Spatial Temporal Conditional Auto-Regressive Model: A New Autoregressive Matrix
DOI:
https://doi.org/10.17713/ajs.v39i3.246Abstract
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 models
relate 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).
References
Antonelli, G., and Chiaverini, S. (2009). Introduzione a SCILAB 5.1 [Computer software manual]. Cassino (FR), Italy.
Besag, J. E. (1974). Spatial interaction and the statistical analysis of lattice systems. Journal of the Royal Statistical Society, Series B, 36, 192–236.
Besag, J. E., York, J. C., and Molliè, A. (1991). Bayesian image restoration, with two applications in spatial statistics (with discussion). Annals of the Institute of Statistical Mathematics, 43, 1–59.
Böhning, D., and Sarol, J. (2000a). Estimating risk difference in multicenter studies under baseline-risk heterogeneity. Biometrics, 56, 157–161.
Böhning, D., and Sarol, J. (2000b). A nonparametric estimator of heterogeneity variance with applications to smr and proportion-data. Biometrical Journal, 42, 321–334.
Carlin, B. P., and Banerjee, S. (2003). Hierarchical multivariate CAR models for spatiotemporally correlated survival data (with discussion). In J. M. Bernardo et al. (Eds.), Bayesian Statistics 7 (pp. 45–63). Oxford: Oxford University Press.
Chellappa, R. (1985). Two-dimensional discrete Gaussian Markov random field models for image processing. In L. N. Kanal and A. Rosenfeld (Eds.), Progress in Pattern Recognition 2 (pp. 79–112).
Consul, P. C. (1989). Generalized Poisson Distributions: Properties and Applications (Vol. 99). New York: Marcel Dekker Inc.
Consul, P. C., and Famoye, F. (1992). Generalized Poisson regression model. Communications in Statistics - Theory and Methods, 21, 89–109.
Cressie, N. (1993). Statistics for Spatial Data (Second ed.). New York: Wiley.
Gelfand, A. E., and Smith, A. F. M. (1990). Sampling-based approaches to calculating marginal densities. Journal of The American Statistical Association, 85, 398–409.
Gelfand, A. E., and Vounatsou, P. (2003). Proper multivariate conditional autoregressive models for spatial data analysis. Biostatistics, 4, 11–25.
Jin, X., Carlin, B. P., and Banerjee, S. (2005). Generalized hierarchical multivariate CAR models for areal data. Biometrics, 61, 950–961.
Kim, H., Sun, D., and Tsutakawa, R. K. (2001). A bivariate Bayes method for improving the estimates of mortality rates with a twofold conditional autoregressive model. Journal of the American Statistical Association, 96, 1506–1521.
Knorr-Held, L., and Best, N. G. (2001). A shared component model for joint and selective clustering of two diseases. Journal of the Royal Statistical Society, Series A, 164(1), 73–85.
Lawson, A., Browne,W., and Vidal Rodeiro, C. (2003). Disease Mapping with WinBUGS and MLwiN. New York: J. Wiley & Sons.
Ma, H., and Carlin, B. P. (2007). Bayesian multivariate areal wombling for multiple disease boundary analysis. Bayesian Analysis, 1(5), 1–22.
Mardia, K. V. (1988). Multi-dimensional multivariate Gaussian Markov random fields with application to image processing. Journal of Multivariate Analysis, 24, 265–284.
Ord, K. (1975). Estimation methods for models of spatial interaction. Journal of the American Statistical Association, 70(349), 120–126.
R Development Core Team. (2008). R: A Language and Environment for Statistical Computing [Computer software manual]. Vienna, Austria.
Sims, C. A. (1980). Macroeconomics and reality. Econometrica, 48, 1–48.
Smith, R. L. (2001). Enviromental Statistics.
Spiegelhalter, D., Thomas, A., Best, N., and Lunn, D. (2005). WinBUGS User Manual [Computer software manual]. UK.
Sun, D., Tsutakawa, R. K., and Speckman, P. (1999). Posterior Distribution of Hierarchical Models Using CAR(1) Distributions (Tech. Rep. No. 96). National Institute of Statistical Sciences.
Thomas, A., Best, N., Lunn, D., Arnold, R., and Spiegelhalter, D. (2004). GeoBUGS User Manual [Computer software manual]. UK.
Vicari, P., Ferillo, A., and Valeri, A. (2009). Classificazione delle Attività Economiche - Ateco 2007. Roma: Istituto Nazionale di Statistica (ISTAT).
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