Symmetry Testing in Time Series using Ordinal Patterns: A U-Statistic Approach

TL;DR

This paper presents a unified, data-driven method for testing various types of temporal symmetry in time series using ordinal patterns and U-statistics, with proven asymptotic properties and demonstrated effectiveness.

Contribution

It introduces a general framework for symmetry testing in time series based on ordinal patterns, extending beyond specific cases like time reversal.

Findings

High sensitivity to structural asymmetries

Asymptotic validity for broad classes of processes

Efficient computation demonstrated on synthetic and real data

Abstract

We introduce a general framework for testing temporal symmetries in time series based on the distribution of ordinal patterns. While previous approaches have focused on specific forms of asymmetry, such as time reversal, our method provides a unified framework applicable to arbitrary symmetry tests. We establish asymptotic results for the resulting test statistics under a broad class of stationary processes. Comprehensive experiments on both synthetic and real data demonstrate that the proposed test achieves high sensitivity to structural asymmetries while remaining fully data-driven and computationally efficient.