Delay Parameter Selection in Permutation Entropy Using Topological Data Analysis

Permutation Entropy (PE) is a powerful tool for quantifying the complexity of a signal which includes measuring the regularity of a time series. Additionally, outside of entropy and information theory, permutations have recently been leveraged as a graph representation, which opens the door for graph theory tools and analysis. Despite the successful application of permutations in a variety of scientific domains, permutations requires a judicious choice of the delay parameter $\tau$ and dimension $n$. However, $n$ is typically selected within an accepted range giving optimal results for the majority of systems. Therefore, in this work we focus on choosing the delay parameter, while giving some general guidance on the appropriate selection of $n$ based on a statistical analysis of the permutation distribution. Selecting $\tau$ is often accomplished using trial and error guided by the expertise of domain scientists. However, in this paper, we show how persistent homology, a commonly used tool from Topological Data Analysis (TDA), provides methods for the automatic selection of $\tau$. We evaluate the successful identification of a suitable $\tau$ from our TDA-based approach by comparing our results to both expert suggested parameters from published literature and optimized parameters (if possible) for a wide variety of dynamical systems.