Applied Bayesian Modelling - Free

300 MODELS FOR SPATIAL OUTCOMESfor the ith regression, one might use the notation y ik ˆ Y k (providing n copies of thedata), and then takey ik N(m ik , t ik ) k ˆ 1, : : n (7:25a)t ik ˆ f i =a ik(7:25b)m ik ˆ b 0i b 1i x 1k b 2i x 2k . . . b pi x pk (7:25c)where f i is an overall variance, homogenous across areas in the ith regression. Since thea ik decline with distance, nearer observations have lower variances (higher precisions).This implies a greater weighting in the ith regression such that observations near thearea or point i have more influence on the parameter estimate b i ˆ (b 0i , b 1i , . . . : b pi ) thanobservations further away.As in the spatial expansion method, one might robustify against outlying areas (interms of the estimated relationships between y ik and x jk ) by taking a scale mixture(heavy tailed) approach. This allows non-constant variances with scaling factors k ikdrawn from a gamma mixture G(0:5n, 0:5n), where n is the degrees of freedom of theStudent density. Thusy ik N(m ik , t ik ) k ˆ 1, : : n (7:26a)t ik ˆ f ik =a ik(7:26b)f ik ˆ f i =k ik(7:26c)Lesage (2001) considers several spatial applications in terms of a choice between n ˆ 30or 40 (essentially a fixed variance over all areas in the ith regression so that f ik ˆ f i )and n varying between 2 and 9, leading to non-constant variances f ik . Under the latter,one may set a higher stage prior on n itself. For example, takingn G(8, 2)would be an informative prior consistent with expected spatial heteroscedasticity, andallow one to discriminate between non-constant variances over space as against nonconstantregression relationships. On the other hand, a model with the full GWRparameterisation {namely the set of parameters h; n; b ji , j ˆ 0, : : p; i ˆ 1, : : n; andf ik , i ˆ 1, : : n; k ˆ 1, : : n) may become subject to relatively weak identifiability.Although not considered in the `classical' GWR literature there seems nothing againstrandom effects models for the b ji (the coefficient on variable j in the ith regression) as away to pool strength and improve identifiability. Either unstructured effects could beused, with the b ji referred to an overall average m:b j or spatially structured as consideredby Lesage (2001).7.5.3 Varying regressions effects via multivariate priorsA third approach to spatially varying regression coefficients, with particular relevanceto discrete outcomes, involves extending the prior of Leyland et al. (2000) and Langfordet al. (1999). Note that this is a single regression model, not involving n fold repetitionlike the GWR method. Thus, consider a model for a count outcome Y i Poi(E i u i ) suchthat the coefficients on two predictors x i and z i were spatially varying. The convolutionmodel is thenlog (u i ) ˆ a u i e i b i x i g i z i (7:27)