From the work of G. Matheron till nowadays, multivariate geostatistics has been dominated by the linear model of coregionalization (with its simpler counterpart, the proportional covariance model, otherwise known as intrinsic correlation model) and by the related "factorial kriging analysis". It is fair to say that the widespread use of this model is mainly due to the convenient characterization of the multivariate spatial autocorrelation structure, as well as to the availability of a widely accepted estimation procedure based on the (eigenvalue) principal component decomposition and on the algorithm due to M. Goulard and M. Voltz. Nevertheless, being this model based only on the form taken by direct and cross variograms, no hypothesis is made about the distribution of the data, and the implied estimation and prediction procedures may be considered adequate only in the case in which the data can be assumed to be multivariate Gaussian. In other cases, particularly in presence of count data, as in many environmental and epidemiological situations, the use of this model can lead to misleading predictions and to erroneous conclusions about the underling factors. To cope with such situations, following the proposals put forward in the literature for the univariate case, it is possible to construct a multivariate hierarchical spatial factor model by building upon a generalization of the classical geostatistical proportional covariance model. Adopting a non-Bayesian inferential framework, estimation of the parameters of this model can be carried out by employing the method of moments, adapting some of the standard procedures used for the linear model of coregionalization. However, although these estimation procedures may easily be implemented, they lack of sufficient optimality properties and more efficient procedures, such as likelihood based procedures, should be considered. Assuming that the number of factors and the spatial autocorrelation function have already been chosen, likelihood inference would require the maximization of the marginal density function of the observed random variables. Though this marginal density is not available, and though the integration required in the E step of the EM algorithm is not easy, we can resort to some stochastic versions of the EM algorithm, like the stochastic EM (StEM) algorithm or the Monte Carlo EM (MCEM) algorithm.
Estimation by stochastic EM in a class of spatial factor models
MINOZZO, Marco
2004-01-01
Abstract
From the work of G. Matheron till nowadays, multivariate geostatistics has been dominated by the linear model of coregionalization (with its simpler counterpart, the proportional covariance model, otherwise known as intrinsic correlation model) and by the related "factorial kriging analysis". It is fair to say that the widespread use of this model is mainly due to the convenient characterization of the multivariate spatial autocorrelation structure, as well as to the availability of a widely accepted estimation procedure based on the (eigenvalue) principal component decomposition and on the algorithm due to M. Goulard and M. Voltz. Nevertheless, being this model based only on the form taken by direct and cross variograms, no hypothesis is made about the distribution of the data, and the implied estimation and prediction procedures may be considered adequate only in the case in which the data can be assumed to be multivariate Gaussian. In other cases, particularly in presence of count data, as in many environmental and epidemiological situations, the use of this model can lead to misleading predictions and to erroneous conclusions about the underling factors. To cope with such situations, following the proposals put forward in the literature for the univariate case, it is possible to construct a multivariate hierarchical spatial factor model by building upon a generalization of the classical geostatistical proportional covariance model. Adopting a non-Bayesian inferential framework, estimation of the parameters of this model can be carried out by employing the method of moments, adapting some of the standard procedures used for the linear model of coregionalization. However, although these estimation procedures may easily be implemented, they lack of sufficient optimality properties and more efficient procedures, such as likelihood based procedures, should be considered. Assuming that the number of factors and the spatial autocorrelation function have already been chosen, likelihood inference would require the maximization of the marginal density function of the observed random variables. Though this marginal density is not available, and though the integration required in the E step of the EM algorithm is not easy, we can resort to some stochastic versions of the EM algorithm, like the stochastic EM (StEM) algorithm or the Monte Carlo EM (MCEM) algorithm.
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