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

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.

Improvement of Statistical Performance of Ordinal Multiscale Entropy Techniques Using Refined Composite Downsampling Permutation Entropy

“…Since ( 24 ) and ( 34 ) are functionally the same equation, the only difference comes from the artifact cross-correlation effect. The non-independent patterns consideration [ 21 ], on the other hand, can explain the difference between the measured rcDPE and the theoretical horizontal PE line shown in Figure 4 b. Finally, Figure 4 c shows the PE bias for each method.…”

“…This, however, do not completely follow the predicted values (shown by the black dotted line). One possible explanation comes from the fact that the pattern counts are not strictly independent between each other [ 21 ]. Nonetheless, this still shows the original MPE, as well as the composite methods, having a time scale dependency, even for uncorrelated white noise, where no such trend is expected.…”

Symmetries in reversible programming: from symmetric rig groupoids to reversible programming languages

“…They can be written as a decimal number in the factorial number system, or as a tuple encoding the digits [Knuth 1997;Laisant 1888]. This gives a convenient way of representing permutations on a computer, partly because they are bitwise-optimal [Berger et al 2019].…”

The Impact of Linear Filter Preprocessing in the Interpretation of Permutation Entropy

“…This measure can effectively assess the amount of “surprise” (new information) contained in any given instance of the result of a random variable with a known distribution function. Originally formulated in the context of telecommunications, researchers have extended the applications of this technique in a wide array of research fields, such as econometry [ 2 ], computer theory [ 3 ], set theory [ 4 ], and medicine [ 5 ].…”