Hostname: page-component-7dd5485656-wxk4p Total loading time: 0 Render date: 2025-10-31T12:05:16.823Z Has data issue: false hasContentIssue false
  • English
  • Français

An innovation approach to non-Gaussian time series analysis

Part of: Inference from stochastic processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Tohru Ozaki  and
Mitsunori Iino
Tohru Ozaki*
Affiliation:
Institute of Statistical Mathematics, Tokyo
Mitsunori Iino*
Affiliation:
Institute of Statistical Mathematics, Tokyo
*
1 Postal address: Institute of Statistical Mathematics, 4-6-7 Minami-Azabu Minato-Ku, Tokyo 106-8569, Japan. Email: ozaki@ism.ac.jp
2 Email: iino@ypri.co.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper shows that the use of both types of random noise, white noise and Poisson noise, can be justified when using an innovations approach. The historical background for this is sketched, and then several methods of whitening dependent time series are outlined, including a mixture of Gaussian white noise and a compound Poisson process: this appears as a natural extension of the Gaussian white noise model for the prediction errors of a non-Gaussian time series. A statistical method for the identification of non-linear time series models with noise made up of a mixture of Gaussian white noise and a compound Poisson noise is presented. The method is applied to financial time series data (dollar-yen exchange rate data), and illustrated via six models.

Keywords

Innovationsnon-linear modelingnon-Gaussian time series analysisdiffusion processpoint processwhite noisePoisson noiseLévy processexchange rate data

MSC classification

Primary: 62M10: Time series, auto-correlation, regression, etc.
Secondary: 60G35: Signal detection and filtering

