Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-01-26T04:02:07.666Z Has data issue: false hasContentIssue false
  • English
  • Français

The theory of KM2O-Langevin equations and its applications to data analysis (III): Deterministic analysis

Published online by Cambridge University Press:  22 January 2016

Yasunori Okabe  and
Toshiyuki Yamane
Yasunori Okabe
Affiliation:
Department of Mathematical Engineering and Information Physics, Graduate School and Faculty of Engineering, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan
Toshiyuki Yamane
Affiliation:
Department of Mathematical Engineering and Information Physics, Graduate School and Faculty of Engineering, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan
Rights & Permissions [Opens in a new window]

Abstract.

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A technique is given for detecting deterministic dynamics in time series. Some stochastic difference equations, called KM2O-Langevin equations, are extracted directly from given data. We can find deterministic dynamics in time series by evaluating the magnitude of innovation part of the KM2O-Langevin equations. We can further find chaotic dynamics in time series by predicting it from the viewpoint of the theory of KM2O-Langevin equations.

We apply our method to the data of measles and chicken pox, which are also treated by G. Sugihara and R.M. May in [1]. The result of numerical experiments indicates that there seem to exist some deterministic dynamics in both time series. It also suggests, however, that the data of measles seems to be chaotic while that of chicken pox not, which corresponds to the result of G. Sugihara and R.M. May.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 152 , December 1998 , pp. 175 - 201
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1998

References

[1] Sugihara, G. and May, R. M., Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series, Nature, 344 (1990), 734–741.Google Scholar
[2] Masani, P. and Wiener, N., Non-linear prediction, Probability and Statistics,, The Harald Cramér, (Grenander, V., ed.), John Wiley, 1959, pp. 190–212.Google Scholar
[3] Okabe, Y., On a stochastic difference equation for the multi-dimensional weakly stationary process with discrete time, Prospect of Algebraic Analysis, (Kashiwara, M. and Kawai, T., eds.), Academic Press, Tokyo, 1988, pp. 601–645.Google Scholar
[4] Okabe, Y. and Nakano, Y., The theory of KM2 O-Langevin equations and its applications to data analysis (I): Stationary analysis, Hokkaido Math. J., 20 (1991), 45–90.Google Scholar
[5] Okabe, Y., Application of the theory of KM2 O-Langevin equations to the linear prediction problem for the multi-dimensional weakly stationary process, J. Math. Soc. of Japan, 45 (1993), 277–294.Google Scholar
[6] Okabe, Y., Langevin equations and causal analysis, SUGAKU Exposition, Math Amer.. Soc. Transl., 161 (1994), 19–50.Google Scholar
[7] Okabe, Y. and Inoue, A., The theory of KM2 O-Langevin equations and its applications to data analysis (II): causal analysis (1), Nagoya Math J., 134 (1994), 1–28.Google Scholar
[8] Okabe, Y. and Ootsuka, T., Applications of the theory of KM2 O-Langevin equations to the nonlinear prediction problem for the one dimensional strictly stationary time series, J. Math. Soc. Japan, 47 (1995), 349–367.Google Scholar
[9] Okabe, Y., Non-linear time series analysis based upon the fluctuation-dissipation theorem, Nonlinear Analysis, Theory, Method & Applications, 30 (1997), 2249-.Google Scholar
[10] Efron, B., Bootstrap methods: Another look at the jackknife, The Annals of Statistics, 7 (1979), 1–28.Google Scholar