Discriminant analysis for locally stationary processes |
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Authors: | Kenji Sakiyama Masanobu Taniguchi |
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Affiliation: | a Research Center for Advanced Science and Technology, The University of Tokyo, Tokyo, Japan;b Department of Mathematical Sciences, School of Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo, 169-8555, Japan |
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Abstract: | In this paper, we discuss discriminant analysis for locally stationary processes, which constitute a class of non-stationary processes. Consider the case where a locally stationary process {Xt,T} belongs to one of two categories described by two hypotheses π1 and π2. Here T is the length of the observed stretch. These hypotheses specify that {Xt,T} has time-varying spectral densities f(u,λ) and g(u,λ) under π1 and π2, respectively. Although Gaussianity of {Xt,T} is not assumed, we use a classification criterion D( f:g), which is an approximation of the Gaussian likelihood ratio for {Xt,T} between π1 and π2. Then it is shown that D( f:g) is consistent, i.e., the misclassification probabilities based on D( f:g) converge to zero as T→∞. Next, in the case when g(u,λ) is contiguous to f(u,λ), we evaluate the misclassification probabilities, and discuss non-Gaussian robustness of D( f:g). Because the spectra depend on time, the features of non-Gaussian robustness are different from those for stationary processes. It is also interesting to investigate the behavior of D( f:g) with respect to infinitesimal perturbations of the spectra. Introducing an influence function of D( f:g), we illuminate its infinitesimal behavior. Some numerical studies are given. |
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Keywords: | Locally stationary vector process Classification criterion Time-varying spectral density matrix Misclassification probability Non-Gaussian robust Least favorable spectral density Influence function |
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