Estimation of Finite Mixture Models

Abstract

A recorded signal frequently results from the mixture of many signals from several classifiable sources. Knowledge of the contribution of the underlying sources to the recorded signal is valuable in several applications, such as remote sensing. Such mixtures may be analyzed using finite mixture models. Historically, finite mixture models decompose a density as the sum of a finite number of component densities. Current methods for estimating the contribution of each component assume a parametric form for the mixture components. Furthermore, these methods assume a collection of samples from the mixture are observed rather than an aggregate representation of the samples, such as a histogram.

This work introduces a method to address the many practical cases where parametric mixture models are insufficient to describe the mixture components. The observed mixture is assumed to occur in an aggregate representation of samples. Thus, the mixture components are represented as finite-length signals or vectors. The proposed method incorporates the first and second order statistics of the mixture components obtained from previously collected samples of the mixture components. The new method is based on the set theoretic method of successive projections onto convex sets (POCS). The set theoretic approach defines a set of feasible solutions as the intersection of sets consistent with the prior knowledge of a desirable solution. POCS is an iterative procedure used to find a point in the set of feasible solutions. This work considers several sets describing the finite mixture model, including a new model set generalizing a set based on the error-in-variables model.

To illustrate the viability of the new method, comparisons are made with the expectation-maximization (EM) algorithm for mixtures with parametric components. Simulations of mixture with nonparametric components emphasize the advantages of the new method, since no other methods address mixtures with nonparametric components. The new method is applied to the problem of resolving hyperspectral data representing the mixture of several component spectra.