Modelling Multivariate Spatial Data

Transcription

1 Modelling Multivariate Spatial Data Sudipto Banerjee 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. June 20th,

2 Point-referenced spatial data often come as multivariate measurements at each location. 2 Bayesian Multivariate Spatial Regression Models

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 Bayesian Multivariate Spatial Regression Models

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 Bayesian Multivariate Spatial Regression Models

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 Bayesian Multivariate Spatial Regression Models

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 Bayesian Multivariate Spatial Regression Models

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 Bayesian Multivariate Spatial Regression Models

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 Bayesian Multivariate Spatial Regression Models

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 Bayesian Multivariate Spatial Regression Models

10 Bivariate Linear Spatial Regression A single covariate X(s) and a univariate response Y (s) At any arbotrary point in the domain, we conceive a linear spatial relationship: E[Y (s) X(s)] = β 0 + β 1 X(s); where X(s) and Y (s) are spatial processes. Regression on uncountable sets: Regress {Y (s) : s D} on {X(s) : s D}. Inference: Estimate β 0 and β 1. Estimate spatial surface {X(s) : s D}. Estimate spatial surface {Y (s) : s D}. 3 Bayesian Multivariate Spatial Regression Models

11 Bivariate spatial process A bivariate distribution [Y, X] will yield regression [Y X]. So why not start with a bivariate process? [ ] ([ ] [ ]) X(s) µx (s) CXX ( ; θ Z(s) = GP Y (s) 2, Z ) C XY ( ; θ Z ) µ Y (s) C Y X ( ; θ Z ) C Y Y ( ; θ Z ) The cross-covariance function: [ ] CXX (s, t; θ C Z (s, t; θ Z ) = Z ) C XY (s, t; θ Z ) C Y X (s, t; θ Z ) C Y Y (s, t; θ Z ), where C XY (s, t) = cov(x(s), Y (t)) and so on. 4 Bayesian Multivariate Spatial Regression Models

12 Cross-covariance functions satisfy certain properties: C XY (s, t) = cov(x(s), Y (t)) = cov(y (t), X(s)) = C Y X (t, s). Caution: C XY (s, t) C XY (t, s) and C XY (s, t) C Y X (s, t). In matrix terms, C Z (s, t; θ Z ) = C Z (t, s; θ Z ) Positive-definiteness for any finite collection of points: n n a i C Z (s i, t j ; θ Z )a j > 0 for all a i R 2 \ {0}. i=1 j=1 5 Bayesian Multivariate Spatial Regression Models

13 Bivariare Spatial Regression from a Separable Process To ensure E[Y (s) X(s)] = β 0 + β 1 X(s), we must have [ ] ([ ] [ ]) X(s) µ1 T11 T Z(s) = N, 12 for every s D Y (s) µ 2 T 12 T 22 Simplifying assumption : C Z (s, t) = ρ(s, t)t = Σ Z = {ρ(s i, s j )T} = R(φ) T. 6 Bayesian Multivariate Spatial Regression Models

14 Then, p(y (s) X(s)) = N(Y (s) β 0 + β 1 X(s), σ 2 ), where β 0 = µ 2 T 12 T 11 µ 1, β 1 = T 12 T 11, σ 2 = T 22 T 2 12 T 11. Regression coefficients are functions of process parameters. Estimate {µ 1, µ 2, T 11, T 12, T 22 } by sampling from p(φ) N(µ δ, V µ) IW (T r, S) N(Z µ, R(φ) T) Immediately obtain posterior samples of {β 0, β 1, σ 2 }. 7 Bayesian Multivariate Spatial Regression Models

15 Misaligned Spatial Data 8 Bayesian Multivariate Spatial Regression Models

16 Bivariate Spatial Regression with Misalignment Rearrange the components of Z to Z = (X(s 1 ), X(s 2 ),..., X(s n ), Y (s 1 ), Y (s 2 ),..., Y (s n )) yields [ ] X N Y ([ ] ) µ1 1, T R (φ). µ 2 1 Priors: Wishart for T 1, normal (perhaps flat) for (µ 1, µ 2 ), discrete prior for φ or perhaps a uniform on (0,.5max dist). Estimation: Markov chain Monte Carlo (Gibbs, Metropolis, Slice, HMC/NUTS); Integrated Nested Laplace Approximation (INLA). 9 Bayesian Multivariate Spatial Regression Models

17 Dew-Shrub Data from Negev Desert in Israel Negev desert is very arid Condensation can contribute to annual water levels Analysis: Determine impact of shrub density on dew duration 1129 locations with UTM coordinates X(s) : Shrub density at location s (within 5m 5m blocks) Y (s) : Dew duration at location s (in 100-th of an hour) Separable model with an exponential correlation function, ρ( s t ; φ) = e φ s t 10 Bayesian Multivariate Spatial Regression Models

