Hierarchical Modelling for Multivariate Spatial Data

Transcription

1 Hierarchical Modelling for Multivariate Spatial Data Geography 890, Hierarchical Bayesian Models for Environmental Spatial Data Analysis February 15,

2 Point-referenced spatial data often come as multivariate measurements at each location. 2 Geography 890

3 Point-referenced spatial data often come as multivariate measurements at each location. Examples: Environmental monitoring: stations yield measurements on ozone, NO, CO, and PM Geography 890

4 Point-referenced spatial data often come as multivariate measurements at each location. Examples: Environmental monitoring: stations yield measurements on ozone, NO, CO, and PM 2.5. Community ecology: assembiages of plant species due to water availibility, temerature, and light requirements. 2 Geography 890

5 Point-referenced spatial data often come as multivariate measurements at each location. Examples: Environmental monitoring: stations yield measurements on ozone, NO, CO, and PM 2.5. Community ecology: assembiages of plant species due to water availibility, temerature, and light requirements. Forestry: measurements of stand characteristics age, total biomass, and average tree diameter. 2 Geography 890

6 Point-referenced spatial data often come as multivariate measurements at each location. Examples: Environmental monitoring: stations yield measurements on ozone, NO, CO, and PM 2.5. Community ecology: assembiages of plant species due to water availibility, temerature, and light requirements. Forestry: measurements of stand characteristics age, total biomass, and average tree diameter. Atmospheric modeling: at a given site we observe surface temperature, precipitation and wind speed 2 Geography 890

7 Point-referenced spatial data often come as multivariate measurements at each location. Examples: Environmental monitoring: stations yield measurements on ozone, NO, CO, and PM 2.5. Community ecology: assembiages of plant species due to water availibility, temerature, and light requirements. Forestry: measurements of stand characteristics age, total biomass, and average tree diameter. Atmospheric modeling: at a given site we observe surface temperature, precipitation and wind speed We anticipate dependence between measurements 2 Geography 890

8 Point-referenced spatial data often come as multivariate measurements at each location. Examples: Environmental monitoring: stations yield measurements on ozone, NO, CO, and PM 2.5. Community ecology: assembiages of plant species due to water availibility, temerature, and light requirements. Forestry: measurements of stand characteristics age, total biomass, and average tree diameter. Atmospheric modeling: at a given site we observe surface temperature, precipitation and wind speed We anticipate dependence between measurements at a particular location 2 Geography 890

9 Point-referenced spatial data often come as multivariate measurements at each location. Examples: Environmental monitoring: stations yield measurements on ozone, NO, CO, and PM 2.5. Community ecology: assembiages of plant species due to water availibility, temerature, and light requirements. Forestry: measurements of stand characteristics age, total biomass, and average tree diameter. Atmospheric modeling: at a given site we observe surface temperature, precipitation and wind speed We anticipate dependence between measurements at a particular location across locations 2 Geography 890

10 Each location contains m spatial regressions Y k (s) = µ k (s) + w k (s) + ɛ k (s), k = 1,..., m. Mean: µ(s) = [µ k (s)] m k=1 = [xt k (s)β k] m k=1 Cov: w(s) = [w k (s)] m k=1 MV GP (0, Γ w(, )) Γ w (s, s ) = [Cov(w k (s), w k (s ))] m k,k =1 Error: ɛ(s) = [ɛ k (s)] m k=1 MV N(0, Ψ) Ψ is an m m p.d. matrix, e.g. usually Diag(τk 2)m k=1. 3 Geography 890

11 Multivariate Gaussian process w(s) MV GP (0, Γ w ( )) with Γ w (s, s ) = [Cov(w k (s), w k (s ))] m k,k =1 Example: with m = 2 ( Γ w (s, s Cov(w1 (s), w ) = 1 (s )) Cov(w 1 (s), w 2 (s ) )) Cov(w 2 (s), w 1 (s )) Cov(w 2 (s), w 2 (s )) For finite set of locations S = {s 1,..., s n }: V ar ([w(s i )] n i=1) = Σ w = [Γ w (s i, s j )] n i,j=1 4 Geography 890

