Permutation entropy revisited

Time-series analysis in terms of ordinal patterns is revisited by introducing a generalized permutation entropy Hp(w, L), which depends on two different window lengths: w, implicitly defining the resolution of the underlying partition; L, playing the role of an embedding dimension, analogously to standard nonlinear time-series analysis. The w-dependence provides information on the structure of the corresponding invariant measure, while the L-dependence helps determining the Kolmogorov–Sinai entropy. We finally investigate the structure of the partition with the help of principal component analysis, finding that, upon increasing w, the single atoms become increasingly elongated.