Abstract
We analyze the effects of noise on the permutation entropy of dynamical systems. We take as numerical examples the logistic map and the Rössler system. Upon varying the noise strength we find a transition from an almost-deterministic regime, where the permutation entropy grows slower than linearly with the pattern dimension, to a noise-dominated regime, where the permutation entropy grows faster than linearly with the pattern dimension. We perform the same analysis on experimental time-series by considering the stochastic spiking output of a semiconductor laser with optical feedback, and find that the permutation entropy always increases faster than linearly. Nevertheless, the analysis allows to detect regularities of the underlying dynamics and model simulations are in a good agreement with the empirical data. By comparing the results of these three different examples, we discuss the possibility of determining from a time series whether the underlying dynamics is dominated by noise or not.
Highlights
Ordinal analysis is a method of time series analysis that consists of computing the probabilities of ordinal patterns, which are defined according to the ordering of D consecutive values in the series [1]
We mostly focus on permutation entropy as the effect of noise on block entropies is fairly well understood, see e.g. [30, 31]
We have studied the influence of noise in the permutation entropy of dynamical systems, considering both, simulated data and experimental data
Summary
Ordinal analysis is a method of time series analysis that consists of computing the probabilities of ordinal patterns, which are defined according to the ordering of D consecutive values in the series [1] The entropy of these probabilities, referred to as permutation entropy, is a tool to detect possible regularities in the time series. Applications to experimental time series analysis include classification and discrimination of dynamical states in normal and epileptic EEG [17,18,19,20] and detection of heart rate variability under different physiological and pathological conditions [21,22,23] Given this growing interest, it is relevant to understand the relation between the permutation entropy and other complexity measures. The Kolmogorov-Sinai entropy is obtained as the rate of growth, for D → ∞ and in the limit of a very refined partition, of the block entropy
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