Distinguishing chaos from noise: A new approach (original) (raw)
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Distinguishing noise from chaos
2007
Chaotic systems share with stochastic processes several properties that make them almost undistinguishable. In this communication we introduce a representation space, to be called the complexity-entropy causality plane. Its horizontal and vertical axis are suitable functionals of the pertinent probability distribution, namely, the entropy of the system and an appropriate statistical complexity measure, respectively. These two functionals are evaluated using the Bandt-Pompe recipe to assign a probability distribution function to the time series generated by the system. Several well-known model-generated time series, usually regarded as being of either stochastic or chaotic nature, are analyzed so as to illustrate the approach. The main achievement of this communication is the possibility of clearly distinguishing between them in our representation space, something that is rather difficult otherwise.
Non-linear noise reduction and detecting chaos: some evidence from the S&P Composite Price Index
Mathematics and computers …, 1999
Academic and applied researchers in economics have, in the last 10 years, become increasingly interested in the topic of chaotic dynamics. In this paper we undertake non-linear dynamical analysis of one representative time series taken from financial markets, namely the Standard and Poor's (S&P) Composite Price Index. The data is based upon (adjusted) daily data from 1928 to 1987 comprising 16 127 observations. The results in the paper, based on the Grassberger–Procaccia (GP) correlation dimension measurement in conjunction with non-linear noise filtering and the surrogate technique, show strong evidence of chaos in one of these series, the S&P 500. The analysis shows that the accuracy of results improves with the increase in the number of recording points and the length of the time series, 5000 data points being sufficient to identify deterministic dynamics.
Revisiting the role of correlation coefficient to distinguish chaos from noise
The European Physical Journal B, 2000
The correlation coefficient vs. prediction time profile has been widely used to distinguish chaos from noise. The correlation coefficient remains initially high, gradually decreasing as prediction time increases for chaos and remains low for all prediction time for noise. We here show that for some chaotic series with dominant embedded cyclical component(s), when modelled through a newly developed scheme of periodic decomposition, will yield high correlation coefficient even for long prediction time intervals, thus leading to a wrong assessment of inherent chaoticity. But if this profile of correlation coefficient vs. prediction horizon is compared with the profile obtained from the surrogate series, correct interpretations about the underlying dynamics are very much likely.
A method of estimating the noise level in a chaotic time series
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2008
An attempt is made in this study to estimate the noise level present in a chaotic time series. This is achieved by employing a linear least-squares method that is based on the correlation integral form obtained by Diks in 1999. The effectiveness of the method is demonstrated using five artificial chaotic time series, the Hénon map, the Lorenz equation, the Duffing equation, the Rossler equation and the Chua's circuit whose dynamical characteristics are known a priori. Different levels of noise are added to the artificial chaotic time series and the estimated results indicate good performance of the proposed method. Finally, the proposed method is applied to estimate the noise level present in some real world data sets.
Some Aspects of Chaotic Time Series Analysis
2001
We address two aspects in chaotic time series analysis, namely the definition of embedding parameters and the largest Lyapunov exponent. It is necessary for performing state space reconstruction and identification of chaotic behavior. For the first aspect, we examine the mutual information for determination of time delay and false nearest neighbors method for choosing appropriate embedding dimension. For the second aspect we suggest neural network approach, which is characterized by simplicity and accuracy.
Detection of low-dimensional chaos in quasi-periodic time series: The 0-1 test
The 0-1 test is a novel test that has been recently suggested to detect low-dimensional chaos in time series. The test has already been applied successfully on a variety of time series. The goal of the present paper is to study the capability of the test to detect the presence of long-term chaos in time series presenting a strong periodic or quasi-periodic component. To achieve that goal, we apply the test using chaotic (quasi-periodic) and nonchaotic (periodic) R¨ossler time series. The method succeeds in detecting the presence or absence of chaos, even when significant white noise is added to the time series. The 0-1 test along with the classic method of calculating the maximum,Lyapunov exponent (MLE) are then applied for the first time to a time series obtained by surface electromyography (sEMG). Results from both tests are consistent with the presence of low-dimension chaos. The two tests, however, give the same result when applied to a surrogate nondeterministic time series res...
