A continuous ordinal patterns-based paradigm for the analysis and comparison of time series

We here introduce a method to compare time series, leveraging continuous ordinal patterns, i.e., a variant of the classical order pattern analysis, and inspired by a next-generation reservoir computing architecture. The former ones are used as the non-linear transformation units in the latter, in place of traditional neural elements; as a final step, the transformed time series are classified using a standard machine learning model. This allows one to calculate the distance between a time series and a reference, e.g., a randomly shuffled or time-reversed version of the same, and hence to test properties such as stochasticity or time irreversibility. The method is validated through synthetic data generated by classical chaotic systems, demonstrating higher sensitivity when compared to classical tests, and through data representing real-world systems, spanning financial, medical, and technological ones. We finally discuss some additional relevant topics, including the generalizability of the obtained results and the sensitivity of the method to the chosen parameters.