Discriminant Analysis for Stationary Time Series

Abstract

The extension of classical discriminant analysis techniques in multivariate analysis to time series data is a problem of practical interest. In Section 7.1 we give a basic formulation of discriminant analysis. We begin with the standard methods from classical multivariate analysis and then introduce the frequency domain approach in time series analysis (Section 7.2). Even if the Gaussianity of the process is not assumed, we can construct the Gaussian likelihood or its spectral version, which is called the Whittle asymptotic likelihood. In Section 7.3 we discuss the non-Gaussian robustness of the discriminant rule based on the Whittle asymptotic likelihood. By the non-Gaussian robustness we mean that two kinds of misclassification probability, P(2|1) and P(1|2), are asymptotically independent of the nonGaussianity of the sequence {U(t)= (U₁(t),..., U_m(t))’}of the i.i.d. errors, which appear in rnvector linear processes. Generally, the asymptotic distribution of the resulting discriminant statistic depends on the fourth order cumulants kabcd = Cum{U _a (1),U _b (1), U _c (1), U _d (1)}, a, b, c,d = 1, ..., m. We usually consider the local analysis in such a way that, as the sample size n tends to infinity, we move the hypothetical spectral density under a category II₂ closer to the one under a category II₁. The Whittle asymptotic log-likelihood can be regarded as a spectral measure. It is natural to define a more general spectral distance, including two important measures like the Kullback—Leibler divergence and the Chernoff information divergence which are, in Section 7.6, derived as spectral approximations. Based on such a general spectral distance, two approaches for the discriminant analysis are possible. One approach is to use a nonparametric kernel spectral density estimator constructed from a realization that we want to classify. Section 7.4 extends the results of Section 7.3 to nonparametric discriminant statistics. The other approach treated in Section 7.5 is based on a fitted parametric spectrum (for example, an AR spectrum). We also discuss the higher orde asymptotics of parametric discriminant statistics for Gaussian stationary processes. Section 7.7 gives a brief discuddion on the discriminant analysis for other stochastic processes such as diffusion processes and nonlinear processes.