Discriminant analysis based on binary time series
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Discriminant analysis based on binary time series Yuichi Goto1
· Masanobu Taniguchi1
Received: 6 January 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract Binary time series can be derived from an underlying latent process. In this paper, we consider an ellipsoidal alpha mixing strictly stationary process and discuss the discriminant analysis and propose a classification method based on binary time series. Assume that the observations are generated by time series which belongs to one of two categories described by different spectra. We propose a method to classify into the correct category with high probability. First, we will show that the misclassification probability tends to zero when the number of observation tends to infinity, that is, the consistency of our discrimination method. Further, we evaluate the asymptotic misclassification probability when the two categories are contiguous. Finally, we show that our classification method based on binary time series has good robustness properties when the process is contaminated by an outlier, that is, our classification method is insensitive to the outlier. However, the classical method based on smoothed periodogram is sensitive to outliers. We also deal with a practical case where the two categories are estimated from the training samples. For an electrocardiogram data set, we examine the robustness of our method when observations are contaminated with an outlier. Keywords Stationary process · Spectral density · Binary time series · Robustness · Discriminant analysis · Misclassification probability Mathematics Subject Classification 62H30 · 62G86
This research supported by Grant-in-Aid for JSPS Research Fellow Grant Number JP201920060 (Yuichi Goto), and the Research Institute for Science & Engineering of Waseda University and JSPS Grant-in-Aid for Scientific Research (S) Grant Number JP18H05290 (Masanobu Taniguchi). Electronic supplementary material The online version of this article (https://doi.org/10.1007/s00184019-00746-1) contains supplementary material, which is available to authorized users.
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Yuichi Goto [email protected] Present Address: Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
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Y. Goto, M. Taniguchi
1 Introduction Binary time series can be derived from an underlying latent process. The relation between binary time series and the original process has been studied by many authors. Rice (1944) gave a pioneer study that shows a relation between the autocorrelation for a scalar Gaussian process in continuous time and that for binary time series of the process. After that, He and Kedem (1989) extended the results to a class of processes whose finite dimensional distributions are elliptically symmetric. Buz and Litan (2012), Keenan (1982), Lomnicki and Zaremba (1955) studied the properties of binary time series. Recently, categorical time series have been studied by Fahrmeir and Kaufmann (1987), Fokianos and Kedem (1998), Fokianos and Kedem (2003), Kaufmann (1987), Kedem and Fokianos (2002).
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