Transcripts and Algebraic Distances in Time Series: Stochastic Properties and Nonparametric Dependence Tests

2026-05-25

AI Summary

This study addresses the challenge of effectively characterizing and testing complex nonlinear dependencies in continuous time series. The authors propose a novel ordinal pattern (OP)-based approach that introduces, for the first time, the concept of a “transcript”—a symbolic sequence derived from the original time series—and combines Cayley and Kendall algebraic edit distances to construct a nonparametric test for dependence. Through theoretical analysis, they elucidate the distinct statistical properties of transcript sequences under different stochastic processes and derive their asymptotic distributions. Simulation experiments demonstrate that the proposed method substantially outperforms existing OP-based tests in terms of statistical power. Furthermore, applications to real-world data confirm its practical effectiveness and interpretability.

Abstract

The use of ordinal patterns (OPs) for analyzing the dependence structure of univariate and continuously distributed processes has gained popularity in recent years. This research goes one step further and considers the transcripts being computed from successive OPs in the time series. Transcripts constitute a kind of ``difference'' between successive OPs and thus naturally relate to two algebraic distances between OPs, the Cayley and Kendall edit distances. The original time series is transformed into a sequence of transcripts or distances, respectively, and important stochastic properties thereof are derived. It is shown that these properties differ substantially among different types of original processes. This motivates the development of various statistics based on transcripts and edit distances in order to investigate the dependence structure of the original process. In particular, the asymptotic distribution of these statistics under the null hypothesis of serial independence is derived, which is then used to implement nonparametric tests for serial dependence. A simulation study shows that these novel dependence tests have appealing power properties, often outperforming former OP-based dependence tests. A concluding real-world data example illustrates the application and interpretation of the proposed approaches in practice.

Problem

• serial dependence
• ordinal patterns
• algebraic distances
• time series
• nonparametric tests

Innovation

• ordinal patterns
• transcripts
• algebraic distances
• nonparametric tests
• serial dependence