Ordinal pattern and statistical complexity analysis of daily stream flow time series

When calculating the Bandt and Pompe ordinal pattern distribution from given time series at depth D, some of the D! patterns might not appear. This could be a pure finite size effect (missing patterns) or due to dynamical properties of the observed system (forbidden patterns). For pure noise, no forbidden patterns occur, contrary to deterministic chaotic maps. We investigate long time series of river runoff for missing patterns and calculate two global properties of their pattern distributions: the Permutation Entropy and the Permutation Statistical Complexity. This is compared to purely stochastic but long-range correlated processes, the k-noise (noise with power spectrum f−k), where k is a parameter determining the strength of the correlations. Although these processes closely resemble runoff series in their correlation behavior, the ordinal pattern statistics reveals qualitative differences, which can be phrased in terms of missing patterns behavior or the temporal asymmetry of the observed series. For the latter, an index is developed in the paper, which may be used to quantify the asymmetry of natural processes as opposed to artificially generated data.