On the automatic parameter selection for permutation entropy

Abstract

Permutation Entropy (PE) is a cost effective tool for summarizing the complexity of a time series. It has been used in many applications including damage detection, disease forecasting, detection of dynamical changes, and financial volatility analysis. However, to successfully use PE, an accurate selection of two parameters is needed: the permutation dimension n and embedding delay τ. These parameters are often suggested by experts based on a heuristic or by a trial and error approach. Both of these methods can be time-consuming and lead to inaccurate results. In this work, we investigate multiple schemes for automatically selecting these parameters with only the corresponding time series as the input. Specifically, we develop a frequency-domain approach based on the least median of squares and the Fourier spectrum, as well as extend two existing methods: Permutation Auto-Mutual Information Function and Multi-scale Permutation Entropy (MPE) for determining τ. We then compare our methods as well as current methods in the literature for obtaining both τ and n against expert-suggested values in published works. We show that the success of any method in automatically generating the correct PE parameters depends on the category of the studied system. Specifically, for the delay parameter τ, we show that our frequency approach provides accurate suggestions for periodic systems, nonlinear difference equations, and electrocardiogram/electroencephalogram data, while the mutual information function computed using adaptive partitions provides the most accurate results for chaotic differential equations. For the permutation dimension n, both False Nearest Neighbors and MPE provide accurate values for n for most of the systems with a value of n=5 being suitable in most cases.

Network Analysis of Time Series: Novel Approaches to Network Neuroscience

“…A significant difficulty of auto-mutual information measures, however, is that they are often not naturally applicable to continuous, real-valued time series, so some binning procedure is likely to be required (for a discussion of mutual information estimators for continuous data, see Papana and Kugiumtzis, 2009). Recently, much more involved algorithms for automatic optimization of d and τ based on frequency-domain analysis have been proposed as well (Myers and Khasawneh, 2020), and debate on this question is likely to be on-going for the foreseeable future. One possibility that has not been explored is selecting parameters that produce surrogate data that best preserves some feature of interest in the original data: (McCullough et al, 2017) provides an algorithm for how a constrained random walk on an OPN can be used to generate synthetic time series that preserve the ordinal partition transition dynamics of the original time series and so d and τ might be selected as the values that create OPNs than produce synthetic data that has (for example) the frequency band profile most like the original data.…”

A look into chaos detection through topological data analysis

“…Additionally, false positives are found when periodic time series are contaminated with substantial noise [5]. The issue of oversampling can be resolved by properly sub-sampling the time series according to [10,11,12], the effects of noise are more difficult to overcome. A method has been developed which allows the 0-1 test to handle noise very effectively if a gauge-value (i.e.…”

Detection of Local Mixing in Time-Series Data Using Permutation Entropy

“…The calculation focuses on the local relationships among a sequence of points by mapping their values to an ordering of the same length. The three-point sequence [7,3,11], for instance, would map to the ordering [1, 0, 2] because 3 < 7 < 11. The statistics of these ordinal sequences or permutations are calculated as follows.…”

“…[6] If the signal were wholly random, the P E would be 1. The value of the P E is a function of the subsequence length , of course; please see [3,7,8] for more discussion of this parameter and its implications.…”