Spatiotemporal Models for Gaussian Areal Data

Abstract

We introduce a class of spatiotemporal models for Gaussian areal data. These models assume a latent random field process that evolves through time with random field convolutions; the convolving fields follow proper Gaussian Markov random field (PGMRF) processes. At each time, the latent random field process is linearly related to observations through an observational equation with errors that also follow a PGMRF. The use of PGMRF errors brings modeling and computational advantages. With respect to modeling, it allows more flexible model structures such as different but interacting temporal trends for each region, as well as distinct temporal gradients for each region. Computationally, building upon the fact that PGMRF errors have proper density functions, we have developed an efficient Bayesian estimation procedure based on Markov chain Monte Carlo with an embedded forward information filter backward sampler (FIFBS) algorithm. We show that, when compared with the traditional oneat-a-time Gibbs sampler, our novel FIFBS-based algorithm explores the posterior distribution much more efficiently. Finally, we have developed a simulation-based conditional Bayes factor suitable for the comparison of nonnested spatiotemporal models. An analysis of the number of homicides in Rio de Janeiro State illustrates the power of the proposed spatiotemporal framework.Supplemental materials for this article are available online in the journal's webpage.

Dynamic Multiscale Spatiotemporal Models for Gaussian Areal Data

“…To investigate the practical importance and appropriateness of our dynamic multiscale spatio temporal model further, we compare our model with one of the most widely used spatio‐temporal models for areal data based on Markov random fields. The model criterion that we use for comparison is the conditional Bayes factor (Viva and Ferreira, 2009). Specifically, we consider the common mean level model where ɛ t ∼ N ( 0 , V t ), w t ∼ N (0, W ) and .…”

“…In particular, we assume that μ 1 ∼ N (0,10 10 ), W ∼IG(1,0.1) and τ ∼gamma(1,1). Further, similarly to Vivar and Ferreira (2009), we have considered a prior for φ i of the form π ( φ i )∝1 if 0< φ i <1, and π ( φ i )∝0 otherwise.…”

“…As a consequence, their scope is limited to modelling processes that are reasonably well behaved in terms of spatiotemporal interactions. More recently, Viva and Ferreira (2009) introduced a class of spatiotemporal models for areal data that allow for more complex spatiotemporal dynamics. Although their approach results in an algorithm where the computational complexity grows linearly with the number of time steps, the computational complexity grows cubically with the number of subregions.…”

Modeling temporal gradients in regionally aggregated California asthma hospitalization data

“…Pace et al (2000) work with simultaneous autoregressive (SAR) models extending them to allow temporal neighbors as well as spatial neighbors. Later examples include the space–time interaction CAR model proposed by Schmid and Held (2004), the dynamic CAR model proposed by Martínez-Beneito, López-Quilez and Botella-Rocamora (2008), the proper Gaussian MRF process models of Vivar and Ferreira (2009) and the latent structure models approach from Lawson et al (2010).…”

Extracting Sunbeams From Cucumbers

“…As one example of how such practice can help, consider Table 4, which is drawn directly from the article by Vivar and Ferreira (2009). One of the authors' aims is to compare the performance of the various models shown in the table.…”