Applied Bayesian Modelling - Free

DIRECT MODELLING OF SPATIAL COVARIATION IN REGRESSION 293So estimation of the variogram typically considering distances between all n(n 1)=2pairs {i, j} of points and then grouping them into a relatively few (say 10 or 20) distancebands. Then the relation between ^g(d) and d is modelled over distance bandsd ˆ 1, : : , D via nonlinear least squares in terms of parametric forms such as Equation(7.20). If n(n 1)=2 is not unduly large, one might fit a nonlinear curve such asEquation (7.20) to all sets of paired differences (Y i Y j ) 2 or (e i e j ) 2 .However, recent Bayesian approaches have tended to focus on spatial interpolationconsequent on direct estimation of the covariance matrix from the likelihood forY i ˆ Y(s i ). Thus, Diggle et al. (1998) consider adaptations for discrete outcomes ofthe canononical modelY i ˆ m e(s i ) u i (7:21)where the errors u i are independently N(0, 2 ), and accounts for the nugget variance,and the vector e(s) N n (0, s 2 R) models the spatial structure in the errors. This can beseen as a continuous version of the convolution model of Equation (7.9). Ecker andGelfand (1997) also adopt this likelihood approach, but consider generalisations of theusual parametric covariance structures such as those of Equations (7.19a)±(7.19c).Whatever the approach used to estimate S(d), prediction of Y new at a new points.new then involves the n 1 vector g of covariances g i ˆ Cov(s new , s i ) between the newpoint and the sample sites s 1 , s 2 . . . s n . For instance, if S(d) ˆ s 2 e 3d=r , then the covariancevector is obtained by plugging in to this parametric form the distancesd 1new ˆ js new s 1 j, d 2new ˆ js new s 2 j, etc. The prediction Y new is a weighted combinationof the existing points with weights l i , i ˆ 1, : : n determined byl ˆ gS 1For example, the prediction Y new under Equation (7.21) is obtained (Diggle et al., 1998,p. 303) asY new ˆ m g(t 2 I s 2 R) 1 (Y m)7.4.4 Conditional specification of spatial errorWhile joint prior specifications are the norm in such applications, one might alsoconsider conditional specifications with the same goals (e.g. estimation of proximityeffects and interpolation) in mind. Defining weights a ij in terms of the parametric formsin Equation (7.19), and then defining w ij ˆ a ij = P j6ˆi a ij, one might specifyY i N(m i , t 2 )m i ˆ me iwhere e i follows a conditional priore i NX j6ˆiw ij e j , f 2 = X j6ˆia ij!This is the ICAR(1) model with non-binary weights, as in Equation (7.11b). Forinstance under the exponential model, a ij ˆ exp ( kd ij ), where k > 0. As above, identifiabilitymay be established by introducing a correlation parameter r, which might, forinstance, be assigned a prior limited to the range [0, 1], so that