Information

Type
Time series analysis
Information
Journal of Applied Probability , Volume 38 , Issue A: Probability, Statistics and Seismology , 2001 , pp. 78 - 92
Copyright
Copyright © Applied Probability Trust 2001 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Akaike, H. (1968). Fitting autoregressive models for prediction. Ann. Inst. Statist. Math. 20, 55–69.CrossRefGoogle Scholar
Akaike, H. (1973). Maximum likelihood identification of Gaussian autoregressive moving average models. Biometrika 60, 255–265.Google Scholar
Akaike, H. (1977). On entropy maximization principle. In Applications of Statistics , ed. Krishnaiah, P. R., North-Holland, Amsterdam, 27–41.Google Scholar
Astrom, K. J. and Kallstrom, C. G. (1973). Application of system identification techniques to the determination of ship dynamics. In Identification and System Parameter Estimation , ed. Eykhoff, P., North-Holland, Amsterdam, 415–424.Google Scholar
Bouchard, J. P. and Cont, R. (1998). A Langevin approach to stock market fluctuations and crashes. European Phys. J. Ser. B 6, 543–550.CrossRefGoogle Scholar
Box, G. E. P. and Jenkins, G. M. (1970). Time Series Analysis, Forecasting and Control. Holden-Day, San Francisco.Google Scholar
Bryson, A. E. and Frazier, M. (1963). Smoothing for linear and nonlinear dynamic systems. In Proc. Optimum System Synthesis Conf. , 354–364. Reprinted in (1977) Linear Least-Squares Estimation (Benchmark Papers in Electrical Eng. Comput. Sci. 17), ed. Kailath, T., Halsted Press, New York, 290–300.Google Scholar
Doob, J. L. (1953). Stochastic Processes. Wiley, New York.Google Scholar
Farmer, J. D. (1998). Market force, ecology, and evolution. Working Paper 98-12-117, Santa Fe Institute.Google Scholar
Feller, W. (1966). An Introduction to Probability Theory and its Applications , Vol. 2. Wiley, New York.Google Scholar
Frost, P. A. and Kailath, T. (1971). An innovation approach to least squares estimation, III: Nonlinear estimation in white Gaussian noise. IEEE Trans. Automatic Control 16, 217–226.Google Scholar
Hannan, E. J. (1969). The identification of vector mixed autoregressive-moving average systems. Biometrika 56, 223–225.Google Scholar
Harrison, P. J. and Stevens, C. F. (1976). Bayesian forecasting (with discussion). J. Roy. Statist. Soc. Ser. B. 68, 205–247.Google Scholar
Ho, Y. C. and Lee, R. C. K. (1964). A Bayesian approach to problems in stochastic estimation and control. IEEE Trans. Automatic Control 9, 333–339.Google Scholar
Iino, M. and Ozaki, T. (1999a). Filtering and prediction of non-Gaussian time series via jump diffusion process. Proc. Inst. Statist. Math. 47, 327–342.Google Scholar
Iino, M. and Ozaki, T. (1999b). Modeling financial time series using state space models. In Proc. 1999 Workshop on Information-Based Induction Sciences. Google Scholar
Iino, M. and Ozaki, T. (2000). A nonlinear model for financial dynamics. In Poster session of the Internat. Symp. Frontiers of Time Series Anal., February 7-9, 2000, The Institute of Statistical Mathematics, Tokyo.Google Scholar
Ikoma, N. (1999). An improvement of the Monte Carlo filter. In Proc. 67th Annual Conf. Japan Statist. Soc. , 367–368 (in Japanese).Google Scholar
Kalman, R. E. (1960). A new approach to linear filtering and prediction probabilities. Trans. ASME J. Basic Eng. D82, 35–45.Google Scholar
Kolmogorov, A. N. (1941). Interpolation and extrapolation of stationary random sequences. Bull. Acad. Sci. USSR, Math. Ser. 5, 3–14 (in Russian). English translation: (1962) Memo RM-3090-PR, RAND Corp., Santa Monica, CA.Google Scholar
Levy, P. (1956). Sur une class de courbes de l'espace de Hilbert et sur une équation intégrale non linéaire. Ann. Sci. école Norm. Sup. 73, 121–156.Google Scholar
Masani, P. and Wiener, N. (1959). Nonlinear prediction. In Probability and Statistics: The Harald Cramér Volume , ed. Grenander, U., Wiley, New York, 190–212.Google Scholar
Meditch, J. S. (1967). On optimal linear smoothing theory. J. Inf. Control 10, 598–615.Google Scholar
Mehra, R. K. (1971). Identification of stochastic linear dynamic systems. Amer. Inst. Aeronaut. Astronaut. J. 9, 28–31.Google Scholar
Nelson, D. B. (1990). ARCH models as diffusion approximations. J. Econometrics 45, 7–39.Google Scholar
Nelson, D. B. and Foster, D. P. (1994). Asymptotic filtering theory for univariate ARCH models. Econometrica 62, 1–41.CrossRefGoogle Scholar
O'Hara, M. (1995). Market Microstructure Theory. Blackwell, Cambridge.Google Scholar
Ozaki, T. (1985). Nonlinear time series models and dynamical systems. In Handbook of Statistics 5, eds Hannan, E. J. et al., North-Holland, Amsterdam, 25–83.Google Scholar
Ozaki, T. (1992a). A bridge between nonlinear time series models and nonlinear stochastic dynamical systems: a local linearization approach. Statistica Sinica 2, 113–135.Google Scholar
Ozaki, T. (1992b). Identification of nonlinearities and non-Gaussianities in time series. In New Directions in Time Series Analysis, Part I (IMA Vol. Math. Applic. 45), eds Brillinger, D. R. et al., Springer, New York, 227–264.Google Scholar
Protter, P. (1990). Stochastic Integration and Differential Equations. Springer, New York.CrossRefGoogle Scholar
Rauch, H. E. (1963). Solution to the linear smoothing problem. IEEE Trans. Automatic Control 8, 371–372.CrossRefGoogle Scholar
Taylor, S. J. (1994). Modelling stochastic volatility: A review and comparative study. Math. Finance 4, 184–204.Google Scholar
Valdes, P., Jimenez, J. C., Riera, J., Biscay, R. and Ozaki, T. (1999). Nonlinear EEG analysis based on a neural mass model. Biol. Cybernetics 81, 415–424.Google Scholar
Whittle, P. (1963). Prediction and Regulation by Linear Least Squares Method. English Universities Press, London.Google Scholar
Wiener, N. (1942). Extrapolation, Interpolation and Smoothing of Stationary Time Series with Engineering Applications. A classified report by MIT Radiation Lab., Cambridge MA, February 1942. Later published (1949), Wiley, New York.Google Scholar
Wiener, N. (1948). Cybernetics. MIT Press, Cambridge, MA.Google Scholar
Wold, H. (1938). A Study in the Analysis of Stationary Time Series. Almqvist and Wiksell, Uppsala.Google Scholar