This paper proposes an elegant, yet straightforward, model for classifying linear mixtures. A linear mixture is defined as a random vector y in which the variable 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.
KEY WORDS
Gas chromatography
Simplex Barycentric co-ordinates Discriminant analysis INTRODUCTION
1.1. Overview
Consider g groups, each of which can be characterized in terms of p particular variables. Suppose that a test observation y is a 'linear mixture', in the sense that each of the p variables associated with y can be characterized as a convex combination of the corresponding variables in these component groups. The weights defining this convex combination will be called 'mixing proportions'. The test observation is 'classified' when the mixture constituents are identified and the mixing proportions are estimated.
In this paper we propose a model which seeks to classify linear mixtures. Section 2 contains the motivation for, and an outline of, the model development. Section 3 contains the details and results of the application of the model to the problem of identifying the constituents in polychlorinated biphenyl samples. Finally, section 4 contains a statement of our conclusions, and section 5 briefly mentions the computer routine used to implement the methodology. So that the presentation will remain concise, technical details are relegated to appendices.