Abstract
In the study of geographical patterns of disease, multivariate areal data models proposed so far in the literature (Ma and Carlin, 2007; Carlin and Banerjee, 2003; Knorr-Held and Best, 2001) have allowed to handle several features of a phenomenon at the same time. In this paper, we propose a new model for areal data, the Spatial Temporal Conditional Auto-Regressive (STCAR) model, that allows to handle the spatial dependence between sites as well as the temporal dependence among the realizations, in the presence ofmeasurements recorded at each spatial location in a time interval. Inspired by the Generalized Multivariate Conditional Auto-Regressive (GMCAR) model published by Jin, Carlin, and Banerjee (2005), the STCAR model reduces the unknown parameters to the single parameter of spatial association estimated at every period considered. Unlike the Vector Auto-Regressive (VAR) model proposed by Sims (1980), in addition, its space-time autoregressive matrix takes into account the spatial localization of the realizations sampled. Moreover, we already know that the main areas of application of these modelsrelate to disease mapping, disease clustering, ecological analysis (Lawson, Browne, and Vidal Rodeiro, 2003). In this work, however, the STCAR model is applied in business, exploiting the analogy between the danger of contracting a particular disease and the risk of falling into bankruptcy, in order to “reconstruct” the spatial temporal distribution of expected bankruptcies of small and medium enterprises of the province of Lecce (Italy).
Highlights
The analysis of data collected in each spatial location and in a time interval, requires the use of Conditional Auto-Regressive (CAR) model that, in addition to spatial dependence between sites, examines the temporal dependence between the different realizations
We propose a new model of space-time, called Spatial Temporal Conditional Auto-Regressive (STCAR) model, which directly specifies the joint distribution of a sequence of Markov random fields (Cressie, 1993) via conditional and marginal distributions, using information derived from temporal evolution of the phenomenon
The STCAR model is constructed through a space-time autoregressive matrix so as to give a temporal coefficient in the same location sampled in different instants, a spatial coefficient in nearby locations sampled in the same instant, the product between a temporal coefficient and a spatial coefficient in nearby locations identified in different instants
Summary
The analysis of data collected in each spatial location and in a time interval, requires the use of Conditional Auto-Regressive (CAR) model that, in addition to spatial dependence between sites, examines the temporal dependence between the different realizations. Unlike the Generalized Multivariate Conditional Auto-Regressive (GMCAR) model proposed in Jin et al (2005), the STCAR model is used, not to treat more features at the same instant, but the same feature recorded in a time interval. This peculiarity reduces the number of parameters of the GMCAR model to a single parameter of spatial association estimated in the respective time: this in turn leads to a significant reduction in the computational burden in hierarchical spatial random effect modeling.
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