Including the Magnitude Variability of a Signal in the Ordinal Pattern Analysis

One of the most popular and innovative methods to analyse signals is by using Ordinal Patterns (OPs). The OP encoding is based on transforming a (univariate) signal into a symbolic sequence of OPs, where each OP represents the number of permutations needed to order a small subset of the signal's magnitudes. This implies that OPs are conceptually clear, methodologically simple to implement, and robust to noise, and that they can be applied to short signals. Moreover, they simplify the statistical analyses that can be carried out on a signal, such as entropy and complexity quantifications. However, because of the relative ordering, information about the magnitude of the signal at each timestamp is lost-this being one of the major drawbacks of this method. Here, we propose a way to use the signal magnitudes discarded in the OP encoding as a complementary variable to its permutation entropy. To illustrate our approach, we analyse synthetic trajectories from logistic and Hénon maps-with and without added noise-and real-world signals, including intracranial electroencephalographic recordings from rats in different sleep-wake states and frequency fluctuations in power grids. Our results show that, when complementing the permutation entropy with the variability in the signal magnitudes, the characterisation of these signals is improved and the results remain explainable. This implies that our approach can be useful for feature engineering and improving AI classifiers, as typical machine learning algorithms need complementary signal features as inputs to improve classification accuracy.