Unsupervised spectral-band feature identification for optimal process discrimination

Changes in real-world dynamic processes are often described in terms of differences in energies $\textbf{E}(\underline{\alpha})$ of a set of spectral-bands $\underline{\alpha}$. Given continuous spectra of two classes $A$ and $B$, or in general, two stochastic processes $S^{(A)}(f)$ and $S^{(B)}(f)$, $f \in \mathbb{R}^+$, we address the ubiquitous problem of identifying a subset of intervals of $f$ called spectral-bands $\underline{\alpha} \subset \mathbb{R}^+$ such that the energies $\textbf{E}(\underline{\alpha})$ of these bands can optimally discriminate between the two classes. We introduce EGO-MDA, an unsupervised method to identify optimal spectral-bands $\underline{\alpha}^*$ for given samples of spectra from two classes. EGO-MDA employs a statistical approach that iteratively minimizes an adjusted multinomial log-likelihood (deviance) criterion $\mathcal{D}(\underline{\alpha},\mathcal{M})$. Here, Mixture Discriminant Analysis (MDA) aims to derive MLE of two GMM distribution parameters, i.e., $\mathcal{M}^* = \underset{\mathcal{M}}{\rm argmin}~\mathcal{D}(\underline{\alpha}, \mathcal{M})$ and identify a classifier that optimally discriminates between two classes for a given spectral representation. The Efficient Global Optimization (EGO) finds the spectral-bands $\underline{\alpha}^* = \underset{\underline{\alpha}}{\rm argmin}~\mathcal{D}(\underline{\alpha},\mathcal{M})$ for given GMM parameters $\mathcal{M}$. 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 $\mathcal{M}$ 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.