Permutation Jensen-Shannon distance: A versatile and fast symbolic tool for complex time-series analysis.

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.

💡 Research Summary

The paper introduces the Permutation Jensen‑Shannon Distance (PJSD), a novel symbolic metric designed to quantify the similarity between two arbitrary time series. PJSD combines two well‑established concepts: (i) the Jensen‑Shannon (JS) divergence, a symmetric and bounded version of the Kullback‑Leibler divergence, whose square root is a true metric on probability distributions; and (ii) the Bandt‑Pompe (BP) ordinal symbolization, which maps a scalar time series into a probability distribution over D! possible permutation patterns (ordinal patterns) by ranking D consecutive (or lag‑τ spaced) points. Because the BP mapping uses only the relative order of points, it is invariant under monotonic transformations, robust to additive noise and outliers, and requires only the choice of two integer parameters (order D and lag τ).

The methodology proceeds as follows. For each series, the chosen D and τ define overlapping vectors of length D. Each vector is replaced by the permutation that sorts its elements; the relative frequencies of the D! permutations provide an empirical probability distribution P (or Q). The JS divergence D_JS(P,Q) is then computed, and the distance is defined as √D_JS(P,Q). By normalizing with the theoretical maximum √ln 2, PJSD values lie in