

24 Estimation of ψ Second-order composite likelihood (given ˆβ): CL 2 (ψ ˆβ) = logρ (2) (u,v Z; ˆβ,ψ) u,v X W u v R u v R ρ (2) (u,v Z; ˆβ,ψ)dudv ρ (2) (, Z;β,ψ) second-order product density for X Z. For additive model: ρ (2) (, Z;β,ψ) = ρ(u Z,β)ρ(v Z,β)+c 0 (u v;ψ) 24/34

25 Environmental variances Z observed on grid G = {u i } i=1,...,m and Z(u) = βz(u) T. and where σ 2 Z = 1 M σ 2 exp Z = 1 M u G 2 { Z(u) ρ Z} u G { exp [ Z(u) ] } 2 ρexp Z = 1 Z(u) ρ ρ Z M exp = 1 exp [ Z(u) ] Z M u G u G 25/34

26 Results for rain forest data Covariates elevation and potassium for Acalypha and Capparis. Nitrogen and phosphorous for Lonchocarpus. Qualitative similar results for additive and log-linear model regarding dependence on covariates (β). Species model for Λ c 0 CL 2 ( ψ β) R 2 log-linear Gaussian Matérn Acalypha LG-Matérn additive Gaussian Matérn LG-Matérn LG-Matérn: c 0 (h) = ρ 2 0 [exp(c(h)) 1] where c( ) Matérn. 26/34

27 Results continued Species model for Λ c 0 CL 2 ( ψ β) R 2 log-linear Gaussian Matérn Loncocharpus LG-Matérn additive Gaussian Matérn LG-Matérn log-linear Gaussian Matérn Capparis LG-Matérn additive Gaussian Matérn LG-Matérn /34

28 Fitted covariance functions (log-linear model) Acalypha Loncocharpus Capparis c(r) Matern LG Matern Cauchy Gaussian c(r) Matern LG Matern Cauchy Gaussian c(r) Matern LG Matern Cauchy Gaussian r r r 28/34

29 Some conclusions Estimated R 2 sensitive to choice of model: especially c 0. Best fit with log-linear model (interpretation in terms of survival). Best fit with (LG)-Matérn (heavy tails for covariance/cluster density). Largest R 2 for Capparis (bird/mammal seed dispersal), smallest for Acalypha (explosive capsules). 29/34

30 Stationary environment? Z: Acalypha (additive) Acalypha (log-linear) 6e Loncho (log-linear) Capparis (log-linear) Handling possible non-stationarity topic for further research. 30/34

31 Composite likelihood obtained from binary random field Random count variables: N i = #X C i number of points in C i. Disjoint subdivision W = n i=1 C i in cells C i. X i = 1[N i > 0] binary random variable. P(X i = 1) = ρ(u i ;β) C i. Bernouilli composite likelihood n P(X i = 1) X i (1 P(X i = 1)) 1 X i i=1 has limit ( C i 0) CL 1 (β) = u X W n ρ(u i ;β) X i (1 ρ(u i ;β) C i ) 1 X i i=1 logρ(u;β) ρ(u; β)du W 31/34

32 Second-order composite likelihood obtained from binary variables associated with joint presence of points in pairs of cells C i and C j : X ij = 1[N i > 0,N j > 0] (= N i N j when C i and C j small ) where P(X ij = 1) = EN i N j = ρ (2) (u,v;β,ψ) C i C j 32/34

33 Alternative: GEE Observation vector Y = [N 1,...,N m ] Mean vector µ = [ ρ(u 1 ) C 1,...,ρ(u m ) C m ] Working covariance matrix V = [V ij ] ij V ii = VarN i = ρ(u i ) C i +ρ(u i ) 2 [g(u i,u i ;ψ) 1] V ij = Cov[N i,n j ] = ρ(u i )ρ(u j )[g(u i,u j ;ψ) 1] where g( ; ψ) working pair correlation function. 33/34

34 GEE [Y µ]v 1 D D = dµt dβ has limiting form (using Neuman series for V 1 ) u X W + v 0 X W W ρ (u;β) ρ(u) 1 ρ(v 0 ) ρ(v 0 ;β) ρ(v 0 ) ρ (u;β) ρ(u;β) W ρ(u) du k ( ρ(vl 1 )[g(v l 1,v l ) 1] ) ρ (v k ;β)dv 1 dv k k=1 k=1 W k l=1 W k l=1 k ( ρ(vl l )[g(v l 1,v l ) 1] ) ρ (v k ;β)dv 1 dv k Topic of current research: truncate infinite sum or approximate integral? 34/34