Primary Noise Reduction Efficiency for Detecting Chaos in High Noisy Pseudoperiodic Time Series

A long-standing fundamental issue in nonlinear time series analysis is to determine whether a complex time series is regular, deterministically chaotic, or random. An accurate identification of the dynamics underlying a complex time series, is of crucial importance in understanding the corresponding physical process, and in turn affects the subsequent model development. A steady stream of efforts has been made, and a number of effective methods have been proposed (also in the latest years – [1]-[3]) to tackle this difficult problem. The vast majority of these methods are based on attractor reconstruction from time series and such characteristics as largest Lyapunov exponents, K2 entropy, and correlation dimension calculation [4], [5]. Since the analysis of chaotic data in terms of dimensions, entropies, and Lyapunov exponents requires access to the small length scales (small-scale fluctuations of the signal), already a moderate amount of measurement noise on data is known to be destructive. One class of time series – pseudoperiodic – has aroused great interest due to their close relation to some important natural and physiological systems. Zhang et all [6] have proposed a method to detect deterministic structure from this certain class of chaotic time series, which can deal with small or moderate amounts of noise. But I have not found any publication, devoted to detecting the deterministic structure from a high noisy pseudoperiodic time series, when the noise level reducing is desirable with expected to preserve the exponential divergence of nearest neighbors. Noise reduction methods designed for signals that can be treated by a linear model fail to eliminate noise from a contaminated chaotic time series because the spectra of the chaotic signal and the noise overlap. Noise reduction based on time delay embedding, which has been widely studied, may be the most promising way to filter the noisy chaotic data [8]-[10]. Several phase space projection methods, based on subspace decomposition, were proposed for application to the problem of additive noise reduction in the context of phase space analysis – the global projections method [10] and the local (nearest neighborhoods) phase spaces method [7], [9], [10]. A two step method is proposed to reduce colored noise [11]. These methods performed well with moderate amounts of noise. Hovewer, in order to distinguish between regular and non-regular dynamics of time series, which exhibit pseudoperiodic behaviour, it is not necessary to eliminate the noise perfectly. Importantly, that signal distortion and noise residual on noise reduction would enable to detect the presence of chaos in a dynamical system by measuring the largest Lyapunov exponents or characteristics, related with the largest Lyapunov exponents. The aim of present paper – to establish, which method fits best for a primary noise reduction of high noisy pseudoperiodic time series and evaluate primary noise reduction efficiency for detecting chaos in these time series. Also the straightforward and relative noisy resistant algorithm to detect chaos in pseudoperiodic time series by using the correlation coefficient as a measure of the distance [6] between vectors of reconstructed phase space is present. Proposed algorithm is principally based on the algorithm of Rosenstein [12] for largest Lyapunov exponent’s calculation, but by over-embedding and an appropriately longer embedding window and by using the correlation coefficient as a measure of the distance instead of Euclidean distance. Throughout the paper, the x component of the well-known Rossler system for illustration, which is chaotic and contain obvious periodic component, is used. The organization of this paper is as follows. In Sec. II, the principle of noise reduction for chaotic data in the global and local phase space is reviewed, and algorithm for detecting chaos in pseudoperiodic time series is described. In Sec. III, the results of calculations are given. Finally, some discussions and conclusions are given in Sec. IV.