## 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## Notes

### Acknowledgements

The authors are grateful to the editor in chief Professor Hajo Holzmann, the anonymous associate editor, and two referees for their instructive comments and kindness. The first author Y.G. thanks Doctor Fumiya Akashi for his encouragements and comments and was supported by Grant-in-Aid for JSPS Research Fellow Grant Number JP201920060. The second author M.T. was supported by the Research Institute for Science & Engineering of Waseda University and JSPS Grant-in-Aid for Scientific Research (S) Grant Number JP18H05290.

### Compliance with ethical standards

### Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

## Supplementary material

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