pp 1–27 | Cite as

Discriminant analysis based on binary time series

  • Yuichi GotoEmail author
  • Masanobu Taniguchi


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.


Stationary process Spectral density Binary time series Robustness Discriminant analysis Misclassification probability 

Mathematics Subject Classification

62H30 62G86 



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

184_2019_746_MOESM1_ESM.pdf (56 kb)
Supplementary material 1 (pdf 56 KB)


  1. Anderson TW (1984) An introduction to multivariate statistical analysis. Wiley, New YorkzbMATHGoogle Scholar
  2. Bagnall A, Janacek G (2005) Clustering time series with clipped data. Mach Learn 58(2–3):151–178CrossRefGoogle Scholar
  3. Billinsley P (1968) Convergence of probability measures. Wiley, New YorkGoogle Scholar
  4. Brillinger DR (1968) Estimation of the cross-spectrum of a stationary bivariate gaussian process from its zeros. J R Stat Soc Ser B Stat Methodol 30:145–159MathSciNetzbMATHGoogle Scholar
  5. Brillinger DR (1981) Time series: data analysis and theory, expanded edn. Holden-Day, San FranciscozbMATHGoogle Scholar
  6. Buz A, Litan C (2012) What properties do clipped data inherit from the generating processes? Studia Universitatis Babes-Bolyai 57(3):85–96Google Scholar
  7. Dau HA, Keogh E, Kamgar K, Yeh CCM, Zhu Y, Gharghabi S, Ratanamahatana CA, Chen Y, Hu B, Begum N, Bagnall A, Mueen A, Batista G (2018) The UCR time series classification archive.
  8. Dette H, Hallin M, Kley T, Volgushev S (2015) Of copulas, quantiles, ranks and spectra: An \(l_1\)-approach to spectral analysis. Bernoulli 21(2):781–831Google Scholar
  9. Fahrmeir L, Kaufmann H (1987) Regression models for non-stationary categorical time series. J Time Ser Anal 8(2):147–160MathSciNetCrossRefGoogle Scholar
  10. Fitzmaurice GM, Lipsitz SR (1995) A model for binary time series data with serial odds ratio patterns. J R Stat Soc Ser C 44(1):51–61zbMATHGoogle Scholar
  11. Fokianos K, Kedem B (1998) Prediction and classification of non-stationary categorical time series. J Multivar Anal 67(2):277–296MathSciNetCrossRefGoogle Scholar
  12. Fokianos K, Kedem B (2003) Regression theory for categorical time series. Stat Sci 18:357–376MathSciNetCrossRefGoogle Scholar
  13. Giraitis L, Kokoszka P, Leipus R (2000) Stationary arch models: dependence structure and central limit theorem. Econom Theory 16(1):3–22MathSciNetCrossRefGoogle Scholar
  14. Gómez E, Gómez-Villegas MA, Marín JM (2003) A survey on continuous elliptical vector distributions. Rev Mat Complut 16:345–361MathSciNetCrossRefGoogle Scholar
  15. Hannan EJ (1970) Multiple time series, vol 38. Wiley, HobokenCrossRefGoogle Scholar
  16. He S, Kedem B (1989) On the Stieltjes–Sheppard orthant probability formula. Tech Rep tr-89-69, Dept. Mathematics, Unvi. Maryland, College ParkGoogle Scholar
  17. Hinich M (1967) Estimation of spectra after hard clipping of gaussian processes. Technometrics 9(3):391–400MathSciNetGoogle Scholar
  18. Hosoya Y, Taniguchi M (1982) A central limit theorem for stationary processes and the parameter estimation of linear processes. Ann Stat 10:132–153MathSciNetCrossRefGoogle Scholar
  19. Ibragimov IA, Rozanov Y (1978) Gaussian random processes, vol 9. Springer, BerlinCrossRefGoogle Scholar
  20. Johnson RA, Wichern DW (1988) Applied multivariable statistical analysis, 2nd edn. Prentice-Hall, Englewood Cliffs zbMATHGoogle Scholar
  21. Kakizawa Y (1996) Discriminant analysis for non-gaussian vector stationary processes. J Nonparametr Stat 7(2):187–203MathSciNetCrossRefGoogle Scholar