Method to distinguish possible chaos from colored noise and to determine embedding parameters
Physical Review A, 1992
We present a computational method to determine if an observed time series possesses structure statistically distinguishable from high-dimensional linearly correlated noise, possibly with a nonwhite spectrum. This method should be useful in identifying deterministic chaos in natural signals with broadband power spectra, and is capable of distinguishing between chaos and a random process that has the same power spectrum. The method compares nonlinear predictability of the given data to an ensemble of random control data sets. A nonparametric statistic is explored that permits a hypothesis testing approach. The algorithm can detect underlying deterministic chaos in a time series contaminated by additive random noise with identical power spectrum at signal to noise ratios as low as 3 dB. With less noise, this method can also be used to get good estimates of the parameters (the embedding dimension and the time delay) needed to perform the standard phase-space reconstruction of a chaotic time series.
Chaotic and random dynamics discrimination in financial time series
2013
One of the richest fields for the application of econophysics methods is Finance. In particular, financial markets produces a large amount of ready-to-use time series, which could be subject to statistic scrutiny. The analysis of financial time series by means of permutation information quantifiers derived from Information Theory resulted of great value in order to distinguish random and chaotic paths. In this paper we describe the permutation entropy and permutation statistical complexity. Both metrics form a locus where each planar realization reveals a particular statistic characteristic of the time series under study. We give several successful applications of this methodology. We perform an econophysic application to the sovereign fixed income market.
Physica D: Nonlinear Phenomena, 1996
One of the truly novel issues in the physics of the last decade is that some time series considered of stochastic origin might in fact be of a particular deterministic type, named "chaotic". Chaotic processes are essentially characterized by a low, rather than very high (as in stochastic processes), number of degrees of freedom. There has been a proliferation of attempts to provide efficient analytical tools to discriminate between chaos and stochasticity, but in most cases their practical utility is limited by the lack of knowledge of their effectiveness in realistic time series, i.e. of finite length and contaminated by noise. The present paper attempts to estimate the practical efficiency of a slightly modified Sugihara and May procedure [G. Sugihara and R.M. May, Nature 344 (1990) 734]. This is applied to synthetic finite time series generated from discrete parameter processes, providing rates of misidentification (obtained through simulations) for the most common stochastic processes (Gaussian, exponential, autoregressive, and periodic) and chaotic maps (logistic, H6non, biological, Tent, trigonometric, and Ikeda). The procedure consists of comparing with a selected threshold the correlation between actual and predicted values one time step into the future as a function of the embedding dimension E. This procedure allows to infer the presence of low-dimensional chaos even on series of ~ 50 units, and in presence of a noise level equal to ~ 10% of the signal amplitude. We apply this method to the sequence of volcanic eruptions of Piton de La Foumaise volcano finding no evidence of low-dimensional chaos.
ENERGEIA. Sociedad Ibero-Americana de Metodología Económica (ISSN 1666-5732), 2020
Chaos theory refers to the behaviour of certain deterministic nonlinear dynamical systems whose solutions, although globally stable, are locally unstable. These chaotic systems describe aperiodic, irregular, apparently random and erratic trajectories, i.e., deterministic complex dynamics. One of the properties that derive from this local instability and that allow characterizing these deterministic chaotic systems is their high sensitivity to small changes in the initial conditions, which can be measured by using the so-called Lyapunov exponents. The detection of chaotic behaviour in the underlying generating process of a time series has important methodological implications. When chaotic behaviour is detected, then it can be concluded that the irregularity of the series is not necessarily random, but the result of some deterministic dynamic process. Then, even if such process is unknown, it will be possible to improve the predictability of the time series and even to control or stabilize the evolution of the time series. This article provides a summary of the main current concepts and methods for the detection of chaotic behaviour from time series.