Entropic and algebraic transcript-based tools in time series analysis

2026-05-28

AI Summary

This study addresses the effective analysis of synchronization and interdependencies between coupled time series. Building upon algebraic representations of time series—such as ordinal patterns—it introduces the concept of a transcript that integrates symmetric group structure with information-theoretic measures (entropy, statistical complexity, mutual information) and algebraic distances (Cayley and Kendall distances). The work proposes a novel similarity metric, the average Kendall distance, which demonstrates superior performance over existing transcript-based tools in generalized synchronization detection tasks. This approach exhibits enhanced discriminative power and robustness, offering a new analytical framework for coupled dynamical systems that combines algebraic rigor with sensitivity to informational content.

Abstract

Algebraic representations of time series are symbolic representations whose symbols belong to a finite group. Precisely, the framework of the present paper is the analysis of coupled time series in algebraic representations and, more generally, group-valued time series. The prototype of an algebraic representation is an ordinal representation, whose symbols are permutations, also called ordinal patterns in the context of time series analysis. In fact, permutations, endowed with function composition, build a group called a symmetric group. A simple way to harness the algebraic structure of the alphabet in such cases is the concept of transcript from one group element to another. Since transcripts involve two group elements, they are very suitable for studying couplings between time series in the same algebraic representation. In this paper, we outline several existing entropic and algebraic transcript-based tools for analyzing coupled time series and systems. In addition to entropy, the entropic tools include divergence, statistical complexity and mutual information. The algebraic tools comprise order classes and, most recently, the Cayley and Kendall distances. We use the detection of generalized synchronization in a well-studied coupled system to compare the performances of some of those tools. To this end, we also provide an alternative tool called the similarity distance between times series, which is a mean Kendall distance. We found that the novel similarity distance outperforms the other tools tested.

Problem

• coupled time series
• generalized synchronization
• algebraic representations
• ordinal patterns
• transcript-based analysis

Innovation

• transcript
• algebraic representation
• similarity distance
• Kendall distance
• coupled time series