The present work is concerned with the analysis of non Gaussian multivariate spatial data and, in particular, with the case in which the available measurements are counts. Following standard geostatistical practice, given a set of data resulting from the measurement of N regionalized variables at n spatial locations, we see this set of N regionalized variables as one particular realization of a set of N random functions. When the number N of random functions involved is high, it becomes paramount to explore feasible ways to model data sets of this kind in terms of a small number of elementary components. Leaving aside any consideration about the distribution of the random functions, and as far as only direct variograms (or direct covariances) and cross-variograms (or cross-covariances) are concerned, the standard modelling tool emploied in presence of multivariate spatial data is a geostatistical spatial factor model, related to the so called ``factorial kriging analysis'', known as the linear model of coregionalization. Although this model, together with the use of factorial cokriging predictions, may be adequate in the case in which the data may be assumed to be (multivariate) Gaussian, in the other cases, particularly when count data are present, as in many environmental data sets, it can lead to very inefficient predictions of the underling factors. However, whereas in the univariate case the problem has already been tackled by adopting a generalized linear mixed model approach, in the multivariate case not much has yet been done. In this work we present a slight generalization of the classical geostatistical proportional covariance model (otherwise known as intrinsic correlation model), and then use this generalization to introduce a hierarchical spatial factor model for Poisson data.
Hierarchical spatial factor models for Poisson count data
MINOZZO, Marco
2002-01-01
Abstract
The present work is concerned with the analysis of non Gaussian multivariate spatial data and, in particular, with the case in which the available measurements are counts. Following standard geostatistical practice, given a set of data resulting from the measurement of N regionalized variables at n spatial locations, we see this set of N regionalized variables as one particular realization of a set of N random functions. When the number N of random functions involved is high, it becomes paramount to explore feasible ways to model data sets of this kind in terms of a small number of elementary components. Leaving aside any consideration about the distribution of the random functions, and as far as only direct variograms (or direct covariances) and cross-variograms (or cross-covariances) are concerned, the standard modelling tool emploied in presence of multivariate spatial data is a geostatistical spatial factor model, related to the so called ``factorial kriging analysis'', known as the linear model of coregionalization. Although this model, together with the use of factorial cokriging predictions, may be adequate in the case in which the data may be assumed to be (multivariate) Gaussian, in the other cases, particularly when count data are present, as in many environmental data sets, it can lead to very inefficient predictions of the underling factors. However, whereas in the univariate case the problem has already been tackled by adopting a generalized linear mixed model approach, in the multivariate case not much has yet been done. In this work we present a slight generalization of the classical geostatistical proportional covariance model (otherwise known as intrinsic correlation model), and then use this generalization to introduce a hierarchical spatial factor model for Poisson data.
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