Abstract
Ordinal patterns are the common basis of various techniques used in the study of dynamical systems and nonlinear time series analysis. The present article focusses on the computational problem of turning time series into sequences of ordinal patterns. In a first step, a numerical encoding scheme for ordinal patterns is proposed. Utilising the classical Lehmer code, it enumerates ordinal patterns by consecutive non-negative integers, starting from zero. This compact representation considerably simplifies working with ordinal patterns in the digital domain. Subsequently, three algorithms for the efficient extraction of ordinal patterns from time series are discussed, including previously published approaches that can be adapted to the Lehmer code. The respective strengths and weaknesses of those algorithms are discussed, and further substantiated by benchmark results. One of the algorithms stands out in terms of scalability: its run-time increases linearly with both the pattern order and the sequence length, while its memory footprint is practically negligible. These properties enable the study of high-dimensional pattern spaces at low computational cost. In summary, the tools described herein may improve the efficiency of virtually any ordinal pattern-based analysis method, among them quantitative measures like permutation entropy and symbolic transfer entropy, but also techniques like forbidden pattern identification. Moreover, the concepts presented may allow for putting ideas into practice that up to now had been hindered by computational burden. To enable smooth evaluation, a function library written in the C programming language, as well as language bindings and native implementations for various numerical computation environments are provided in the supplements.
Highlights
The article Permutation Entropy: A Natural Complexity Measure for Time Series by Christoph Bandt and Bernd Pompe [1] pioneered a novel approach towards nonlinear time series analysis
Three different algorithms were discussed that all analyse the ordinal patterns
Of a given time series, and encode them in a computationally advantageous way, such that the ordinal patterns {Π1, Π2, . . . , Πm! } of order m are compactly represented by the set of non-negative integers
Summary
The article Permutation Entropy: A Natural Complexity Measure for Time Series by Christoph Bandt and Bernd Pompe [1] pioneered a novel approach towards nonlinear time series analysis. The time series of interest is embedded in an m-dimensional phase space, each delay vector is discretised according to the ordinal ranks among its m components. This procedure yields a sequence of symbols synonymously called order patterns or ordinal patterns, whereas the parameter m is either referred to as the embedding dimension, or the order of the ordinal pattern. Permutation entropy (PeEn) is in turn defined as the Shannon entropy [2,3] of a marginal probability distribution of such ordinal patterns.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have