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.
- Publication:
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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:
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- Nonlinear Sciences - Chaotic Dynamics
- E-Print:
- doi:10.1016/j.physd.2005.07.006