Delay Parameter Selection in Permutation Entropy Using Topological Data Analysis

TL;DR

Persistent homology, a commonly used tool from topological data analysis (TDA), provides methods for the automatic selection of persistent homology, while giving some general guidance on the appropriate selection of n based on a statistical analysis of the permutation distribution.

Abstract

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 $τ$ 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 $τ$ 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 $τ$. We evaluate the successful identification of a suitable $τ$ 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.

Figures (14)

• Figure 1: All possible permutation configurations (motifs) for n = 3, where $[\pi_1 \ldots \pi_6] = [(0, 1, 2) \ldots (2,1,0)]$.
• Figure 2: Abundance of each permutation from example data set $X$.
• Figure 3: Example formulation of a persistence diagram based on $0$-D sublevel sets.
• Figure 4: Example formation of a permutation sequence from the time series $x(t) = 2\sin(t)$ with sampling frequency $f_s = 20$ Hz, permutation dimension $n=3$ and delay $\tau = 40$. The corresponding time-delay embedded vectors from $x(t)$ with the permutation binnings ($\pi_1, \ldots, \pi_6$) in the state space are shown in the bottom figure.
• Figure 5: Example comparing first minima of mutual information and first maxima of multi-scale permutation entropy, which demonstrates the correspondence between the two. On the left are the $n=3$ time delayed state space reconstructions with an inaccurately chosen $\tau = 1$ and appropriate $\tau = 14$. On the right shows the permutation distribution as $\tau$ increases and the associated multi-scale permutation entropy and mutual information plots.