12 Multivariate Gaussian process Properties: Γ w (s, s) = Γ T w(s, s ) lim s s Γ w (s, s ) is p.d. and Γ w (s, s) = V ar(w(s)). For sites in any finite collection S = {s 1,..., s n }: n i=1 j=1 n u T i Γ w (s i, s j )u j 0 for all u i, u j R m. Any valid Γ W must satisfy the above conditions. The last property implies that Σ w is p.d. In complete generality: Γ w (s, s ) need not be symmetric. Γ w (s, s ) need not be p.d. fors s. 5 Geography 890

13 Modelling cross-covariances Moving average or kernel convolution of a process: Let Z(s) GP (0, ρ( )). Use kernels to form: w j (s) = κ j (u)z(s + u)du = κ j (s s )Z(s )ds Γ w (s s ) has (i, j)-th element: [Γ w (s s )] i,j = κ i (s s + u)κ j (u )ρ(u u )dudu Convolution of Covariance Functions: ρ 1, ρ 2,...ρ m are valid covariance functions. Form: [Γ w (s s )] i,j = ρ i (s s t)ρ j (t)dt. 6 Geography 890

14 Modelling cross-covariances Constructive approach Let v k (s) GP (0, ρ k (s, s )), for k = 1,..., m be m independent GP s with unit variance. Form the simple multivariate process v(s) = [v k (s)] m k=1 : v(s) MV GP (0, Γ v (, )) with Γ v (s, s ) = Diag(ρ k (s, s )) m k=1. Assume w(s) = A(s)v(s) arises as a space-varying linear transformation of v(s). Then: Γ w (s, s ) = A(s)Γ v (s, s )A T (s ) is a valid cross-covariance function. 7 Geography 890

15 Modelling cross-covariances Constructive approach, contd. When s = s, Γ v (s, s) = I m, so: Γ w (s, s) = A(s)A T (s) A(s) identifies with any square-root of Γ w (s, s). Can be taken as lower-triangular (Cholesky). A(s) is unknown! Should we first model A(s) to obtain Γ w (s, s)? Or should we model Γ w (s, s ) first and derive A(s)? A(s) is completely determined from within-site associations. 8 Geography 890

16 Modelling cross-covariances Constructive approach, contd. If A(s) = A: w(s) is stationary when v(s) is. Γ w (s, s ) is symmetric. Γ v (s, s ) = ρ(s, s )I m Γ w = ρ(s, s )AA T Last specification is called intrinsic and leads to separable models: Σ w = H(φ) Λ; Λ = AA T 9 Geography 890

17 Hierarchical modelling Let y = [Y(s i )] n i=1 and w = [W(s i)] n i=1. First stage: n y β, w, Ψ MV N ( Y(s i ) X(s i ) T β + w(s i ), Ψ ) i=1 10 Geography 890

18 Hierarchical modelling Let y = [Y(s i )] n i=1 and w = [W(s i)] n i=1. First stage: n y β, w, Ψ MV N ( Y(s i ) X(s i ) T β + w(s i ), Ψ ) i=1 Second stage: w θ MV N(0, Σ w (Φ)) where Σ w (Φ) = [Γ W (s i, s j ; Φ)] n i,j=1. 10 Geography 890

19 Hierarchical modelling Let y = [Y(s i )] n i=1 and w = [W(s i)] n i=1. First stage: n y β, w, Ψ MV N ( Y(s i ) X(s i ) T β + w(s i ), Ψ ) i=1 Second stage: w θ MV N(0, Σ w (Φ)) where Σ w (Φ) = [Γ W (s i, s j ; Φ)] n i,j=1. Third stage: Priors on Ω = (β, Ψ, Φ). 10 Geography 890

20 Hierarchical modelling Let y = [Y(s i )] n i=1 and w = [W(s i)] n i=1. First stage: n y β, w, Ψ MV N ( Y(s i ) X(s i ) T β + w(s i ), Ψ ) i=1 Second stage: w θ MV N(0, Σ w (Φ)) where Σ w (Φ) = [Γ W (s i, s j ; Φ)] n i,j=1. Third stage: Priors on Ω = (β, Ψ, Φ). Marginalized likelihood: y β, θ, Ψ MV N(Xβ, Σ w (Φ) + I Ψ) 10 Geography 890

