Detecting nonlinearity in short and noisy time series using the permutation entropy

Permutation entropy contains the information about the temporal structure associated with the underlying dynamics of a time series. Its estimation is simple, and because it is based on the comparison of neighboring values, it becomes significantly robust to noise. It is also computationally efficient and invariant with respect to nonlinear monotonous transformations. For all these reasons, the permutation entropy seems to be particularly suitable as a discriminative measure for unveiling nonlinear dynamics in arbitrary real-world data. In this paper, we study the efficacy of a conventional surrogate method with a linear stochastic process as the null hypothesis but implementing the permutation entropy as a nonlinearity measure. Its discriminative power is tested by implementing several analyses on numerical signals whose dynamical properties are known a priori (linear discrete and continuous models, chaotic regimes of discrete and continuous systems). The performance of the proposed approach in real-world applications (chaotic laser data, monthly smoothed sunspot index and neuro-physiological recordings) is also demonstrated. The results obtained allow us to conclude that this symbolic tool is very useful for discriminating nonlinear characteristics in very short and noisy data.