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Discontinuity, Nonlinearity, and Complexity 15(1) (2026) 15--22 | DOI:10.5890/DNC.2026.03.002

Kazimieras Pukenas

Department of Health Promotion and Rehabilitation, Lithuanian Sports University, Sporto 6, LT-44221, Kaunas, Lithuania

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Abstract

This paper proposes a modified multivariate permutation entropy (MPE) algorithm to quantify the complexity of the multi-dimensional time series. The novelty of the algorithm lies in a multivariate ordinal pattern representation which requires a significantly smaller number of combinations of obtained multivariate ${d}$-order permutations for ${m}$-dimensional time series, comparing the direct application of a univariate permutation entropy to the multi-dimensional time series and resulting in the value of $d!{}^{m}$ combinations. More specifically, the number of combinations of multivariate ordinal patterns is reduced to $d!{}^{k}$ with ${k}<<{m}$ by obtaining the patterns from \textit{k}-dimensional time series, randomly selected with multiple repetitions from the ${m}$-dimensional time series. The final distribution of the multivariate ordinal patterns is obtained by the summation of these patterns across all repetitions. The capabilities of the proposed method are verified based on numerical experiments and empirical multidimensional data sets. The method is specifically applied to i) a multivariate deterministic system (a network of ten chaotic R\"{o}ssler oscillators diffusively coupled to their nearest neighbors); ii) a multivariate noise field that is colored in space and white in time; and iii) multi-channel electroencephalogram (EEG) recordings.

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