The role of the likelihood function in the estimation of chaos models

Journal of Time Series Analysis

Volumn 21, Issue 4, 2000, Pages 363-387

  Ozaki, T ^a   Jimenez, J C ^b   Haggan Ozaki, V ^c  

^a INSTITUTE OF STATISTICAL MATHEMATICS   (Japan)

^b INSTITUTO DE CIBERNÉTICA   (Cuba)

^c SOPHIA UNIVERSITY   (Japan)

Author keywords

Chaos; Likelihood; Local linearization filter; Nonlinear Kalman filter; State space representation

Indexed keywords

EID: 0000212566     PISSN: 01439782     EISSN: None     Source Type: Journal    
DOI: 10.1111/1467-9892.00189     Document Type: Article

References (28)

1. AKAIKE, H. (1977) On entropy maximization principle. In Application of Statistics (ed. P. R. Krishnaiah) Amsterdam: North-Holland, pp. 27-42.
2. BARTLETT, M. S. (1990) Chance and chaos, a discussion paper. J. R. Stat. Soc. Ser. A 153, 321-47.
3. BERLINER, L. M. (1991) Likelihood and Bayesian prediction of chaotic systems. J. Am. Stat. Soc. 86, 938-52.
4. BISCAY, R., JIMENEZ, J. C., RIERA, J. J. and VALDES, P. A. (1996) Local linearization method for the numerical solution of stochastic differential equations. Ann. Inst. Stat. Math. 48, 631-44.
5. FARMER, J. D. and SIDOROWICH, J. J. (1991) Optimal shadowing and noise reduction. Physica D 47, 373-92.
6. FROST, P. A. and KAILATH, T. (1971) An innovation approach to least squares estimation - part III: Nonlinear estimation in white Gaussian noise. IEEE Trans. Autom. Control. AC-16, 217-26.
7. ITO, K. (1980) Chaos in the Rikitake two-disc dynamo system. Earth Planet. Sci. Lett 51, 451-56.
8. JIMENEZ, J. C., SHOJI, I. and OZAKI, T. (1999) Simulation of stochastic differential equations through the local linearization method: a comparative study. J. Stat. Phys. 94, 587-602.
9. KALMAN, R. E. (1960) A new approach to linear filtering and prediction probabilities. Trans. ASME J. Basic Eng. D82, 35-45.
10. KLOEDEN, P. E. and PLATEN, E. (1995) Numerical Solution of Stochastic Differential Equations. New York: Springer.
11. KOSTELICH, E. J. and YORKE, J. A. (1988) Noise reduction in dynamical systems. Phys. Rev. A. 38, 1649-52.
12. _ and _ (1990) Noise reduction: finding the simplest dynamical system consistent with the data. Physica D 41, 183-96.
13. KUSHNER, H. J. (1962) On the differential equations satisfied by conditional probability densities of Markov processes, with applications. J. SIAM Control, Ser. A. 2, 106-19.
14. LORENZ, E. N. (1963) Deterministic non-periodic flow. J. Atmos. Sci 20, 130-41.
15. MIL'STEIN, G. N. (1995) Numerical Integration of Stochastic Differential Equations. Dordrecht: Kluwer Academic.
16. ODA, H., OZAKI, T. and YAMANOUCHI, Y. (1987) A nonlinear system identification in the analysis of offshore structure dynamics in a random sea. Stochastic engineering systems. In Proceedings of IUTAM Symposium, Innsbruck/Iglis, Austria, 1987, June 21-26 (eds. F. Ziegler and G. I. Schuëller). New York: Springer.
17. OZAKI, T. (1985a) Nonlinear time series models and dynamic systems. In Handbook of Statistics, Vol. 5 (ed. E. J. Hannan and others). Amsterdam: North-Holland, pp. 25-83.
18. _ (1985b) Statistical identification of storage models with application to stochastic hydrology. Water Resour. Bull 21, 663-75.
19. _ (1992) A bridge between nonlinear time series models and nonlinear stochastic dynamical systems: a local linearization approach. Stat. Sinica 2, 113-35.
20. _ (1993) A local linearization approach to nonlinear filtering. Int. J. Control 57, 75-96.
21. _, JIMENEZ, J. C. and HAGGAN-OZAKI, V. (1996) The role of the likelihood function in the estimation of chaos models. Technical Report 1996/11, University of Manchester/U.M.I.S.T., Manchester Centre for Statistical Science.
22. RIKITAKE, T. (1958) Oscillations of a system of disk dynamos. Proc. Camb. Philos. Soc. 54, 89-105.
23. SHOJI, J. (1998) Approximation of a continuous time stochastic process by a local linearization method. Math. Comput. 67, 287-98.
24. _ and OZAKI, T. (1997) Comparative study of estimation methods for a continuous time stochastic process. J. Time Ser. Anal. 18, 485-506.
25. _ and _ (1998a) Estimation for nonlinear stochastic differential equations by a local linearization approach. Stochastic Anal. Appl. 16, 733-52.
26. _ and _ (1998b) A statistical method of estimation and simulation for systems of stochastic differential equations. Biometrika 85, 240-43.
27. THOMPSON, J. M. T., BOKAIAN, A. R. and GHAFFARI, R. (1984) Subharmonic and chaotic motions of compliant offshore structures and articulated mooring towers. ASME J. Energy Resour. Tech. 106, 191-98.
28. WAGNER, W. and PLATEN, E. (1978) Approximation of Ito Integral Equations. Preprint ZIMM, Akad. Wissenschaft, Berlin.