Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T23:53:45.041Z Has data issue: false hasContentIssue false
  • English
  • Français

The theory of KM2O-Langevin equations and applications to data analysis (II): Causal analysis (1)

Published online by Cambridge University Press:  22 January 2016

Yasunori Okabe  and
Akihiko Inoue
Yasunori Okabe
Affiliation:
Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060Japan
Akihiko Inoue
Affiliation:
Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060Japan
Rights & Permissions [Opens in a new window]

Extract

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.

It is not too much to say that the problem of finding a cause-and-effect relationship is a fascinating and eternal theme in both natural and social sciences. It is often difficult to decide whether one is the cause of another in two related phenomena, but it is an important problem. It is related to the internal structure of phenomena which generate deterministic or random changes as time passes. We note that the phenomena to be considered are often not deterministic but random. For example, in physical systems such as quantum mechanics or chaotic classical mechanics, it is well known that certain probabilistic reasonings are indispensable.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 134 , June 1994 , pp. 1 - 28
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1994

References

[1] Akaike, H. and Nakagawa, T., Statistical analysis and control for a dynamical system, Science Company, 1973 (in Japanese).Google Scholar
[2] Box, G.E.P. and Jenkins, G.M., Time series analysis, Forcasting and Control, Holden-Day, 1976.Google Scholar
[3] Durbin, J., The fitting of time series model, Rev. Int. Stat, 28 (1960), 233–244.CrossRefGoogle Scholar
[4] Elton, C. and Nicholson, M., The ten-year cycle in numbers of the Lynx in Canada, J. Animal Ecology, 11 (1942), 215–244.CrossRefGoogle Scholar
[5] Granger, C. W. J., Investigating causal relations by econometric models and cross-spectral methods, Econometrica, 37 (1969), 424–438.CrossRefGoogle Scholar
[6] Jevons, W. S., Commercial crises and sun-spots, Nature, 15 (1878), 33–37.CrossRefGoogle Scholar
[7] Kubo, R., Statistical mechanical theory of irreversible processes I, general theory and simple applications to magnetic and conduction problem, J. Phys. Soc. Japan, 12 (1957), 570–586.CrossRefGoogle Scholar
[8] Levinson, N., The Wiener RMS error criterion in filter design and prediction, J. Math. Phys., 25 (1947), 261–278.CrossRefGoogle Scholar
[9] Mata, C.G. and Shaffner, F.I., Solar and economic relationships; a preliminary report, The Quarterly Journal of Economics, Nov. (1934).Google Scholar
[10] Moran, P.A.P., The statistical analysis of the sunspot and Lynx cycles, J. Animal Ecology, 18 (1949), 115–116.CrossRefGoogle Scholar
[11] Moran, P.A.P., The statistical analysis of the Canadian Lynx cycle I, structure and prediction, J. Animal Ecology, 22 (1951), 163–173.Google Scholar
[12] Nakano, Y. and Okabe, Y., On a multi-dimensional [α, β, γ]-Langevin equation, Proc. Japan Acad., 59 (1983), 171–173.Google Scholar
[13] Okabe, Y., On a stationary Gaussian process with T-positivity and its associated Langevin equation and S-matrix, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 26 (1979), 115–165.Google Scholar
[14] Okabe, Y., On a stochastic differential equation for a stationary Gaussian process with T-positivity and the fluctuation-dissipation theorem, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28 (1981), 169–213.Google Scholar
[15] Okabe, Y., On a stochastic differential equation for a stationary Gaussian process with finite multiple Markovian property and the fluctuation-dissipation theorem, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28 (1982), 793–804.Google Scholar
[16] Okabe, Y., On a Langevin equation, Sûgaku, 33 (1981), 306–324 (in Japanese).Google Scholar
