Refined scale-dependent permutation entropy to analyze systems complexity

Multiscale entropy (MSE) has become a prevailing method to quantify the complexity of systems. Unfortunately, MSE has a temporal complexity in O(N2), which is unrealistic for long time series. Moreover, MSE relies on the sample entropy computation which is length-dependent and which leads to large variance and possible undefined entropy values for short time series. Here, we propose and introduce a new multiscale complexity measure, the refined scale-dependent permutation entropy (RSDPE). Through the processing of different kinds of synthetic data and real signals, we show that RSDPE has a behavior close to the one of MSE. Furthermore, RSDPE has a temporal complexity in O(N). Finally, RSDPE has the advantage of being much less length-dependent than MSE. From all this, we conclude that RSDPE over-performs MSE in terms of computational cost and computational accuracy.