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Open AccessArticle

Teaching Ordinal Patterns to a Computer: Efficient Encoding Algorithms Based on the Lehmer Code

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Department of Anaesthesiology and Intensive Care, Klinikum rechts der Isar der Technischen Universität München, 81675 Munich, Germany
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Institute of Geomatics Engineering, University of Applied Sciences and Arts Northwestern Switzerland, 4132 Muttenz, Switzerland
*
Author to whom correspondence should be addressed.
Entropy 2019, 21(10), 1023; https://doi.org/10.3390/e21101023
Received: 25 July 2019 / Revised: 17 October 2019 / Accepted: 18 October 2019 / Published: 21 October 2019
(This article belongs to the Section Signal and Data Analysis)
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. View Full-Text
Keywords: Lehmer code; ordinal patterns; symbolic dynamics; permutation entropy; symbolic transfer entropy Lehmer code; ordinal patterns; symbolic dynamics; permutation entropy; symbolic transfer entropy
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Berger, S.; Kravtsiv, A.; Schneider, G.; Jordan, D. Teaching Ordinal Patterns to a Computer: Efficient Encoding Algorithms Based on the Lehmer Code. Entropy 2019, 21, 1023.

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