Multivariate spatial modeling Spatial data often come as multivariate measurements in each location Examples Environmental monitoring stations yield measurements on ozone NO CO PM2 5 etc In atmospheric modeling at a given site we observe surface temperature precipitation and wind speed In real estate modeling for an individual property we observe selling price and total rental income We anticipate dependence between measurements at a particular location across locations Hierarchical Modelling and Analysis for Spatial Data p 1 1 Basic issues Y s denotes a p 1 vector of random variables at s We seek to model Y s s D again specifying finite dimensional distribution for Y Y s1 Y sn Crucial object the cross covariance C s s Cov Y s Y s a p p matrix that need not be symmetric i e cov Yj s Yj s need not equal cov Yj s Yj s C s s is not positive definite except in a limiting sense as s approaches s C s s becomes the covariance matrix associated with Y s our primary focus Gaussian processes and simple modeling of C s s Hierarchical Modelling and Analysis for Spatial Data p 2 1 Separable models A popular specification is the separable model C s s s s T where is a valid univariate correlation function and T is a p p positive definite matrix T is like a non spatial covariance matrix controls spatial association based upon proximity Easily to verify that Y H T where Hij si sj and is the Kronecker product Y is positive definite since H and T are Y is convenient since Y H p T n and Y 1 H 1 T 1 Hierarchical Modelling and Analysis for Spatial Data p 3 1 Application Bivariate spatial regression A single covariate X s and a univariate response Y s Treat this as a bivariate process X s Z s N s T Y s Simplifying assumptions separable cross covariance for Z s s 1 2 is coordinate free p Y s x s N 0 1 x s 2 where 2 T12 T T12 1 1 and 2 T22 12 0 2 T11 T11 T11 Hierarchical Modelling and Analysis for Spatial Data p 4 1 Bivariate spatial regression cont d Rearrangement of the components of W to e X s1 X s2 X sn Y s1 Y s2 Y sn W yields X 1 1 N T H Y 2 1 Priors Wishart for T 1 vague but proper normal for 1 2 uniform or other suitable choice for Full conditionals for Gibbs sampler again Wishart for T 1 bivariate normal for 1 2 nonconjugate for so need Metropolis sampling here Hierarchical Modelling and Analysis for Spatial Data p 5 1 Example with dew shrub data 1129 locations with UTM coordinates Y s shrub density at location s X s Dew duration at location s Bayesian analysis under a separability assumption was carried out assuming an exponential correlation function h e h conjugate priors for T as above prior for has infinite variance and suggests a range 3 of 125 km roughly half the maximum pairwise distance in the region 1 2 T11 T12 T22 updated directly updated via Metropolis 0 1 2 samples automatically determined as functions of the others Hierarchical Modelling and Analysis for Spatial Data p 6 1 Sites yielding dew and shrub data 1031550 1031500 1031450 1031400 1031350 Northings 128100 128150 128200 128250 128300 Eastings Hierarchical Modelling and Analysis for Spatial Data p 7 1 Parameter estimation dew shrub data Parameter 1 2 T11 T12 T22 0 1 2 T12 T11 T22 2 5 50 97 5 73 12 73 89 74 67 5 20 5 38 5 572 95 10 105 22 117 69 4 46 2 42 0 53 5 56 6 19 6 91 0 01 0 03 0 21 5 72 7 08 8 46 0 04 0 02 0 01 5 58 6 22 6 93 0 17 0 10 0 02 surprising significant negative association between dew duration and shrub density Hierarchical Modelling and Analysis for Spatial Data p 8 1 Benefits and limitations of separability Benefits Easier interpretability decomposition of variance structure Substantial computational benefits Limitations Symmetry in cross covariance matrix Imposes same spatial range for every measurement Solutions Spatial delay models Coregionalization models Hierarchical Modelling and Analysis for Spatial Data p 9 1 Linear Model of Coregionalization For point referenced data Y s Av s where v s v1 s v2 s vp s p independent spatial processes with stationary correlation functions j s s j 1 2 p If j for all j separable case and AAT In general the cross covariance matrix is C s s p X j 1 j s s aj aTj Approach is constructive and hence immediately valid still stationary and delivers a distinct covariance function for each component Hierarchical Modelling and Analysis for Spatial Data p 10 1 Linear Model of Coregionalization More frisky Y s A s v s Spatially varying LMC model A s model s A s AT s s g X s s is a spatial process e g 1 s is a spatial Wishart process Hierarchical Modelling and Analysis for Spatial Data p 11 1 Other Approaches Moving average or kernel convolution of a process Z Z Yj s kj u Z s u du kj s s Z s ds where Z s is a univariate spatial process and kj are kernel functions j 1 2 p Yields the cross covariance Z Z Cij s s ki s s u kj u u u dudu Convolution of Covariance Functions Suppose C1 C2 C R p are valid covariance functions Define Cij s Ci s t Cj t dt Then the p p matrix C s Cij s is a valid cross covariance function Hierarchical Modelling and Analysis for Spatial Data p 12 1 Multivariate Areal Data Models Now areal units e g counties instead of points Need to model dependence within and across units As in univariate case often handled using spatial random effects ij where again i 1 n indexes region but now j 1 p indexes variable e g cancer type within region Suppose we observe Yi Yi1 Yi2 Yip Then g E Yij xTij j ij where i i1 ip and 1 n Link function g useful for modeling rate e g Poisson or survival e g Weibull or Cox model data Hierarchical Modelling and Analysis for Spatial Data p 13 1 Multivariate CAR MCAR models Again local or neighbor idea conditioning CAR Two strategies multivariate CAR MCAR p i j j 6 i two fold CAR p ij ij For MCAR X p i j6 i i N Bij j i i 1 n j As earlier Brook s Lemma yields p improper etc Simplification Bij bij I bij wij wi i w1i To make proper add or perhaps j j 1 p Hierarchical Modelling and Analysis for Spatial Data p 14 1