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 December 13, /25

2 Tropical rain forest example: Capparis Frondosa observation window W = 1000 m 500 m seed dispersal clustering environment inhomogeneity Elevation Potassium content in soil. How much variation due to environmental variables and how much due to seed dispersal? Framework: spatial Cox point processes. 2/25

3 Poisson and Cox processes Poisson process with intensity function ρ( ): counts N() independent and Poisson distributed with mean EN() = ρ(u)du A Cox process X with random intensity function Λ: conditional on Λ = λ, X Poisson process with intensity function λ. 3/25

4 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) not due to 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. 4/25

5 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. 5/25

6 Example: Cox/cluster process: Inhomogeneous Thomas process X 0 Λ0 shot-noise process X0 cluster process: Offspring distributed around Poisson parents mothers according to Gaussian density 6/25

7 Example: Cox/cluster process: Inhomogeneous Thomas process X 0 Λ0 shot-noise process X0 cluster process: Offspring distributed around Poisson parents mothers according to Gaussian density Inhomogeneity: offspring in X 0 survive according to probability p(u) exp[βz(u) T ] depending on covariates (independent thinning). X /25

8 Decomposition of variance for a count Prediction of count N() given Λ : ˆN() = E[N() Λ] = Λ(u)du A 8/25

9 Decomposition of variance for a count Prediction of count N() given Λ : ˆN() = E[N() Λ] = Λ(u)du A VarN() = variation = 9/25

10 Decomposition of variance for a count Prediction of count N() given Λ : ˆN() = E[N() Λ] = Λ(u)du A VarN() = VarˆN() variation = variation Λ 10/25

11 Decomposition of variance for a count Prediction of count N() given Λ : ˆN() = E[N() Λ] = Λ(u)du VarN() = VarˆN()+Var [ N() ˆN() ] variation = variation Λ + Poisson noise A Further, ˆΛ(u) = E[Λ(u) Z] 11/25

12 Decomposition of variance for a count Prediction of count N() given Λ : ˆN() = E[N() Λ] = Λ(u)du VarN() = VarˆN()+Var [ N() ˆN() ] variation = variation Λ + Poisson noise A Further, ˆΛ(u) = E[Λ(u) Z] VarΛ(u) = structured variation = SST = 12/25

13 Decomposition of variance for a count Prediction of count N() given Λ : ˆN() = E[N() Λ] = Λ(u)du VarN() = VarˆN()+Var [ N() ˆN() ] variation = variation Λ + Poisson noise A Further, ˆΛ(u) = E[Λ(u) Z] VarΛ(u) = VarˆΛ(u) structured variation = variation due to environment SST = SSR 13/25

14 Decomposition of variance for a count Prediction of count N() given Λ : ˆN() = E[N() Λ] = Λ(u)du VarN() = VarˆN()+Var [ N() ˆN() ] variation = variation Λ + Poisson noise A Further, ˆΛ(u) = E[Λ(u) Z] VarΛ(u) = VarˆΛ(u)+Var [ Λ(u) ˆΛ(u) ] structured variation = variation due to environment + other sources SST = SSR+SSE 14/25

15 Measure of influence of environmental covariates Z: R 2 = SSR SST = VarE[Λ(u) Z] VarΛ(u) (right hand side does not depend on u in case of stationary environment) 15/25

16 R 2 for additive and log-linear models Additive: R 2 = σ2 Z σ 2 Z +σ 2 0 σ 2 Z = Var Z(u) and σ 2 0 = VarΛ 0 (u) Log-linear: R 2 = σ 2 exp Z ] σ 2 exp 0[ Z +σ2 σ 2 exp Z +ρ2 exp Z σ 2 exp Z = Varexp[ Z(u)] and ρ exp Z = Eexp[ Z(u)] 16/25

17 Estimation: environmental variances Z observed on grid G = {u i } i=1,...,m and where σ 2 exp Z = 1 M σ 2 Z = 1 Z(u) 2 M u G u G { exp [ Z(u) ] } 2 ρexp Z ρ exp = 1 exp [ Z(u) ] Z M u G and Z(u) = βz(u) T. 17/25

18 Estimation: β Estimate β conditioning on Z. First-order log composite likelihood: CL 1 (β) = logρ(u Z;β) u X W ρ(u Z;β)du ρ( Z,β) = E[Λ(u) Z] intensity function for X Z. For additive model: ρ(u Z,β) = ρ 0 +βz(u) T ρ 0 = EΛ 0 (u) 18/25

19 Parametric models for c 0 (u v) = Cov[Λ 0 (u),λ 0 (v)] 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 { } 0 exp[cov(y(u),y(v))] 1 where Y Gaussian field and ρ 0 = EΛ 0 (u). Shot-noise: Λ 0 (u) = v C αk(u v) c 0 (u v) = κα 2 R 2 k(w)k(w +v u)dw where C homogeneous Poisson with intensity κ and k( ) probability density. 19/25

20 essel shot-noise/matérn covariance Λ 0 essel shot-noise process: sum of K (ν 1)/2 essel densities centered around points of homogeneous Poisson process. Matérn covariance function: c 0 (h) = σ 2 0 ( h /η) ν K ν ( h /η) 2 ν 1 Γ(ν) ν = 1/2: exponential model ν = : Gaussian c 0 (h) = σ 2 0 exp( h /η) c 0(h) = σ 2 0 exp[ ( h /η)2 ] exp. covariance Gau. covariance t t 20/25

21 Estimation of ψ Second-order log composite likelihood (given ˆβ, conditioning on Z): CL 2 (ψ ˆβ) = u,v X u v R u v R logρ (2) (u,v Z; ˆβ,ψ) ρ (2) (u,v Z; ˆβ,ψ)dudv ρ (2) (u,v Z;β,ψ) = E[Λ(u)Λ(v) Z] second-order product density for X Z. For additive model: ρ (2) (u,v Z;β,ψ) = ρ(u Z,β)ρ(v Z,β)+c 0 (u v;ψ) 21/25

22 Three species with different modes of seed dispersal: Acalypha Diversifolia explosive capsules Loncocharpus Heptaphyllus wind Capparis Frondosa bird/mammal 22/25

23 Results for rain forest data Species model for Λ c 0 CL 2 ( ψ β) R 2 log-linear Gaussian Acalypha Matérn additive Gaussian Matérn log-linear Gaussian Loncocharpus Matérn additive Gaussian Matérn log-linear Gaussian Capparis Matérn additive Gaussian Matérn /25

24 Some conclusions Covariance functions for loncocharpus est fit with Matérn (heavy tails for covariance/cluster density). c(r) Matern LG Matern Cauchy Gaussian est fit with log-linear model (interpretation in terms of survival). Largest R 2 for Capparis (bird/mammal seed dispersal), smallest for Acalypha (explosive capsules). r 24/25

25 Advertisement Ph.D.-scholarship in Spatial Statistics Department of Mathematics, Aalborg University, Denmark Info: Rasmus Waagepetersen Jesper Møller 25/25