A program to perform Ward's clustering method on several regionalized variables

Abstract

There are many statistical techniques that allow finding similarities or differences among data and variables. Cluster analysis encompasses many diverse techniques for discovering structure within complex sets of data. The objective of cluster analysis is to group either the data or the variables into clusters such that the elements within a cluster have a high degree of “natural association” among themselves while clusters are “relatively distinct” from one another. To do so, many criteria have been described: partitioning methods, arbitrary origin methods, mutual similarity procedures and hierarchical clustering techniques. One of the most widespread hierarchical clustering methods is the Ward's method. Earth science studies deal in general with multivariate and regionalized observations which may be compositional, i.e. data such as percentages, concentrations, mg/kg (ppm). Sometimes, it is interesting to know whether these data have to be divided into different subpopulations. This problem cannot be studied with traditional Ward's method because samples are not independent. In that case, an extension of Ward's clustering method to spatially dependent samples can be used. This methodology is based on a generalized Mahalanobis distance, which uses the covariance and cross-covariance (or variogram and cross-variogram) matrices. This paper describes a refinement of this method previously defined, which was iterative and tedious, as it was necessary to re-estimate the spatial covariance structure at each step. In this paper, we stay within the same theoretical framework, but we improve the methodology using the fast fourier Transform method to find the covariance structure. Thus, we obtain a generalization to several variables of adapted Ward's clustering method.