Permutation Entropy and Order Patterns in Long Time Series
While ordinal techniques are commonplace in statistics, they have been introduced to time series fairly recently by Hallin and coauthors. Permutation entropy, an average of frequencies of order patterns, was suggested by Bandt and Pompe in 2002 and used by many authors as a complexity measure in physics, medicine, engineering, and economy. Here a modified version is introduced, the "distance to white noise." For datasets with tens of thousands or even millions of values, which are becoming standard in many fields, it is possible to study order patterns separately, determine certain differences of their frequencies, and define corresponding autocorrelation type functions. In contrast to classical autocorrelation, these functions are invariant with respect to nonlinear monotonic transformations of the data. For order three patterns, a variance analytic "Pythagoras formula" combines the different autocorrelation functions with our new version of permutation entropy. We demonstrate the use of such correlation type functions in sliding window analysis of biomedical and environmental data.