One of the most important aspects of time series is their degree of stochasticity vs. chaoticity. Since the discovery of chaotic maps, many algorithms have been proposed to discriminate between these two alternatives and assess their prevalence in real-world time series. Approaches based on the combination of “permutation patterns” with different metrics provide a more complete picture of a time series’ nature, and are especially useful to tackle pathological chaotic maps. Here, we provide a review of such approaches, their theoretical foundations, and their application to discrete time series and real-world problems. We compare their performance using a set of representative noisy chaotic maps, evaluate their applicability through their respective computational cost, and discuss their limitations.

The main motivation of this paper is to introduce the permutation Jensen-Shannon distance, a symbolic tool able to quantify the degree of similarity between two arbitrary time series. This quantifier results from the fusion of two concepts, the Jensen-Shannon divergence and the encoding scheme based on the sequential ordering of the elements in the data series. The versatility and robustness of this ordinal symbolic distance for characterizing and discriminating different dynamics are illustrated through several numerical and experimental applications. Results obtained allow us to be optimistic about its usefulness in the field of complex time-series analysis. Moreover, thanks to its simplicity, low computational cost, wide applicability, and less susceptibility to outliers and artifacts, this ordinal measure can efficiently handle large amounts of data and help to tackle the current big data challenges.

We introduce a representation space to contrast chaotic with stochastic dynamics. Following the complex network representation of a time series through ordinal pattern transitions, we propose to assign each system a position in a two-dimensional plane defined by the permutation entropy of the network (global network quantifier) and the minimum value of the permutation entropy of the nodes (local network quantifier). The numerical analysis of representative chaotic maps and stochastic systems shows that the proposed approach is able to distinguish linear from non-linear dynamical systems by different planar locations. Additionally, we show that this characterization is robust when observational noise is considered. Experimental applications allow us to validate the numerical findings and to conclude that this approach is useful in practical contexts.