  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.
