A MODEL FOR CLASSIFYING LINEAR MIXTURES (BARYCENTRIC COORDINATES, CHEMOMETRICS, GAS CHROMATOGRAPHY)

This article proposes an elegant, yet straightforward model for classifying linear mixtures. A linear mixture is defined as a random vector y in which the variables are a (nonnegative) weighted average of corresponding variables, assumed to characterize g component groups. These weights are referred to as "mixing proportions". The model seeks to identify the mixture constituents and estimate the mixing proportions. It is demonstrated within the context of high resolution gas chromatography and the problem of identifying the constituents in polychlorinated biphenyl mixtures.; The methodology used to construct the model extends the applicability of classical discriminant techniques to linear mixtures. This extension is accomplished through the use of simplices and barycentric coordinates, the latter providing a formal estimate of the unknown mixing proportions. In addition, residual vectors are developed for use in monitoring the performance of the model.; The demonstration of the model proves remarkably successful, clearly supporting the importance of the statistical methodology. Classical discriminant techniques are principally concerned with classifying a test observation into a single group. More recent techniques - eg. SIMCA, (Wold, 1984) and "classification trees", (Brieman, 1984) are likewise concerned with a one-group classification. Statements concerning the probability of belonging to a class are available in these methods. However, it is not clear how to translate this uncertainty into a statement about mixing proportions. The model presented in this paper operates with a more focused purpose, offering rigorous and easily interpretable results.