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Multivariate spatial modeling Point-referenced spatial data often come as multivariate measurements at each location Chapter 7: Multivariate Spatial Modeling p. 1/21

Multivariate spatial modeling Point-referenced spatial data often come as multivariate measurements at each location Examples: Chapter 7: Multivariate Spatial Modeling p. 1/21

Multivariate spatial modeling Point-referenced spatial data often come as multivariate measurements at each location Examples: Environmental monitoring stations yield measurements on ozone, NO, CO, PM 2.5, 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 Chapter 7: Multivariate Spatial Modeling p. 1/21

Basic issues Y(s) denotes a p 1 vector of random variables at s Chapter 7: Multivariate Spatial Modeling p. 2/21

Basic issues Y(s) denotes a p 1 vector of random variables at s We seek to model Y(s) : s D, again specifying a finite dimensional distribution for Y = (Y(s 1 ),...,Y(s n )) Chapter 7: Multivariate Spatial Modeling p. 2/21

Basic issues Y(s) denotes a p 1 vector of random variables at s We seek to model Y(s) : s D, again specifying a finite dimensional distribution for Y = (Y(s 1 ),...,Y(s n )) 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(y j (s),y j (s )) need not equal cov(y j (s),y j (s )) C(s, s ) is not positive definite except in a limiting sense: C(s, s) is the covariance matrix associated with Y(s). Chapter 7: Multivariate Spatial Modeling p. 2/21

Separable models A popular specification is the separable model C(s, s ) = ρ(s, s ) T where ρ is a valid (univariate) correlation function and T is a p p positive definite matrix Chapter 7: Multivariate Spatial Modeling p. 3/21

Separable models A popular specification is the separable model C(s, s ) = ρ(s, s ) T where ρ is a valid (univariate) correlation function and T is a p p positive definite matrix T is the non-spatial or local" covariance matrix ρ controls spatial association based upon proximity Easy to verify that Σ Y = H T, where H ij = ρ(s i, s j ) and is the Kronecker product. Chapter 7: Multivariate Spatial Modeling p. 3/21

Application: Bivariate spatial regression A single covariate X(s) and a univariate response Y (s) Chapter 7: Multivariate Spatial Modeling p. 4/21

Application: Bivariate spatial regression A single covariate X(s) and a univariate response Y (s) To enable joint estimation and prediction, treat this as a bivariate process, ( ) X(s) Z(s) = N(µ(s),T) Y (s) Simplifying assumptions: separable cross-covariance for Z(s) µ(s) = (µ 1,µ 2 ) is coordinate-free. Chapter 7: Multivariate Spatial Modeling p. 4/21

Bivariate spatial regression (cont d) Rearrangement of 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 Y ) N (( µ 1 1 µ 2 1 ), T H (φ) ), Chapter 7: Multivariate Spatial Modeling p. 5/21

Bivariate spatial regression (cont d) Rearrangement of 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 Y ) N (( µ 1 1 µ 2 1 ), T H (φ) Priors: Wishart for T 1, vague but proper normal for (µ 1,µ 2 ), uniform or other suitable choice for φ Full conditionals for Gibbs sampler: again Wishart for T 1, bivariate normal for (µ 1,µ 2 ); nonconjugate for φ (so need Metropolis sampling here) ), Chapter 7: Multivariate Spatial Modeling p. 5/21

Example with dew-shrub data 1129 locations with UTM coordinates Chapter 7: Multivariate Spatial Modeling p. 6/21

Example with dew-shrub data 1129 locations with UTM coordinates Y (s) : shrub density (% of land cover) at location s Chapter 7: Multivariate Spatial Modeling p. 6/21

Example with dew-shrub data 1129 locations with UTM coordinates Y (s) : shrub density (% of land cover) at location s X(s) : dew duration (minutes from 8 am) at location s Chapter 7: Multivariate Spatial Modeling p. 6/21

Example with dew-shrub data 1129 locations with UTM coordinates Y (s) : shrub density (% of land cover) at location s X(s) : dew duration (minutes from 8 am) at location s Perform an illustrative Bayesian analysis assuming separability and assuming an exponential correlation function, ρ(h;φ) = e φh conjugate priors for µ, T as above; prior for φ has infinite variance and suggests a range (3/φ) of 125 km, roughly half the maximum pairwise distance in the region Chapter 7: Multivariate Spatial Modeling p. 6/21

Dew and shrub data sites 128100 128150 128200 128250 128300 1031350 1031400 1031450 1031500 1031550 Eastings Northings 128100 128150 128200 128250 128300 1031350 1031400 1031450 1031500 1031550 Chapter 7: Multivariate Spatial Modeling p. 7/21

