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

Annika Betken, Giorgio Micali, Manuel Ruiz Marín

TL;DR

This work introduces a unified, data-driven framework for testing temporal symmetry in time series by analyzing the distribution of ordinal patterns. The authors construct a symmetry test based on partitions of the pattern space $\mathcal{S}_d$, with the estimator $\hat{D}_2(\mathcal{G})$ expressed as a $U$-statistic that is degenerate under $\mathcal{H}_0$, leading to a generalized chi-square limit. They establish asymptotic results under $\mathcal{H}_0$ and $\mathcal{H}_1$, derive a consistent variance estimation via a kernel-based long-run covariance approach, and provide a plug-in method for critical values using estimated group probabilities. Simulations demonstrate good size control and power across Gaussian and non-Gaussian settings, while real-data applications (S&P 500 returns and RR intervals) reveal substantial temporal irreversibility consistent with prior findings. Overall, the framework offers a robust, computationally efficient, model-free tool for diagnosing symmetry properties and detecting nonlinear or non-Gaussian structure in time series.

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.

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

TL;DR

This work introduces a unified, data-driven framework for testing temporal symmetry in time series by analyzing the distribution of ordinal patterns. The authors construct a symmetry test based on partitions of the pattern space , with the estimator expressed as a -statistic that is degenerate under , leading to a generalized chi-square limit. They establish asymptotic results under and , derive a consistent variance estimation via a kernel-based long-run covariance approach, and provide a plug-in method for critical values using estimated group probabilities. Simulations demonstrate good size control and power across Gaussian and non-Gaussian settings, while real-data applications (S&P 500 returns and RR intervals) reveal substantial temporal irreversibility consistent with prior findings. Overall, the framework offers a robust, computationally efficient, model-free tool for diagnosing symmetry properties and detecting nonlinear or non-Gaussian structure in time series.

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.
Paper Structure (10 sections, 13 theorems, 165 equations, 7 figures, 5 tables)

This paper contains 10 sections, 13 theorems, 165 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Main Results
  3. Test Statistic
  4. Limiting Distributions
  5. Variance Estimation
  6. Simulations
  7. Real Dataset
  8. Outlook and Discussion
  9. Auxiliary Results
  10. Proofs

Key Result

Theorem 1

Let $t = d! - m$, and let $(\lambda_1, g^{(1)}), \ldots, (\lambda_t, g^{(t)})$ denote the eigenvalue–-eigenfunction pairs of the Hilbert--Schmidt operator eq:operator associated with the kernel $h$. Assume that $F_i \ll F \times F$ for all $i\ge 1$, and that there exists $\delta>0$ such that $\mathb where $W_1, \ldots, W_t$ are centered Gaussian random variables with covariances and where

Figures (7)

  • Figure 1: Simulated distribution of $S_t(\mathbf{p})+c$ under the null hypothesis obtained from $N=2000$ Gaussian draws with covariance estimated from $\hat{\Omega}_n$, for $n=1000$. The red line shows the kernel density estimate. Left: mean 0.3510, variance 0.2321. Right: mean 0.3604, variance 0.2356
  • Figure 2: On the left: simulated values of $n \hat{D}_2(\mathcal{G})$ together with the density of $S_t(\mathbf{p})$ under $\mathcal{G}$. On the right: simulated values of $\sqrt{n}\,\hat{D}_2(\mathcal{G})$ and the corresponding kernel density estimate, which exhibits approximately normal behavior. Each histogram is based on $N=2000$ independent replications, with time series length $n=1000$.
  • Figure 3: Left: Example of a time series obtained as a subordinated Gaussian process $X_t = g(Y_t)$ with Pareto marginal distribution. Right: Density of the asymptotic null distribution together with the empirical distribution of $n \hat{D}_2(\mathcal{G})$.
  • Figure 4: Empirical ordinal patterns distributions for an MA(1) process with different innovation distributions. Each panel corresponds to a different choice of innovations. The similarity across some panels reflects the fact that the considered innovation distributions are themselves quite similar in shape, leading to nearly indistinguishable ordinal patterns distributions.
  • Figure 5: Daily closing values of the S&P 500 index from January 1990 to August 2011. Logarithmic returns computed as $\log(X_{t+1})-\log(X_t)$ over the same sample period. The plot highlights short-term volatility fluctuations around a stable mean.
  • ...and 2 more figures

Theorems & Definitions (34)

  • Definition 1: U-statistics
  • Definition 2
  • Theorem 1: under $\mathcal{H}_0$
  • Remark 1
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Theorem 2: Under $\mathcal{H}_1$
  • Corollary 1
  • ...and 24 more