Extensions to spatial factor methods with an illustration in geochemistry

Many applications involving spatial data require several layers of information to be simultaneously analyzed in relation to underlying geography and topographic detail. This in turn generates a need for forms of multivariate analysis particularly oriented to spatial problems and designed to handle spatial structure and dependency both within and between spatially indexed multivariate responses. In this paper we focus on one group of such methods sometimes referred to as spatial factor analysis. Use of these techniques has so far been mostly restricted to applications in the geosciences and in some forms of image processing, but the methods have potential for wider use outside these fields. They are concerned with identifying components of a multivariate data set with a spatial covariance structure that predominantly acts over a particular spatial range or zone of influence. We review the various forms of spatial factor analysis that have been proposed and emphasize links between them and with the linear model of coregionalization employed in geostatistics. We then introduce extensions to such methods that may prove useful in exploratory spatial analysis, both generally and more specifically in the context of multivariate spatial prediction. Application of our proposed exploratory techniques is demonstrated on a small but illustrative geochemical data set involving multielement measurements from stream sediments.