Parameter estimation, dew-shrub data Parameter 2.5% 50% 97.5% µ 1 73.12 73.89 74.67 µ 2 5.20 5.38 5.572 T 11 95.10 105.22 117.69 T 12 4.46 2.42 0.53 T 22 5.56 6.19 6.91 φ 0.01 0.03 0.21 β 0 5.72 7.08 8.46 β 1 0.04 0.02 0.01 σ 2 5.58 6.22 6.93 T 12 / T 11 T 22 0.17 0.10 0.02 surprising significant negative association between dew duration and shrub density! (also emerges with a nonspatial model) Chapter 7: Multivariate Spatial Modeling p. 8/21

Benefits and limitations of separability Benefits: Easy interpretation (decomposition of variance structure) Substantial computational benefits Chapter 7: Multivariate Spatial Modeling p. 9/21

Benefits and limitations of separability Benefits: Easy interpretation (decomposition of variance structure) Substantial computational benefits Limitations: Symmetry in cross-covariance matrix Imposes same spatial range for every measurement (only one correlation function) Chapter 7: Multivariate Spatial Modeling p. 9/21

Linear Model of Coregionalization For point referenced data, assume Y(s) = Aw(s) where w(s) = (w 1 (s),w 2 (s),...,w p (s)). Chapter 7: Multivariate Spatial Modeling p. 10/21

Linear Model of Coregionalization For point referenced data, assume Y(s) = Aw(s) where w(s) = (w 1 (s),w 2 (s),...,w p (s)). p independent spatial processes with stationary correlation functions ρ j (s s ),j = 1, 2,...,p If ρ j = ρ for all j separable case with AA = T In general, the cross covariance matrix is C(s s ) = p j=1 ρ j (s s )a j a j where a j is the jth column of A. Chapter 7: Multivariate Spatial Modeling p. 10/21

Linear Model of Coregionalization More frisky: Y(s) = A(s)w(s) = Spatially varying LMC! Chapter 7: Multivariate Spatial Modeling p. 11/21

Linear Model of Coregionalization More frisky: Y(s) = A(s)w(s) = Spatially varying LMC! model A(s) model T(s) = A(s)A (s) Chapter 7: Multivariate Spatial Modeling p. 11/21

Linear Model of Coregionalization More frisky: Y(s) = A(s)w(s) = Spatially varying LMC! model A(s) model T(s) = A(s)A (s) Possibilities for T(s): Chapter 7: Multivariate Spatial Modeling p. 11/21

Linear Model of Coregionalization More frisky: Y(s) = A(s)w(s) = Spatially varying LMC! model A(s) model T(s) = A(s)A (s) Possibilities for T(s): T(s) = g(x(s)) T Chapter 7: Multivariate Spatial Modeling p. 11/21

Linear Model of Coregionalization More frisky: Y(s) = A(s)w(s) = Spatially varying LMC! model A(s) model T(s) = A(s)A (s) Possibilities for T(s): T(s) = g(x(s)) T T(s) is a spatial process (e.g., T 1 (s) is a spatial Wishart process) Chapter 7: Multivariate Spatial Modeling p. 11/21

Other Approaches Moving average or kernel convolution of a process: Y j (s) = k j (u)z(s + u)du = k j (s s )Z(s )ds where Z(s) is a univariate spatial process and k j are kernel functions, j = 1, 2,...,p. Yields the cross covariance C ij (s s ) = k i (s s + u)k j (u )ρ(u u )dudu Convolution of Covariance Functions: Suppose C 1,C 2,...C p are valid covariance functions. Define C ij (s) = C i (s t)c j (t)dt. Then the p p matrix C(s) = {C ij (s)} is a valid cross covariance function. Chapter 7: Multivariate Spatial Modeling p. 12/21

Multivariate Areal Data Models Now areal units (e.g., counties) instead of points Chapter 7: Multivariate Spatial Modeling p. 13/21

Multivariate Areal Data Models Now areal units (e.g., counties) instead of points Need to model dependence within and across units Chapter 7: Multivariate Spatial Modeling p. 13/21

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 φ ij, where again i = 1,...,n indexes region but now j = 1,...,p indexes variable (e.g., cancer type) within region Suppose we observe Y i = (Y i1,y i2,...y ip ), Then g(e(y ij )) = x T ij β j + φ ij, where φ i = (φ i1,...,φ ip ) and φ = (φ 1,...,φ n ). Link function g useful for modeling rate (e.g., Poisson) or survival (e.g., Weibull or Cox model) data. Chapter 7: Multivariate Spatial Modeling p. 13/21

Univariate vs. Multivariate Modeling We can combine CAR and traditional models to handle a broad range of problems. Chapter 7: Multivariate Spatial Modeling p. 14/21

Univariate vs. Multivariate Modeling We can combine CAR and traditional models to handle a broad range of problems. Example: For breast cancer control model in county i, let Y 1i = observed age-adjusted mortality rate / 100,000 Y 2i = observed age-adjusted incidence rate / 100,000 Y 3i = observed percent of late diagnoses Y 4i = % of surveyed women over the age of 40 who have not had a mammogram in two years Chapter 7: Multivariate Spatial Modeling p. 14/21

