A Spatial Analysis of Multivariate Output from Regional ... - IMAGe

Figure 1: Examples of different types of neighborhoods. The left frame shows a withinvariablespatial neighborhood, while the middle frame shows a within-location neighborhood.The right frame demonstrates the neighborhood associated with cross-variable connections.See also Eqn. (2).The key feature of this approach is that it still falls within the original univariateframework of Besag (1974) outlined in Section 2.1. Let y ij denote the jth variable observedat the ith location on the lattice. Then, for each Gaussian conditional distribution, themean and variance need to be specified. With sums on the right-hand side correspondingto specific types of neighborhoods in Figure 1, the conditional mean is given byE[y ij |y −{ij} ] = µ ij + ∑ b ijkj (y kj − µ kj ) + ∑ b ijil (y il − µ il ) + ∑b ijkl (y kl − µ kl ) (2)k≠il≠jk,l≠i,jwith the conditional variance given by Var[y ij |y −{ij} ] = τij, 2 for all lattice points i = 1, . . . , n,variables j = 1, . . . , p, and with y −{ij} denoting all components of y except for the (i, j)-th.In the conditional mean, the coefficients in the first summation, {b ijkj ; i = 1, . . . , n, k ∈N i , j = 1, . . . , p}, represent connections within a particular layer and control conditionaldependence between the ith lattice point and neighboring points for the jth variable (leftframe in Figure 1). The coefficients in the second summation, {b ijil ; i = 1, . . . , n, j ≠l = 1, . . . , p}, represent connections across layers at the same lattice point and controlconditional dependence between variables j and l at the ith lattice point (middle framein Figure 1). Finally, the coefficients in the third summation, {b ijkl ; i = 1, . . . , n, k ∈N i , j ≠ l = 1, . . . , p}, represent connections between locations across layers for different9