The use of BDS statistics for estimating the parameters of chaotic mappings and regular signals in the presence of noise
DOI:
https://doi.org/10.3103/S073527271103006XKeywords:
estimate of parameters, BDS statistic, analysis of accuracy of estimateAbstract
A new method has been proposed in this paper for estimating the parameters of chaotic and regular signals by their observation against the background of white noise under conditions of the a priori uncertainty about the distribution of its values. The method is based on using the nonparametric BDS statistic revealing the sensitivity toward topological properties of the attractors of chaotic, regular and random processes that are characterized by the correlation dimension. The results of numerical simulation of the method proposed for the estimation of parameters of one-dimensional and two-dimensional chaotic mappings and also the harmonic oscillation frequency for the noise with uniform and Gaussian distribution at different levels of its intensity have been presented. This study also includes the analysis of accuracy of estimating the harmonic oscillation frequency by the proposed method and its comparison with potentially attainable values.
References
S. E. Fal’kovich and P. Yu. Kostenko, The Basic Statistical Theory of Radio Engineering Systems. Tutorial (NAU KhAI, Kharkov, 2005) [in Russian].
A. V. Oppenheim, G. W. Wornell, S. H. Isabelle, and K. M. Cuomo, “Signal processing in the context of chaotic signals,” Proc. IEEE ICASSP IV, 117 (1992).
A. S. Dmitriev, A. I. Panas, and S. O. Starkov, “Dynamic Chaos as a Paradigm of Modern Communications Systems,” Zarubezhnaya Radioelektronika, No. 10, 4 (1997).
G. I. Tuzov, V. A. Sivov, V. I. Prytkov, et al., Noise Immunity of Radio Systems with Compound Signals (Radio i Svyaz’, Moscow, 1985) [in Russian].
P. Yu. Kostenko, A. N. Barsukov, K. S. Vasiuta, and S. N. Symonenko, “Detection of the Chaotic Process Distorted by the White Noise Using BDS Statistics,” Izv. Vyssh. Uchebn. Zaved., Radioelektron. 52(11), 41 (2009) [Radioelectron. Commun. Syst. 52(11), 599 (2009)].
P. Yu. Kostenko, S. I. Sivashchenko, and T. P. Kostenko, “Interpolation of Sequences Generated by Nonlinear Dynamic System,” Radiotekhnika (Kharkiv), No. 128, 166 (2002).
D. A. Smirnov, V. S. Vlaskin, and V. I. Ponomarev, “Method of Estimating the Parameters of One-Dimensional Mappings Using Chaotic Time Series,” Pisma v ZhTF 31, No. 3, 18 (2005).
K. Judd, “Chaotic–time–series reconstruction by the Bayesian paradigm: Right results by wrong methods,” Phys. Rev. E. 67, pp. 206–212.
W. A. Brock, W. D. Dechert, and J. A. Scheinkman, “A Test for Independence Based on the Correlation Dimension,” Working Paper No. 8702, Department of Economics, University of Wisconsin (1987).
S. Borovkova, R. Burton, and H. Dehling, “Consistency of the Takens estimator for the correlation dimension,” Annals of Applied Probability 9, No. 2, 376 (1999).
L. Kanzler, Very Fast and Correctly Sized Estimation of the BDS Statistic (Christ Church and Department of Economics, University of Oxford, 1999).
Н. Kantz and T. Schreiber, Nonlinear Time Series Analysis, 2nd ed. (The United Kingdom at the University Press, Cambridge, 2004).