[17] Okabe, Y., On a wave equation associated with prediction errors for a stationary Gaussian process, Lecture Notes in Control and Information Sciences, 49 (1983), 215–226.CrossRefGoogle Scholar
[18] Okabe, Y., A generalized fluctuation-dissipation theorem for the one-dimensional diffusion process, Commum. Math. Phys., 98 (1985), 449–468.CrossRefGoogle Scholar
[19] Okabe, Y., On KMO-Langevin equations for stationary Gaussian process with T-positivity, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 33 (1986), 1–56.Google Scholar
[20] Okabe, Y., On the theory of Brownian motion with the Alder-Wainwright effect, J. Stat. Phys., 45 (1986), 953–981.CrossRefGoogle Scholar
[21] Okabe, Y., KMO-Langevin equation and fluctuation-dissipation theorem (I), Hokkaido Math. J., 15 (1986), 163–216.Google Scholar
[22] Okabe, Y., KMO-Langevin equation and fluctuation-dissipation theorem (II), Hokkaido Math. J., 15 (1986), 317–355.Google Scholar
[23] Okabe, Y., Stokes-Boussinesq-Langevin equation and fluctuation-dissipation theorem, Prob. Theory and Math. Stat. (ed. by Prohorov, et al.), VNU Science Press, 1986, vol. 2, 431–436.Google Scholar
[24] Okabe, Y., On long time tails of correlation functions for KMO-Langevin equations, Proceeding of the Fourth Japan-USSR symposium on Probability theory, Kyoto, July, 1986, Lecture Notes in Mathematics, Springer Vol. 1299, 391–397.Google Scholar
[25] Okabe, Y., On the theory of discrete KMO-Langevin equations with reflection positivity (I), Hokkaido Math. J., 16 (1987), 315–341.CrossRefGoogle Scholar
[26] Okabe, Y., On the theory of discrete KMO-Langevin equations with reflection positivity (II), Hokkaido Math. J., 17 (1988). 1–44.CrossRefGoogle Scholar
[27] Okabe, Y., On the theory of discrete KMO-Langevin equations with reflection positivity (III), Hokkaido Math. J., 18 (1989), 149–174.CrossRefGoogle Scholar
[28] Okabe, Y., On stochastic difference equations for the multi-dimentional weakly stationary time series, Prospect of Algebraic Analysis (ed. by Kashiwara, M. and Kawai, T.), Academic Press, 1988, vol. 2, 601–645.CrossRefGoogle Scholar
[29] Okabe, Y., Langevin equation and Fluctuation-dissipation theorem, Stochastic Processes and their Applications (ed. by Albeverio, S. et al.), Kluwer Academic Publishers, 1990, 275–299.CrossRefGoogle Scholar
[30] Okabe, Y., Application of the theory of KM2O-Langevin equations to the linear prediction problem for the multi-dimensional weakly stationary time series, J. Math. Soc. of Japan, 45 (1993), 277–294.CrossRefGoogle Scholar
[31] Okabe, Y. and Nakano, Y., The theory of KM2O-Langevin equations and its applications to data analysis (I): Stationary analysis, Hokkaido Math. J., 20 (1991), 45–90.CrossRefGoogle Scholar
[32] Okabe, Y. and Ootsuka, T., Application of the theory of KM2O-Langevin equations to the non-linear prediction problem for the one-dimensional strictly stationary time series, to appear in J. Math. Soc. of Japan.Google Scholar
[33] Okabe, Y. and Ootsuka, T., The theory of KM2O-Langevin equations and its applications to data analysis (III): Prediction analysis (in preparation).Google Scholar
[34] Okabe, Y., The theory of KM2O-Langevin equations and its applications to data (IV): Causal analysis (2) (in preparation).Google Scholar
[35] Priestly, M. B., Non-linear and non-stationary time series, Academic Press, 1988.Google Scholar
[36] Walker, G., On periodicity in series of related terms, Proc. Roy. Soc. Ser. A, 131 (1931), 518–532.Google Scholar
[37] Whittle, P., On the fitting of multivariate autoregressions, and the approximate canonical factorization of a spectral density matrix, Biometrika, 50 (1963), 129–134.CrossRefGoogle Scholar
[38] Wiggins, R.A. and Robinson, E.A., Recursive solution to the multichannel fitting problem, J. Geophys. Res., 70 (1965), 1885–1891.CrossRefGoogle Scholar
[39] Yule, G.U., On a method of investigating periodicities in disturbed series, with special references to Wolfer’s sunspot numbers, Phil. Trans. Roy. Soc. London Ser. A, 226 (1927), 268–298.Google Scholar