21 Bayesian computations Choice: Fit as [y Ω] [Ω] or as [y β, w, Ψ] [w Φ] [Ω]. 11 Geography 890

22 Bayesian computations Choice: Fit as [y Ω] [Ω] or as [y β, w, Ψ] [w Φ] [Ω]. Conditional model: Conjugate distributions are available for Ψ and other variance parameters. Easy to program. 11 Geography 890

23 Bayesian computations Choice: Fit as [y Ω] [Ω] or as [y β, w, Ψ] [w Φ] [Ω]. Conditional model: Conjugate distributions are available for Ψ and other variance parameters. Easy to program. Marginalized model: need Metropolis or Slice sampling for most variance-covariance parameters. Harder to program. But reduced parameter space (no w s) results in faster convergence Σ w (Φ) + I Ψ is more stable than Σ w (Φ). 11 Geography 890

24 Bayesian computations Choice: Fit as [y Ω] [Ω] or as [y β, w, Ψ] [w Φ] [Ω]. Conditional model: Conjugate distributions are available for Ψ and other variance parameters. Easy to program. Marginalized model: need Metropolis or Slice sampling for most variance-covariance parameters. Harder to program. But reduced parameter space (no w s) results in faster convergence Σ w (Φ) + I Ψ is more stable than Σ w (Φ). But what about Σ 1 w (Φ)?? Matrix inversion is EXPENSIVE O(n 3 ). 11 Geography 890

25 Bayesian computations Recovering the w s? Interest often lies in the spatial surface w y. 12 Geography 890

26 Bayesian computations Recovering the w s? Interest often lies in the spatial surface w y. They are recovered from [w y, X] = [w Ω, y, X] [Ω y, X]dΩ using posterior samples: 12 Geography 890

27 Bayesian computations Recovering the w s? Interest often lies in the spatial surface w y. They are recovered from [w y, X] = [w Ω, y, X] [Ω y, X]dΩ using posterior samples: Obtain Ω (1),..., Ω (G) [Ω y, X] 12 Geography 890

28 Bayesian computations Recovering the w s? Interest often lies in the spatial surface w y. They are recovered from [w y, X] = [w Ω, y, X] [Ω y, X]dΩ using posterior samples: Obtain Ω (1),..., Ω (G) [Ω y, X] For each Ω (g), draw w (g) [w Ω (g), y, X] 12 Geography 890

29 Bayesian computations Recovering the w s? Interest often lies in the spatial surface w y. They are recovered from [w y, X] = [w Ω, y, X] [Ω y, X]dΩ using posterior samples: Obtain Ω (1),..., Ω (G) [Ω y, X] For each Ω (g), draw w (g) [w Ω (g), y, X] NOTE: With Gaussian likelihoods [w Ω, y, X] is also Gaussian. With other likelihoods this may not be easy and often the conditional updating scheme is preferred. 12 Geography 890

30 Spatial prediction Often we need to predict Y (s) at a new set of locations { s 0,..., s m } with associated predictor matrix X. Sample from predictive distribution: [ỹ y, X, X] = [ỹ, Ω y, X, X]dΩ = [ỹ y, Ω, X, X] [Ω y, X]dΩ, [ỹ y, Ω, X, X] is multivariate normal. Sampling scheme: Obtain Ω (1),..., Ω (G) [Ω y, X] For each Ω (g), draw ỹ (g) [ỹ y, Ω (g), X, X]. 13 Geography 890

31 Illustration Illustration from: Finley, A.O., S. Banerjee, A.R. Ek, and R.E. McRoberts. (2008) Bayesian multivariate process modeling for prediction of forest attributes. Journal of Agricultural, Biological, and Environmental Statistics, 13: Geography 890

32 Illustration Bartlett Experimental Forest Slight digression why we fit a model: Association between response and covariates, β, (e.g., ecological interpretation) Residual spatial and/or non-spatial associations and patterns (i.e., given covariates) Subsequent prediction 15 Geography 890

33 Illustration Bartlett Experimental Forest Study objectives: Evaluate methods for multi-source forest attribute mapping Find the best model, given the data Produce maps of biomass and uncertainty, by tree species 16 Geography 890