18 11 Bayesian Multivariate Spatial Regression Models

19 Parameter 2.5% 50% 97.5% µ µ T T T φ β β σ T 12 / T 11 T Bayesian Multivariate Spatial Regression Models

20 Hierarchical approach (Royle and Berliner, 1999; Cressie and Wikle, 2011) Y (s) and X(s) observed over a finite set of locations S = {s 1, s 2,..., s n }. Y and X are n 1 vectors of observed Y (s i ) s and X(s i ) s, respectively. How do we model Y X? No conditional process meaningless to talk about the joint distribution of Y (s i ) X(s i ) and Y (s j ) X(s j ) for two distinct locations s i and s j. Can model using [X] [Y X] but can we interpolate/predict at arbitrary locations? 13 Bayesian Multivariate Spatial Regression Models

21 Hierarchical approach (contd.) X(s) GP (µ X (s), C X ( ; θ X )). Therefore, X N(µ X, C X (θ X )). C X (θ X ) is n n with entries C X (s i, s j ; θ X ). e(s) GP (0, C e ( ; θ e )); C e is analogous to C X. Y (s i ) = β 0 + β 1 X(s i ) + e(s i ), for i = 1, 2,..., n. Joint distribution of Y and X: ( ) ([ ] [ X µx CX(θ X) β 1C X(θ X) N, Y µ Y β 1C X(θ X) C e(θ e) + β1c 2 X(θ X) ]), where µ Y = β β 1 µ X. 14 Bayesian Multivariate Spatial Regression Models

22 This joint distribution arises from a bivariate spatial process: [ ] [ ] X(s) µ W(s) = and E[W(s)] = µ Y (s) W (s) = X (s) β 0 + β 1 µ X (s). and cross-covariance [ ] C W (s, s CX (s, s ) = ) β 1 C X (s, s ) β 1 C X (s, s ) β1c 2 X (s, s ) + C e (s, s ), where we have suppressed the dependence of C X (s, s ) and C e (s, s ) on θ X and θ e respectively. This implies that E[Y (s) X(s)] = β 0 + β 1 X(s) for any arbitrary location s, thereby specifying a well-defined spatial regression model for an arbitrary s. 15 Bayesian Multivariate Spatial Regression Models

23 Coregionalization (Wackernagel) Separable models assume one spatial range for both X(s) and Y (s). Coregionalization helps to introduce a second range parameter. Introduce two latent independent GP s, each having its own parameters: v 1 (s) GP (0, ρ 1 ( ; φ 1 )) and v 2 (s) GP (0, ρ 2 ( ; φ 2 )) Construct a bivariate process as the linear transformation: w 1 (s) = a 11 v 1 (s) w 2 (s) = a 21 v 1 (s) + a 22 v 2 (s) 16 Bayesian Multivariate Spatial Regression Models

24 Coregionalization Short form: [ ] [ ] a11 0 v1 (s) w(s) = = Av(s) a 21 a 22 v 2 (s) Cross-covariance of v(s): [ ] ρ1 (s, t; φ C v (s, t) = 1 ) 0 0 ρ 2 (s, t; φ 2 ) Cross-covariance of w(s): C w (s, t) = AC v (s, t)a. It is a valid cross-covariance function (by construction). If s = t, then C w (s, s) = AA. No loss of generality to speficy A as (lower) triangular. 17 Bayesian Multivariate Spatial Regression Models

25 If v 1 (s) and v 2 (s) have identical correlation functions, then ρ 1 (s, t) = ρ 2 (s, t) and C w (s) = ρ(s, t; φ)aa = separable model Coregionalized Spatial Linear Model [ ] [ ] [ ] X(s) µx (s) w1 (s) = + Y (s) µ Y (s) w 2 (s) + [ ] ex (s), e Y (s) where e X (s) and e Y (s) are independent white-noise processes [ ] ([ ] [ ]) ex (s) 0 τ 2 N e Y (s) 2, X τy 2 for every s D. 18 Bayesian Multivariate Spatial Regression Models

26 Generalizations Each location contains m spatial regressions Y k (s) = µ k (s) + w k (s) + ɛ k (s), k = 1,..., m. Let v k (s) GP (0, ρ k (s, s )), for k = 1,..., m be m independent GP s with unit variance. Assume w(s) = A(s)v(s) arises as a space-varying linear transformation of v(s). Then: C w (s, t) = A(s)C v (s, t)a (t) is a valid cross-covariance function. 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. 19 Bayesian Multivariate Spatial Regression Models

27 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. 20 Bayesian Multivariate Spatial Regression Models

28 Modelling cross-covariances Other approaches for cross-covariance models Latent dimension approach: Apanasovich and Genton (Biometrika, 2010). Apanasovich et al. (JASA, 2012). Multivariate Matérn family Gneiting et al. (JASA, 2010). Nonstationary variants of Coregionalization Space-varying: Gelfand et al. (Test, 2010); Guhaniyogi et al. (JABES, 2012). Multi-resolution: Banerjee and Johnson (Biometrics, 2006). Dimension-reducing: Ren and Banerjee (Biometrics, 2013). Variogram modeling: De Iaco et al. (Math. Geo., 2003). 21 Bayesian Multivariate Spatial Regression Models

