Revisiting the role of correlation coefficient to distinguish chaos from noise (original) (raw)
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It has been shown that some chaotic time series has cyclostationary characteristic. In this paper, this characteristic is exploited for applications to modeling and prediction of chaotic time series. To this aim, a vector-autoregressive-modelbased model is developed. The model first transforms the scalar chaotic time series into a vector time series based on ployphase decomposition of cyclostationary time series, and then uses the vector autoregressive model for modeling and prediction purposes. The application of the proposed model to simulated data from the periodically perturbed Logistic map is carried out and the results show that the model works well for modeling and long-term prediction in comparison with other models. I.
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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.
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We address the calculation of correlation dimension, the estimation of Lyapunov exponents, and the detection of unstable periodic orbits, from transient chaotic time series. Theoretical arguments and numerical experiments show that the Grassberger-Procaccia algorithm can be used to estimate the dimension of an underlying chaotic saddle from an ensemble of chaotic transients. We also demonstrate that Lyapunov exponents can be estimated by computing the rates of separation of neighboring phase-space states constructed from each transient time series in an ensemble. Numerical experiments utilizing the statistics of recurrence times demonstrate that unstable periodic orbits of low periods can be extracted even when noise is present. In addition, we test the scaling law for the probability of finding periodic orbits. The scaling law implies that unstable periodic orbits of high period are unlikely to be detected from transient chaotic time series.