34 Illustration Bartlett Experimental Forest Study objectives: Evaluate methods for multi-source forest attribute mapping Find the best model, given the data Produce maps of biomass and uncertainty, by tree species Study area: USDA FS Bartlett Experimental Forest (BEF), NH 1,053 ha heavily forested Major tree species: American beech (BE), eastern hemlock (EH), red maple (RM), sugar maple (SM), and yellow birch (YB) 16 Geography 890

35 Illustration Bartlett Experimental Forest Bartlett Experimental Forest Image provided by 17 Geography 890

36 Illustration Bartlett Experimental Forest Response variables: Metric tons of total tree biomass per ha Measured on ha plots Models fit using random subset of 218 plots Prediction at remaining 219 plots Inventory plots Latitude (meters) BE EH Latitude (meters) RM SM Longitude (meters) Latitude (meters) YB Longitude (meters) Geography 890

37 Illustration Bartlett Experimental Forest Covariates DEM derived elevation and slope Spring, Summer, Fall Landsat ETM+ Tasseled Cap features (brightness, greeness, wetness) Latitude (meters) Elevation Slope Longitude (meters) Longitude (meters) 19 Geography 890

38 Illustration Bartlett Experimental Forest Candidate models Each model includes 55 covariates and 5 intercepts, therefore, X T is Geography 890

39 Illustration Bartlett Experimental Forest Candidate models Each model includes 55 covariates and 5 intercepts, therefore, X T is Different specifications of variance structures: 1 Non-spatial multivariate Diag(Ψ) = τ 2 20 Geography 890

40 Illustration Bartlett Experimental Forest Candidate models Each model includes 55 covariates and 5 intercepts, therefore, X T is Different specifications of variance structures: 1 Non-spatial multivariate Diag(Ψ) = τ 2 2 Diag(K), same φ, Diag(Ψ) 20 Geography 890

41 Illustration Bartlett Experimental Forest Candidate models Each model includes 55 covariates and 5 intercepts, therefore, X T is Different specifications of variance structures: 1 Non-spatial multivariate Diag(Ψ) = τ 2 2 Diag(K), same φ, Diag(Ψ) 3 K, same φ, Diag(Ψ) 20 Geography 890

42 Illustration Bartlett Experimental Forest Candidate models Each model includes 55 covariates and 5 intercepts, therefore, X T is Different specifications of variance structures: 1 Non-spatial multivariate Diag(Ψ) = τ 2 2 Diag(K), same φ, Diag(Ψ) 3 K, same φ, Diag(Ψ) 4 Diag(K), different φ, Diag(Ψ) 20 Geography 890

43 Illustration Bartlett Experimental Forest Candidate models Each model includes 55 covariates and 5 intercepts, therefore, X T is Different specifications of variance structures: 1 Non-spatial multivariate Diag(Ψ) = τ 2 2 Diag(K), same φ, Diag(Ψ) 3 K, same φ, Diag(Ψ) 4 Diag(K), different φ, Diag(Ψ) 5 K, different φ, Diag(Ψ) 20 Geography 890

44 Illustration Bartlett Experimental Forest Candidate models Each model includes 55 covariates and 5 intercepts, therefore, X T is Different specifications of variance structures: 1 Non-spatial multivariate Diag(Ψ) = τ 2 2 Diag(K), same φ, Diag(Ψ) 3 K, same φ, Diag(Ψ) 4 Diag(K), different φ, Diag(Ψ) 5 K, different φ, Diag(Ψ) 6 K, different φ, Ψ 20 Geography 890

45 Illustration Bartlett Experimental Forest Model comparison Deviance Information Criterion (DIC): D(Ω) = 2 log L(Data Ω) D(Ω) = E Ω Y [D(Ω)] p D = D(Ω) D( Ω); Ω = E Ω Y [Ω] DIC = D(Ω) + p D. Lower DIC is better. 21 Geography 890

46 Illustration Bartlett Experimental Forest Model comparison Deviance Information Criterion (DIC): D(Ω) = 2 log L(Data Ω) D(Ω) = E Ω Y [D(Ω)] p D = D(Ω) D( Ω); Ω = E Ω Y [Ω] DIC = D(Ω) + p D. Lower DIC is better. Model p D DIC Geography 890

