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Parametric estimation and spectral analysis of piecewise linear maps of the interval

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Artur Lopes  and
Sílvia Lopes
Artur Lopes*
Affiliation:
Universidade Federal do Rio Grande do Sul
Sílvia Lopes*
Affiliation:
Universidade Federal do Rio Grande do Sul
*
∗ Postal address: Instituto de Matemática, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves 9500, Porto Alegre, RS-91540-000, Brasil.
∗ Postal address: Instituto de Matemática, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves 9500, Porto Alegre, RS-91540-000, Brasil.
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Abstract

We present an estimation procedure and analyse spectral properties of stochastic processes of the kind Zt = Xt + ξt = ϕ(Tt(ψ)) + ξt, for t ∊ Z, where T is a deterministic map, ϕ is a given function and ξt is a noise process. The examples considered in this paper generalize the classical harmonic model Zt = Acos(ω0t + ψ) + ξt, for t ∊ Z. Two examples are developed at length. In the first one, the spectral measure is discrete and in the second it is continuous. In the second example, the time series is obtained from a chaotic map. These two examples exhibit the extremal cases of different possibilities for the spectral measure of time series and they are both associated with ergodic deterministic transformations with noise. We present a method for obtaining explicitly the spectral density function (second example) and the autocorrelation coefficients (first example). In the first example the rotation number plays an important role. We also consider large deviation properties of the estimated parameters of the model.

Keywords

Spectral analysischaotic time seriesrotation number

MSC classification

Primary: 60G10: Stationary processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 3 , September 1998 , pp. 757 - 776
Copyright
Copyright © Applied Probability Trust 1998 

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References

Bogomolny, E. and Carioli, M. (1993). Quantum maps from transfer operators. Physica D 67, 88–112.Google Scholar
Bloomfield, P. (1976). Fourier Analysis of Time Series: An Introduction. Wiley, New York.Google Scholar
Brockwell, P. J. and Davis, R.A. (1987). Time Series: Theory and Methods. Springer, New York.Google Scholar
Carmona, S., Landim, C., Lopes, A. and Lopes, S. (1998). A level-2 large deviations principle for the autocorrelation function of uniquely ergodic transformations with noise. J. Statist. Phys. 91, 395–421.Google Scholar
Coelho, Z., Lopes, A. and Rocha, L. F. (1994). Absolutely continuous invariant measures for a class of affine interval exchange maps. Proc. Amer. Math. Soc. 123, 3533–3542.Google Scholar
Cornfeld, I. P., Fomin, S. V. and Sinai, Ya. G. (1982). Ergodic Theory. Springer, New York.Google Scholar
Dembo, A. and Zeitouni, O. (1993). Large Deviations Techniques and Applications. Jones and Bartlett, Boston.Google Scholar
Devaney, R.L. (1989). An Introduction to Chaotic Dynamical System. Addison-Wesley, Redwood City, CA.Google Scholar
Ding, M. Z., Grebogi, C., Ott, E., Sauer, T. and Yorke, J. A. (1993). Estimating correlation dimension from a chaotic time series: when does a plateau onset occur? Physica D 69, 404–424.Google Scholar
Ellis, R. (1989). Entropy, Large Deviations and Statistical Mechanics. Springer, New York.Google Scholar
Gradshteyn, I. S. and Ryzhik, I. M. (1965). Table of Integrals, Series and Products. Academic Press, New York.Google Scholar
Kostelich, E. and Yorke, J. A. (1990). Noise reduction: finding the simplest dynamical system consistent with data. Physica D 41, 183–196.Google Scholar
Lasota, A. and Yorke, J. A. (1973). On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186, 481–488.Google Scholar
Lopes, A. (1994). Entropy, pressure and large deviation. Cellular Automata, Dynamical Systems and Neural Networks. ed. Goles, E. and Martinez, S.. Kluwer, Massachusetts, pp. 79–146.Google Scholar
Lopes, A. and Lopes, S. (1995). Parametric estimation and spectral analysis of chaotic time series. Technical Report, UFRGS, 42, Series A.Google Scholar
Lopes, A. and Lopes, S. (1996). Unique ergodicity, large deviations and parametric estimation. Preprint.Google Scholar
Lopes, A., Lopes, S. and Souza, R. (1997a). On the spectral density of a class of chaotic time series. J. Time Series Anal. 18, 465–474.Google Scholar
Lopes, A., Lopes, S. and Souza, R. (1997b). Spectral analysis of expanding one-dimensional chaotic transformations. Random and Computational Dynamics 15, 65–80.Google Scholar
Orey, S. (1986). Large deviations in ergodic theory. In Seminar on Stochastic Processes, ed. Chung, K. L. et al. Birkhäuser, Boston, pp 195–249.Google Scholar
Parry, W. and Pollicott, M. (1990). Zeta functions and the periodic orbit structure of hyperbolic dynamics. Asterisque 187–188.Google Scholar
Pollicott, M. (1986). Distribution of closed geodesics on the modular surface and quadratic irrationals. Bull. Soc. Math. France. 114, 431–446.CrossRefGoogle Scholar
Sakai, H. and Tokumaru, H. (1980). Autocorrelations of a certain chaos. IEEE Trans. Acoust., Speech and Signal Processing 28, 588–590.Google Scholar
Takens, F. (1994). Analysis of non-linear time series, a survey. Preprint.Google Scholar
Tong, H. (1990). Non-linear Time Series: A Dynamical System Approach. Clarendon Press, Oxford.Google Scholar
Varadhan, S. R. S. (1988). Large deviations and applications. Lecture Notes in Mathematics 1362, ed. P. Diaconis et al.al. Springer, Berlin, pp. 1–49.Google Scholar
Walters, P. (1981). An Introduction to Ergodic Theory. Springer, New York.Google Scholar