Permutation entropy of finite-length white-noise time series

Permutation entropy (PE) is commonly used to discriminate complex structure from white noise in a time series. While the PE of white noise is well understood in the long time-series limit, analysis in the general case is currently lacking. Here the expectation value and variance of white-noise PE are derived as functions of the number of ordinal pattern trials, N, and the embedding dimension, D. It is demonstrated that the probability distribution of the white-noise PE converges to a χ^{2} distribution with D!-1 degrees of freedom as N becomes large. It is further demonstrated that the PE variance for an arbitrary time series can be estimated as the variance of a related metric, the Kullback-Leibler entropy (KLE), allowing the qualitative N≫D! condition to be recast as a quantitative estimate of the N required to achieve a desired PE calculation precision. Application of this theory to statistical inference is demonstrated in the case of an experimentally obtained noise series, where the probability of obtaining the observed PE value was calculated assuming a white-noise time series. Standard statistical inference can be used to draw conclusions whether the white-noise null hypothesis can be accepted or rejected. This methodology can be applied to other null hypotheses, such as discriminating whether two time series are generated from different complex system states.