47 Illustration Bartlett Experimental Forest Selected model Model 5: K, different φ, Diag(Ψ) Parameters: K = 15, φ = 5, Diag(Ψ) = 5 22 Geography 890

48 Illustration Bartlett Experimental Forest Selected model Model 5: K, different φ, Diag(Ψ) Parameters: K = 15, φ = 5, Diag(Ψ) = 5 Focus on spatial cross-covariance matrix K (for brevity). Posterior inference of cor(k), e.g., 50 (2.5, 97.5) percentiles: BE EH BE 1 EH 0.16 (0.13, 0.21) 1 RM (-0.23, -0.15) 0.45 (0.26, 0.66) SM (-0.22, -0.17) (-0.16, -0.09) YB 0.07 (0.04, 0.08) 0.22 (0.20, 0.25) These relationships expressed in mapped random spatial effects, w. 22 Geography 890

49 Illustration Bartlett Experimental Forest Selected model Model 5: K, different φ, Diag(Ψ) Parameters: K = 15, φ = 5, Diag(Ψ) = 5 Focus on spatial cross-covariance matrix K (for brevity). Posterior inference of cor(k), e.g., 50 (2.5, 97.5) percentiles: BE EH BE 1 EH 0.16 (0.13, 0.21) 1 RM (-0.23, -0.15) 0.45 (0.26, 0.66) SM (-0.22, -0.17) (-0.16, -0.09) YB 0.07 (0.04, 0.08) 0.22 (0.20, 0.25) These relationships expressed in mapped random spatial effects, w. 22 Geography 890

50 Illustration Bartlett Experimental Forest E[w Data] Inventory plots Latitude (meters) BE EH Latitude (meters) RM SM Longitude (meters) Latitude (meters) YB Longitude (meters) Geography 890

51 Illustration Bartlett Experimental Forest E[w Data] Inventory plots Latitude (meters) BE EH Latitude (meters) RM SM Longitude (meters) Latitude (meters) YB Longitude (meters) E[w Data] Validation plots Latitude (meters) BE EH Latitude (meters) RM SM Longitude (meters) Latitude (meters) YB Longitude (meters) Geography 890

52 Illustration Bartlett Experimental Forest E[Y Data] Validation plots Latitude (meters) BE EH Latitude (meters) RM SM Longitude (meters) Latitude (meters) YB Longitude (meters) Geography 890

53 Illustration Bartlett Experimental Forest E[Y Data] Validation plots Latitude (meters) BE EH Latitude (meters) RM SM Longitude (meters) Latitude (meters) YB Longitude (meters) P (2.5 < Y < 97.5 Data) Validation plots Latitude (meters) BE EH Latitude (meters) RM SM Longitude (meters) Latitude (meters) YB Longitude (meters) Geography 890

54 Illustration Bartlett Experimental Forest Summary Proposed Bayesian hierarchical spatial methodology: Partition sources of uncertainty Provides hypothesis testing Reveal spatial patterns and missing covariates 25 Geography 890

55 Illustration Bartlett Experimental Forest Summary Proposed Bayesian hierarchical spatial methodology: Partition sources of uncertainty Provides hypothesis testing Reveal spatial patterns and missing covariates Allow flexible inference Access parameters posterior distribution Access posterior predictive distribution 25 Geography 890

56 Illustration Bartlett Experimental Forest Summary Proposed Bayesian hierarchical spatial methodology: Partition sources of uncertainty Provides hypothesis testing Reveal spatial patterns and missing covariates Allow flexible inference Access parameters posterior distribution Access posterior predictive distribution Provide consistent prediction of multiple variables Maintains spatial and non-spatial association 25 Geography 890

57 Illustration Bartlett Experimental Forest Summary Proposed Bayesian hierarchical spatial methodology: Partition sources of uncertainty Provides hypothesis testing Reveal spatial patterns and missing covariates Allow flexible inference Access parameters posterior distribution Access posterior predictive distribution Provide consistent prediction of multiple variables Maintains spatial and non-spatial association Extendable model template: Cluster plot sample design multiresolution models Non-continuous response general linear models Obs. over time and space spatiotemporal models 25 Geography 890