Univariate vs. Multivariate Modeling Writing θ k = (θ k1,...,θ ki ), let θ k CAR(λ k ). Chapter 7: Multivariate Spatial Modeling p. 15/21

Univariate vs. Multivariate Modeling Writing θ k = (θ k1,...,θ ki ), let θ k CAR(λ k ). We then borrow strength across cancer indicators: λ k iid G(a,b) and σ 2 k iid IG(a,b), where G and IG denote the gamma and inverse gamma distributions, respectively. Parameter of interest: For a set of weights α k such that 4 k=1 α k = 1, form the breast cancer control variable, η i = 4 k=1 α k θ ki. Chapter 7: Multivariate Spatial Modeling p. 15/21

Univariate vs. Multivariate Modeling A sample from the posterior distribution of η = (η 1,...,η I ), p(η y), is easily obtained in WinBUGS. Chapter 7: Multivariate Spatial Modeling p. 16/21

Univariate vs. Multivariate Modeling A sample from the posterior distribution of η = (η 1,...,η I ), p(η y), is easily obtained in WinBUGS. Using α k = 1/4 for all k, compare the posterior medians of the η i with naive averages: Chapter 7: Multivariate Spatial Modeling p. 16/21

Multivariate CAR (MCAR) models Again, local or neighbor idea, conditioning, CAR Two strategies: Chapter 7: Multivariate Spatial Modeling p. 17/21

Multivariate CAR (MCAR) models Again, local or neighbor idea, conditioning, CAR Two strategies: multivariate CAR (MCAR): p(φ i φ j,j i) Chapter 7: Multivariate Spatial Modeling p. 17/21

Multivariate CAR (MCAR) models Again, local or neighbor idea, conditioning, CAR Two strategies: multivariate CAR (MCAR): p(φ i φ j,j i) two-fold CAR: p(φ ij φ (ij) ) Chapter 7: Multivariate Spatial Modeling p. 17/21

Multivariate CAR (MCAR) models Again, local or neighbor idea, conditioning, CAR Two strategies: multivariate CAR (MCAR): p(φ i φ j,j i) two-fold CAR: p(φ ij φ (ij) ) For MCAR, p(φ i φ j i, Σ i ) = N j B ij φ j, Σ i, i = 1,...,n As earlier, Brook s Lemma yields p(φ), improper, etc. Chapter 7: Multivariate Spatial Modeling p. 17/21

Application to spatial frailty modeling From the NCI s SEER (Surveillance, Epidemiology, and End Results) database (seer.cancer.gov), we can obtain information on patients suffering from possibly multiple cancers by county for several U.S. states. For instance, cancers of 5 GI-related organs: colorectal, gall bladder, pancreas, small intestine, and stomach. t ijk, the time to death/censoring for patient k with primary cancer j in Iowa county i for diagnosis years 1992-98 Chapter 7: Multivariate Spatial Modeling p. 18/21

Application to spatial frailty modeling Hierarchical model: The likelihood is p i=1 n sij { ( )} δijk j=1 k=1 h tijk ; x ijk ( ) ( )} exp { H 0i tijk exp x T ijk β + zt ijk θ + φ ij, where h ( t ijk ; x ijk ) = h0i ( tijk ) exp (x T ijk β + zt ijk θ + φ ij H 0i ( tijk ) = tijk 0 h 0i (u) du. φ i = (φ i1,φ i2,...,φ in ) T ( ) T and Φ = φ T 1,...,φ T p MCAR (α, Λ). ) Chapter 7: Multivariate Spatial Modeling p. 19/21

Application to spatial frailty modeling We take flat priors on β and θ, and model the county-specific baseline hazard semiparametrically Chapter 7: Multivariate Spatial Modeling p. 20/21

Application to spatial frailty modeling We take flat priors on β and θ, and model the county-specific baseline hazard semiparametrically Alternate approach: Random coefficients model: x T ijk β + zt ijk θ i, β flat, but now Θ ( θ T 1,...,θ T p ) T MCAR (α, Λ). Next slide maps the fitted spatially varying coefficients Not residuals, but the effects of the presence of the primary cancer on the death rate Chapter 7: Multivariate Spatial Modeling p. 20/21

Fitted spatial effects, Iowa SEER data Colorectal Cancer Stomach Cancer 0.15 0.20 0.20 0.25 0.25 0.30 0.30-0.35 0.35-0.40 0.40 0.45 0.45-0.50 1.0-1.05 1.05 1.10 1.10 1.15 1.15 1.20 1.20 1.25 1.25 1.30 1.30-1.35 Pancreas Cancer Small Intestine Cancer 1.55 1.60 1.60 1.65 1.65 1.70 1.70 1.75 1.75 1.80 1.80 1.85 1.85-1.90 0.20 0.25 0.25 0.30 0.30 0.35 0.35 0.40 0.40 0.45 0.45 0.50 0.50-0.55 Chapter 7: Multivariate Spatial Modeling p. 21/21