The permutation entropy rate equals the metric entropy rate for ergodic information sources and ergodic dynamical systems

Abstract

Permutation entropy quantifies the diversity of possible orderings of the values a random or deterministic system can take, as Shannon entropy quantifies the diversity of values. We show that the metric and permutation entropy rates—measures of new disorder per new observed value—are equal for ergodic finite-alphabet information sources (discrete-time stationary stochastic processes). With this result, we then prove that the same holds for deterministic dynamical systems defined by ergodic maps on n-dimensional intervals. This result generalizes a previous one for piecewise monotone interval maps on the real line [C. Bandt, G. Keller, B. Pompe, Entropy of interval maps via permutations, Nonlinearity 15 (2002) 1595–1602.] at the expense of requiring ergodicity and using a definition of permutation entropy rate differing modestly in the order of two limits. The case of non-ergodic finite-alphabet sources is also studied and an inequality developed. Finally, the equality of permutation and metric entropy rates is extended to ergodic non-discrete information sources when entropy is replaced by differential entropy in the usual way.