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 ratesmeasures of new disorder per new observed valueare equal for ergodic finitealphabet information sources (discretetime stationary stochastic processes). With this result, we then prove that the same holds for deterministic dynamical systems defined by ergodic maps on ndimensional 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) 15951602.] 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 nonergodic finitealphabet sources is also studied and an inequality developed. Finally, the equality of permutation and metric entropy rates is extended to ergodic nondiscrete information sources when entropy is replaced by differential entropy in the usual way.
 Publication:

Physica D Nonlinear Phenomena
 Pub Date:
 October 2005
 DOI:
 10.1016/j.physd.2005.07.006
 arXiv:
 arXiv:nlin/0503044
 Bibcode:
 2005PhyD..210...77A
 Keywords:

 Nonlinear Sciences  Chaotic Dynamics
 EPrint:
 doi:10.1016/j.physd.2005.07.006