Zanin, M., Rodríguez-González, A., Menasalvas Ruiz, E., & **Papo, D. **

*Entropy*, **20**:665 (2018)

Time irreversibility, i.e. the lack of invariance of the statistical properties of a system under time reversal, is a fundamental property of all systems operating out of equilibrium. Time reversal symmetry is associated with important statistical and physical properties and is related to the predictability of the system generating the time series. Over the past ﬁfteen years, various methods to quantify time irreversibility in time series have been proposed, but these can be computationally expensive. Here we propose a new method, based on permutation entropy, which is essentially parameter-free, temporally local, yields straightforward statistical tests, and has fast convergence properties. We apply this method to the study of ﬁnancial time series, showing that stocks and indices present a rich irreversibility dynamics. We illustrate the comparative methodological advantages of our method with respect to a recently proposed method based on visibility graphs, and discuss the implications of our results for ﬁnancial data analysis and interpretation.

[Read more in *Entropy*]

*Keywords*: time-reversal symmetry; permutation entropy; entropy production; visibility graphs; out-of-equilibrium system; time series analysis; predictability; Efficient Market Hypothesis