29 Modelling cross-covariances Big Multivariate Spatial Data Covariance tapering (Furrer et al. 2006; Zhang and Du, 2007; Du et al. 2009; Kaufman et al., 2009). Approximations using GMRFs: INLA (Rue et al. 2009; Lindgren et al., 2011). Nearest-neighbor models (processes) (Vecchia 1988; Stein et al. 2004; Datta et al., 2014) Low-rank approaches (Wahba, 1990; Higdon, 2002; Lin et al., 2000; Kamman & Wand, 2003; Paciorek, 2007; Rasmussen & Williams, 2006; Stein 2007, 2008; Cressie & Johannesson, 2008; Banerjee et al., 2008; 2010; Sang et al., 2011; Ren and Banerjee, 2013). 22 Bayesian Multivariate Spatial Regression Models

30 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: Bayesian Multivariate Spatial Regression Models

31 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 24 Bayesian Multivariate Spatial Regression Models

32 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) 24 Bayesian Multivariate Spatial Regression Models

33 Illustration Bartlett Experimental Forest Bartlett Experimental Forest Image provided by 25 Bayesian Multivariate Spatial Regression Models

34 Illustration Bartlett Experimental Forest 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) 26 Bayesian Multivariate Spatial Regression Models

35 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) Bayesian Multivariate Spatial Regression Models

36 Illustration Bartlett Experimental Forest Coregionalized model with dependence within locations Parameters: A (15), φ (5), Diag{τ 2 i } (5). 28 Bayesian Multivariate Spatial Regression Models

37 Illustration Bartlett Experimental Forest Coregionalized model with dependence within locations Parameters: A (15), φ (5), Diag{τ 2 i } (5). Focus on spatial cross-covariance matrix AA (for brevity). Posterior inference of corr(aa ), 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) 28 Bayesian Multivariate Spatial Regression Models

38 Illustration Bartlett Experimental Forest Coregionalized model with dependence within locations Parameters: A (15), φ (5), Diag{τ 2 i } (5). Focus on spatial cross-covariance matrix AA (for brevity). Posterior inference of corr(aa ), 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) 28 Bayesian Multivariate Spatial Regression Models

39 Illustration Bartlett Experimental Forest Parameters 50% (2.5%, 97.5%) Parameters 50% (2.5%, 97.5%) BE Model RM Model Intercept ( , ) Intercept (16.92, ) ELEV 0.17 (0.09, 0.24) ELEV (-0.14, 0.00) AprTC (0.69, 2.74) SLOPE (-2.75, -0.77) AprTC (-1.93, -0.06) AprTC (-1.42, -0.30) AugTC (2.25, 4.51) AugTC (0.43, 2.14) AugTC (-0.25, 3.19) OctTC (-1.61, -0.29) OctTC (-1.83, 0.25) SM Model EH Model Intercept ( , 0.62) Intercept (-364.6, 21.45) SLOPE 1.11 (0.48, 1.74) SLOPE (-1.69, -0.17) AugTC (0.71, 1.37) AprTC (0.54, 3.66) AugTC (-1.1, 0.21) AprTC (-1.85, 0.13) YB Model AprTC (0.38, 3.13) Intercept ( , ) AugTC (-1.11, -0.18) ELEV 0.08 (0.01, 0.13) AugTC (0.60, 2.51) SLOPE 0.01 (-0.76, 0.82) OctTC (-2.76, -0.73) AugTC (-0.26, 0.77) OctTC (0.63, 2.45) AugTC (0.47, 2.21) OctTC (-2.24, -0.31) 29 Bayesian Multivariate Spatial Regression Models

40 Illustration Bartlett Experimental Forest Response Estimates: 50% (2.5%, 97.5%) BE (200.15, ) EH (466.23, ) RM (504.08, ) SM (207.13, ) YB (253.71, ) Table: Distance in meters at which the spatial correlation drops to 0.05 for each of the response variables. Distance calculated by solving the Matérn correlation function for d using ρ = 0.05 and response specific φ and ν parameters estimates from co-regionalized model. 30 Bayesian Multivariate Spatial Regression Models

41 Illustration Bartlett Experimental Forest E[Y Data] Validation plots Latitude (meters) BE EH Latitude (meters) RM SM Longitude (meters) Latitude (meters) YB Longitude (meters) Bayesian Multivariate Spatial Regression Models

42 Illustration Bartlett Experimental Forest E[Y Data] Validation plots Latitude (meters) BE EH Latitude (meters) RM SM Longitude (meters) Latitude (meters) YB Longitude (meters) IQR for Y Validation plots Latitude (meters) BE EH Latitude (meters) RM SM Longitude (meters) Latitude (meters) YB Longitude (meters) Bayesian Multivariate Spatial Regression Models

43 Software Available software for modeling and visualization spbayes R package for Bayesian hierarchical modeling for univariate (splm) and multivariate (spmvlm) spatial data MBA R package for surface approximation/interpolation with multilevel B-splines Manuscripts and course notes at sudiptob Also check out Andrew Finley s website: 32 Bayesian Multivariate Spatial Regression Models

44 Software Thank You! 33 Bayesian Multivariate Spatial Regression Models