Conditional entropy of ordinal patterns

In this paper we investigate a quantity called conditional entropy of ordinal patterns, akin to the permutation entropy. The conditional entropy of ordinal patterns describes the average diversity of the ordinal patterns succeeding a given ordinal pattern. We observe that this quantity provides a good estimation of the Kolmogorov-Sinai entropy in many cases. In particular, the conditional entropy of ordinal patterns of a finite order coincides with the Kolmogorov-Sinai entropy for periodic dynamics and for Markov shifts over a binary alphabet. Finally, the conditional entropy of ordinal patterns is computationally simple and thus can be well applied to real-world data

Forbidden ordinal patterns in higher dimensional dynamics

Forbidden ordinal patterns are ordinal patterns (or `rank blocks') that cannot appear in the orbits generated by a map taking values on a linearly ordered space, in which case we say that the map has forbidden patterns. Once a map has a forbidden pattern of a given length $L_{0}$, it has forbidden patterns of any length $L\ge L_{0}$ and their number grows superexponentially with $L$. Using recent results on topological permutation entropy, we study in this paper the existence and some basic properties of forbidden ordinal patterns for self maps on n-dimensional intervals. Our most applicable conclusion is that expansive interval maps with finite topological entropy have necessarily forbidden patterns, although we conjecture that this is also the case under more general conditions. The theoretical results are nicely illustrated for n=2 both using the naive counting estimator for forbidden patterns and Chao's estimator for the number of classes in a population. The robustness of forbidden ordinal patterns against observational white noise is also illustrated.Comment: 19 pages, 6 figure

Detection of time reversibility in time series by ordinal patterns analysis

Time irreversibility is a common signature of nonlinear processes, and a fundamental property of non-equilibrium systems driven by non-conservative forces. A time series is said to be reversible if its statistical properties are invariant regardless of the direction of time. Here we propose the Time Reversibility from Ordinal Patterns method (TiROP) to assess time-reversibility from an observed finite time series. TiROP captures the information of scalar observations in time forward, as well as its time-reversed counterpart by means of ordinal patterns. The method compares both underlying information contents by quantifying its (dis)-similarity via Jensen-Shannon divergence. The statistic is contrasted with a population of divergences coming from a set of surrogates to unveil the temporal nature and its involved time scales. We tested TiROP in different synthetic and real, linear and non linear time series, juxtaposed with results from the classical Ramsey's time reversibility test. Our results depict a novel, fast-computation, and fully data-driven methodology to assess time-reversibility at different time scales with no further assumptions over data. This approach adds new insights about the current non-linear analysis techniques, and also could shed light on determining new physiological biomarkers of high reliability and computational efficiency.Comment: 8 pages, 5 figures, 1 tabl

Analysis of noise-induced temporal correlations in neuronal spike sequences

This is a copy of the author 's final draft version of an article published in the journal European physical journal. Special topics. The final publication is available at Springer via http://dx.doi.org/10.1140/epjst/e2016-60024-6We investigate temporal correlations in sequences of noise-induced neuronal spikes, using a symbolic method of time-series analysis. We focus on the sequence of time-intervals between consecutive spikes (inter-spike-intervals, ISIs). The analysis method, known as ordinal analysis, transforms the ISI sequence into a sequence of ordinal patterns (OPs), which are defined in terms of the relative ordering of consecutive ISIs. The ISI sequences are obtained from extensive simulations of two neuron models (FitzHugh-Nagumo, FHN, and integrate-and-fire, IF), with correlated noise. We find that, as the noise strength increases, temporal order gradually emerges, revealed by the existence of more frequent ordinal patterns in the ISI sequence. While in the FHN model the most frequent OP depends on the noise strength, in the IF model it is independent of the noise strength. In both models, the correlation time of the noise affects the OP probabilities but does not modify the most probable pattern.Peer ReviewedPostprint (author's final draft