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Extracting relevant properties of empirical signals generated by nonlinear,stochastic, and high-dimensional systems is a challenge of complex systemsresearch. Open questions are how to differentiate chaotic signals fromstochastic ones, and how to quantify nonlinear and/or high-order temporalcorrelations. Here we propose a new technique to reliably address bothproblems. Our approach follows two steps: first, we train an artificial neuralnetwork (ANN) with flicker (colored) noise to predict the value of theparameter, $\alpha$, that determines the strength of the correlation of thenoise. To predict $\alpha$ the ANN input features are a set of probabilitiesthat are extracted from the time series by using symbolic ordinal analysis.Then, we input to the trained ANN the probabilities extracted from the timeseries of interest, and analyze the ANN output. We find that the $\alpha$ valuereturned by the ANN is informative of the temporal correlations present in thetime series. To distinguish between stochastic and chaotic signals, we exploitthe fact that the difference between the permutation entropy (PE) of a giventime series and the PE of flicker noise with the same $\alpha$ parameter issmall when the time series is stochastic, but it is large when the time seriesis chaotic. We validate our technique by analysing synthetic and empirical timeseries whose nature is well established. We also demonstrate the robustness ofour approach with respect to the length of the time series and to the level ofnoise. We expect that our algorithm, which is freely available, will be veryuseful to the community.

Discriminating chaotic and stochastic time series using permutation entropy and artificial neural networks