Bayesian Local Influence for Spatial Autoregressive Models with Heteroscedasticity

Abstract

This paper studies Bayesian local influence analysis for the spatial autoregressive models with heteroscedasticity (heteroscedastic SAR models). Two local diagnostic procedures using curvature-based and slope-based methods are proposed in the framework of Bayesian perspective. The curvature-based diagnostic are obtained by maximizing the normal curvature of an influence graph based on Kullback-Leibler divergence measure and slope-based diagnostic use the first order derivative of Bayesian factor defined for perturbation. Three perturbation schemes under the heteroscedastic SAR models are suggested and the diagnostic measures are derived respectively. The computations for the proposed diagnostic measures can be easily obtained using Markov Chain Monte Carlo sampler. The proposed methodologies are illustrated using two real examples.

A Bayesian multistage spatio‐temporally dependent model for spatial clustering and variable selection

“…Combining the cluster indicators in Figure 8, it is easy to find that population density is selected in clusters (1,5), where most large and medium-sized cities are situated. The social development index is chosen to be a significant factor in 3 clusters: (1,4,5), showing that SDI, the composite measure of neighborhood environments, plays a more important role in developed regions. Unemployment rate is only related to birthweight in clusters with relatively high concentration (generally, >8% unemployment rate), which is consistent with the finding of Pearl et al 8 Low level of maternal education is another risk factor for low birthweight in clusters (3,4) mainly located in rural Georgia.…”

“…The temporal parameters are specified as (𝜌 𝜆 1 , … , 𝜌 𝜆 S ) = (0.8, 0.7, 0.6, 0.5, 0.4, 0.3) for the covariates to obtain distinct temporal patterns in the clusters. For example, the first covariate X 1 is designed to have significant effects in cluster (1,2,4,6), and the solid lines in Figure 4 present the designed variation of 𝜆 st1 , s = 1, 2, 4, 6, the true temporal profiles of the first covariate effect in the selected clusters 1, 2, 4, and 6. The temporal patterns of the other three covariate effects for X 2 , X 3 and X 4 have similar dynamic structures and are omitted for brevity.…”

“…The social development index is chosen to be a significant factor in 3 clusters: (1,4,5), showing that SDI, the composite measure of neighborhood environments, plays a more important role in developed regions. Unemployment rate is only related to birthweight in clusters with relatively high concentration (generally, >8% unemployment rate), which is consistent with the finding of Pearl et al 8 Low level of maternal education is another risk factor for low birthweight in clusters (3,4) mainly located in rural Georgia. Thus, the subset of significant covariates is spatially varying across clusters due to different regional conditions, indicating the necessity of variable selection for each cluster.…”

“…Especially, the association between covariates and health outcomes is getting much more attention 1‐3 . Many existing spatial methods are based on global modeling, with restrictive assumptions of constant covariate effects across the entire domain of interest 4 . However, the impacts of risk factors on the disease are likely to be spatio‐temporally varying.…”

“…[1][2][3] Many existing spatial methods are based on global modeling, with restrictive assumptions of constant covariate effects across the entire domain of interest. 4 However, the impacts of risk factors on the disease are likely to be spatio-temporally varying. Therefore, discovering the spatio-temporal patterns of the covariate effects is essential and helpful to understand disease aetiology and further monitor healthcare access.…”