  22. Kakizawa Y (1997) Higher order asymptotic theory for discriminant analysis in Gaussian stationary processes. J Jpn Stat Soc 27(1):19–35MathSciNetCrossRefGoogle Scholar
  23. Kaufmann H (1987) Regression models for nonstationary categorical time series: asymptotic estimation theory. Ann Stat 15:79–98MathSciNetCrossRefGoogle Scholar
  24. Kedem B (1994) Time series analysis by higher order crossings. IEEE press, New YorkzbMATHGoogle Scholar
  25. Kedem B, Fokianos K (2002) Regression models for time series analysis. Wiley, New YorkCrossRefGoogle Scholar
  26. Kedem B, Li T (1989) Higher order crossings from a parametric family of linear filters. Tech Rep tr-89-47, Dept. Mathematics, Unvi. Maryland, College ParkGoogle Scholar
  27. Kedem B, Slud E (1982) Time series discrimination by higher order crossings. Ann Stat 10(3):786–794 MathSciNetCrossRefGoogle Scholar
  28. Keenan DM (1982) A time series analysis of binary data. J Am Stat Assoc 77(380):816–821MathSciNetCrossRefGoogle Scholar
  29. Kley T (2014) Quantile-based spectral analysis. Ph.D. thesis, Ruhr-Universitut BochumGoogle Scholar
  30. Li TH (2008) Laplace periodogram for time series analysis. J Am Stat Assoc 103(482):757–768MathSciNetCrossRefGoogle Scholar
  31. Liggett W Jr (1971) On the asymptotic optimality of spectral analysis for testing hypotheses about time series. Ann Stat 42(4):1348–1358MathSciNetCrossRefGoogle Scholar
  32. Lomnicki Z, Zaremba S (1955) Some applications of zero-one processes. J R Stat Soc Ser B Stat Methodol 17:243–255MathSciNetzbMATHGoogle Scholar
  33. Olszewski R T (2001) Generalized feature extraction for structural pattern recognition in time-series datas. Ph.D. thesis, Carnegie Mellon UniversityGoogle Scholar
  34. Panagiotakis C, Tziritas G (2005) A speech/music discriminator based on RMS and zero-crossings. IEEE Trans Multimed 7(1):155–166CrossRefGoogle Scholar
  35. Petrantonakis PC, Hadjileontiadis LJ (2010) Emotion recognition from brain signals using hybrid adaptive filtering and higher order crossings analysis. IEEE Trans Affect Comput 1(2):81–97CrossRefGoogle Scholar
  36. Petrantonakis PC, Hadjileontiadis LJ (2010) Emotion recognition from eeg using higher order crossings. IEEE Trans Inf Technol Biomed 14(2):186–197CrossRefGoogle Scholar
  37. Rice SO (1944) Mathematical analysis of random noise. Bell Syst Tech J 23:282–332MathSciNetCrossRefGoogle Scholar
  38. Robinson PM (1991) Automatic frequency domain inference on semiparametric and nonparametric models. Econometrica 59:1329–1363MathSciNetCrossRefGoogle Scholar
  39. Sakiyama K, Taniguchi M (2004) Discriminant analysis for locally stationary processes. J Multivar Anal 90(2):282–300MathSciNetCrossRefGoogle Scholar
  40. Shumway R, Unger A (1974) Linear discriminant functions for stationary time series. J Am Stat Assoc 69(348):948–956MathSciNetCrossRefGoogle Scholar
  41. Tanaka M, Shimizu K (2001) Discrete and continuous expectation formulae for level-crossings, upcrossings and excursions of ellipsoidal processes. Stat Prob Lett 52(3):225–232MathSciNetCrossRefGoogle Scholar
  42. Taniguchi M (1987) Minimum contrast estimation for spectral densities of stationary processes. J R Stat Soc Ser B Stat Methodol 49:315–325MathSciNetzbMATHGoogle Scholar
  43. Taniguchi M, Kakizawa Y (2000) Asymptotic theory of statistical inference for time series. Springer, New YorkCrossRefGoogle Scholar
  44. Zhang G, Taniguchi M (1995) Nonparametric approach for discriminant analysis in time series. J Nonparametr Stat 5(1):91–101MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Waseda UniversityTokyoJapan

Personalised recommendations