Ambiguities in Bandt–Pompe’s methodology for local entropic quantifiers

The Bandt–Pompe (BP) prescription for building up probability densities [C. Bandt, B. Pompe, Permutation entropy: a natural complexity measure for time series, Phys. Rev. Lett. 88 (2002) 174102] constituted a significant advance in the treatment of time-series. However, as we show here, ambiguities arise in applying the BP technique with reference to the permutation of ordinal patterns. This happens if one wishes to employ the BP-probability density to construct local entropic quantifiers that would characterize time-series generated by nonlinear dynamical systems. Explicit evidence of this fact is presented by comparing two different procedures, frequently found in the literature, that generate sequences of ordinal patterns. In opposition to the case of global quantifiers in the orthodox Shannon fashion, the proper order of the pertinent symbols turns out to be not uniquely predetermined for local entropic indicators. We advance the idea of employing the Fisher–Shannon information plane as a tool to resolve the ambiguity and give illustrative examples.