Discriminant analysis based on binary time series

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.

Full text

Metrika (2020) 83:569–595
https://doi.org/10.1007/s00184-019-00746-1
Discriminant analysis based on binary time series
Yuichi Goto1·Masanobu Taniguchi1
Received: 6 January 2019 / Published online: 12 October 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 misclassifica-
tion probability tends to zero when the number of observation tends to infinity, that
is, the consistency of our discrimination method. Further, we evaluate the asymp-
totic 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/s00184-
019-00746- 1) contains supplementary material, which is available to authorized users.
BYuichi Goto
yuu510@fuji.waseda.jp
1Present Address: Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
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