Causality and the entropy–complexity plane: Robustness and missing ordinal patterns

We deal here with the issue of determinism versus randomness in time series. One wishes to identify their relative weights in a given time series. Two different tools have been advanced in the literature to such effect, namely, (i) the “causal” entropy–complexity plane [O.A. Rosso, H.A. Larrondo, M.T. Martín, A. Plastino, M.A. Fuentes, Distinguishing noise from chaos, Phys. Rev. Lett. 99 (2007) 154102] and (ii) the estimation of the decay rate of missing ordinal patterns [J.M. Amigó, S. Zambrano, M.A.F. Sanjuán, True and false forbidden patterns in deterministic and random dynamics, Europhys. Lett. 79 (2007) 50001; L.C. Carpi, P.M. Saco, O.A. Rosso, Missing ordinal patterns in correlated noises. Physica A 389 (2010) 2020–2029]. In this work we extend the use of these techniques to address the analysis of deterministic finite time series contaminated with additive noises of different degree of correlation. The chaotic series studied here was via the logistic map (r=4) to which we added correlated noise (colored noise with f−k Power Spectrum, 0≤k≤2) of varying amplitudes. In such a fashion important insights pertaining to the deterministic component of the original time series can be gained. We find that in the entropy–complexity plane this goal can be achieved without additional computations. ► The concept of forbidden/missing ordinal patterns is investigated. ► We linked it to localization effects in the entropy–complexity plane. ► Research focuses on the decay rate of missing ordinal patterns in noisy time series. ► Typical behaviors allow identification of time series deterministic component.