Journal ArticleDOI
An effect of closure on the structure of principal components
Felix Chayes,Jürgen Trochimczyk +1 more
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The principal components transformation as discussed by the authors provides an elegant escape from closure correlation if a petrographic problem can be restated entirely in terms of component scores, but not if a physical interpretation of the component vectors is required.Abstract:
The principal components transformation generates, from any data array, a new set of variables—the scores of the components—characterized by a total variance exactly equal to that of the initial set. It is in this sense that the transformed variables are said to “contain,” “preserve,” or “account for,” the variance of the original set. The scores, however, are uncorrelated. In the course of the transformation, what becomes of the strong interdependence of variance and covariance so characteristic of closed arrays? The question seems to have attracted little attention; we are aware of no study of it in the earth sciences. Experimental work reported here shows quite clearly that the overall equivalence of variance and covariance imposed by closure, though absent from the component scores,may emerge in relations between the coefficientsof each of the lower-order components; if the raw data are “complete” rock analyses, the sum of all the covariances of the coefficients of such a component is negative, and is very nearly equal to the sum of all the variances in absolute value. (In all cases so far examined, the absolute value of the first sum is a little less than that of the second.) The principal components transformation provides an elegant escape from closure correlation if a petrographic problem can be restated entirely in terms of component scores, but not if a physical interpretation of the component vectors is required.read more
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Quantitative comparison of large sets of geochronological data using multivariate analysis: a provenance study example from Australia
TL;DR: In this paper, principal components derived from the data set allow the construction of provenance models that are independent of expected age components, in this instance the principal components are interpreted in terms of existing knowledge about Australian geology, in particular the provenance relationship between Phanerozoic-aged and late Proterozoic protosources.
Journal ArticleDOI
Reducing the dimensionality of compositional data sets
John Aitchison,John Aitchison +1 more
TL;DR: In this article, a new definition of the covariance structure of a composition is proposed to measure the extent to which a subcomposition retains the pattern of variability of the whole composition and so provides a criterion for the selection of suitable subcompositions.
Journal ArticleDOI
MODAN: an interactive computer program for estimating mineral quantities based on bulk composition
TL;DR: MODAN as discussed by the authors is an interactive computer program written in Microsoft QuickBASIC to estimate mineral quantities (modes) based on bulk and mineralogical compositions of rock, soil and tailings samples.
Journal ArticleDOI
Recent and possible future mathematical developments in quantitative palaeoecology
TL;DR: Two contrasting approaches to quantitative palaeoecology are distinguished as mentioned in this paper, a geological approach that aims to reconstruct past environments and an ecological approach that attempts to test ecological hypotheses in time rather than space.
References
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Journal ArticleDOI
An Approximate Statistical Test for Correlations between Proportions
Felix Chayes,William Kruskal +1 more
TL;DR: In this paper, the mean and variances of data occurring as proportions or percentages are used to estimate analogous parameters of a theoretical open array, X, which, on closure, yields a new array, Y, whose means and variance are exactly those of the observed data, but in which the covariances have been generated entirely by closure.
Journal ArticleDOI
Sampling variation of principal components
J. Trochimczyk,F. Chayes +1 more
TL;DR: In this paper, the authors used 10 samples of approximately 25, 50, 100, and 200 items each were randomly drawn, with replacement, from a source sample consisting of 2086 subalkaline asalt analyses.