Unsupervised spectral-band feature identification for optimal process discrimination

Changes in real-world dynamic processes are often described in terms of differences in energies E(underlineα) of a set of spectral-bands underlineα. Given continuous spectra of two classes A and B, or in general, two stochastic processes S(A)(f) and S(B)(f), f ∈ ℝ^+, we address the ubiquitous problem of identifying a subset of intervals of f called spectral-bands underlineα subset ℝ^+ such that the energies E(underlineα) of these bands can optimally discriminate between the two classes. We introduce EGO-MDA, an unsupervised method to identify optimal spectral-bands underlineα^* for given samples of spectra from two classes. EGO-MDA employs a statistical approach that iteratively minimizes an adjusted multinomial log-likelihood (deviance) criterion D(underlineα,M). Here, Mixture Discriminant Analysis (MDA) aims to derive MLE of two GMM distribution parameters, i.e., M^* = undersetMrm argmin~D(underlineα, M) and identify a classifier that optimally discriminates between two classes for a given spectral representation. The Efficient Global Optimization (EGO) finds the spectral-bands underlineα^* = undersetunderlineαrm argmin~D(underlineα,M) for given GMM parameters M. For pathological cases of low separation between mixtures and model misspecification, we discuss the effect of the sample size and the number of iterations on the estimates of parameters M and therefore the classifier performance. A case study on a synthetic data set is provided. In an engineering application of optimal spectral-banding for anomaly tracking, EGO-MDA achieved at least 70% improvement in the median deviance relative to other methods tested.