Compositional Data Analysis of Geological Variability and Process: A Case Study
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Compositional Data Analysis of Geological Variability and Process: A Case Study1 Christopher W. Thomas2 and John Aitchison3 Developments in the statistical analysis of compositional data over the last two decades have made possible a much deeper exploration of the nature of variability and the possible processes associated with compositional data sets from many disciplines. In this paper, we concentrate on geochemical data. First, we explain how hypotheses of compositional variability may be formulated within the natural sample space, the unit simplex, including useful hypotheses of sub-compositional discrimination and specific perturbational change. Then we develop through standard methodology, such as generalised likelihood ratio tests, statistical tools to allow the systematic investigation of a lattice of such hypotheses. Some of these tests are simple adaptations of existing multivariate tests but others require special construction. We comment on the use of graphical methods in compositional data analysis and on the ordination of specimens. The recent development of the concept of compositional processes is then explained, together with the necessary tools for a staying-in-the-simplex approach, such as the singular value decomposition of a compositional data set. All these statistical techniques are illustrated for a substantial compositional data set, consisting of 209 major oxide and trace element compositions of metamorphosed limestones from the Grampian Highlands of Scotland. Finally, we discuss some unresolved problems in the statistical analysis of compositional processes. KEY WORDS: lattice hypothesis testing, logistic discriminant analysis, metamorphosed limestones, ordination, perturbation of a composition, singular value decomposition.
INTRODUCTION Geochemical compositional data are at the heart of research undertaken to understand and explain the chemical and physical evolution of the Earth (e.g. Rollinson, 1993). Initially, geochemical data were used in a descriptive role, alongside field and petrological observations. However, improved and more rapid methods of analysis in the latter part of the 20th century provided the means to acquire large amounts of data cheaply and consistently. Thus, uses of data moved from description, through classification, into modelling. In this way, geochemical data have 1Received
26 February 2004; accepted 28 February 2005. Geological Survey, West Main Road, Edinburgh, EH9 3LA, UK; e-mail: [email protected]. 3Department of Statistics, University of Glasgow, Glasgow, G12 8QQ, UK; e-mail: john.aitchison@ btinternet.com. 2British
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been used not only to describe and characterise geological variability in rocks, but also to elucidate the processes by which that variability has arisen. It is now widely recognised that the special nature of such compositional data, specifically their positiveness and ‘constant sum,’ requires a special statistical methodology
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