On a quantitative method to analyze dynamical and measurement noise (original) (raw)
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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.
Physical Review E
In the field of nonlinear dynamics, many methods have been proposed to tackle the issue of optimally setting embedding dimension and lag in order to analyze sampled scalar signals. However, intrinsic statistical uncertainties due to the finiteness of input sequences severely hinder a general solution to the problem. A more achievable approach consists of assessing sets of dimension and lag pairs that are equivalently suitable to embed a time series. Here we present a method to identify these sets of embedding pairs, under the hypothesis that the time series of interest is generated by a chaotic, finite-dimensional dynamical system. We first introduce a "distance gauge transformation" based on the analytical forms of correlation integrals corresponding to a Gaussian white noise source. We show that in this new distance gauge, correlation integrals generated by chaotic, finitedimensional systems are characterized by distinctive features, whose absence is a marker of the unsuitability of the underlying embedding choice. By means of a new estimator of the correlation dimension that relies on the new distance gauge, sets of suitable embedding pairs are finally identified by looking at the uniformity of the estimation. The method is completely automatic and was successfully tested on both synthetic and experimental time series. It also provides a tool to estimate the redundance and irrelevance timescales of the system that underlie the time series as well as a lower constraint to the sampling rate. The method is suitable for applications in research fields where a chaotic behavior has to be identified, such as neuroscience, geophysics, and economics.
Chaos, Solitons & Fractals, 2012
In this work we propose a non-iterative method to determine the noise level of chaotic time series. For this purpose, we use the gaussian noise functional derived by Schreiber in 1993. It is shown that the noise function could be approximated by a stretched exponential decay form. The decay function is then used to construct a linear least squares approach where global solution exists. We have developed a software basis to calculate the noise level which is based on TISEAN algorithms. A practical way to exclude the outlying observations for small length scales has been proposed to prevent estimation bias. The algorithm is tested on well known chaotic systems including Henon, Ikeda map and Lorenz, Rössler, Chua flow data. Although the results of the algorithm obtained from simulated discrete dynamics are not satisfactory, we have shown that it performs well on flow data even for extreme level of noise. The results that are obtained from the real world financial and biomedical time series have been interpreted.
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.
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.
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.