Permutation entropy: One concept, two approaches

Since C. Bandt and B. Pompe introduced permutation entropy in 2002 for piecewise strictly monotonous self-maps of one-dimensional intervals, this concept has been generalized to ever more general settings by means of two similar, though not equivalent, approaches. The first one keeps the original spirit in that it uses "sharp" dynamics and the corresponding ordinal partitions. The second uses symbolic (or "coarse-grained" dynamics with respect to arbitrary finite partitions, as in the conventional approach to the Kolmogorov-Sinai entropy of dynamical systems. Precisely, one of the main questions along these two avenues refers to the relation between permutation entropy and Kolmogorov-Sinai entropy. In this paper the authors will explain the underpinnings of both approaches and the latest theoretical results on permutation entropy. The authors also discuss some remaining open questions.