An Overview of Approaches to the Analysis and Modelling of Multivariate Geostatistical Data
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An Overview of Approaches to the Analysis and Modelling of Multivariate Geostatistical Data Trevor C. Bailey · Wojtek J. Krzanowski
Received: 14 December 2009 / Accepted: 24 August 2011 / Published online: 25 October 2011 © International Association for Mathematical Geosciences 2011
Abstract We give an overview of existing approaches for the analysis of geostatistical multivariate data, namely spatially indexed multivariate data where the indexing is continuous across space. These approaches are divided into two classes: factor models and spatial random field models. Factor models may be further subdivided into a descriptive sub-class, where the factors are directly obtainable as linear combinations of the manifest variables, and an inferential subclass, where the factors are latent quantities that have to be estimated from the data. Spatial random field models include a variety of different types, the most prominent being the proportional correlation model, the linear coregionalisation model, and several convolution-based models. We provide an overview of the different approaches, and draw out some connections between them. Keywords Coregionalisation · Convolution models · Factor models · Intrinsic models
1 Introduction Multivariate spatial data are increasingly encountered, not only in geosciences but also in many disciplines such as ecology, agriculture, biology, epidemiology, environmental science, and atmospheric science. The defining feature of such data is the availability of measurements on a set of different and potentially related response variables at each spatial location in the region studied. Often, there is also an associated vector of potential explanatory variables measured at each of these sites. Such multivariate spatial data may exhibit not only correlations between variables at each T.C. Bailey · W.J. Krzanowski () College of Engineering, Mathematics and Physical Sciences, University of Exeter, Harrison Building, North Park Road, Exeter EX4 4QF, UK e-mail: [email protected]
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Math Geosci (2012) 44:381–393
site, but also spatial autocorrelation within each variable, and spatial cross-correlation between variables, at neighbouring sites. Any analysis or modelling must therefore allow for dependency structures that are both complex and inevitably confounded in the observed data. Moreover, if repeat observations are present on the response vector, they often refer to different points in time and add temporal autocorrelations or cross-correlations into the already complex mix of potential correlation structures. In recent years, a range of different approaches has been proposed for handling such complexities, and the literature on the subject is now extensive. One broad distinction in these approaches relates to the spatial indexing of the data under investigation. Leaving aside point process data, where the spatial indexing itself is a random set, other forms of spatial data may in general be categorised into that which is continuously indexed across space (conceptually the phenom
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