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Autocorrelation Test under Frequent Mean Shifts

Ziyang Liu, Ning Hao, Yue Selena Niu, Han Xiao, Hongxu Ding

TL;DR

This work tackles autocorrelation testing in nonstationary time series with frequent mean shifts by developing the Shift-Immune Portmanteau (SIP) framework, which uses mean-robust quadratic forms built from circulant Toeplitz matrices. The SIP statistic, supported by asymptotic normality and chi-square limits, provides reliable inference under piecewise-constant means and offers two practical variants (SIP1, SIP2) with different estimator choices for the long-run variance. A Shift-Immune ACF plot is proposed to visualize dependence patterns obscured by mean shifts, and the method is demonstrated via simulations and a nanopore sequencing dataset, where it uncovers autocorrelation not detectable by classical tools. Overall, SIP delivers robust type I error control and competitive power in settings where frequent mean shifts would otherwise invalidate standard diagnostics, with direct applicability to high-throughput sequencing data and other nonstationary time series.

Abstract

Testing for the presence of autocorrelation is a fundamental problem in time series analysis. Classical methods such as the Box-Pierce test rely on the assumption of stationarity, necessitating the removal of non-stationary components such as trends or shifts in the mean prior to application. However, this is not always practical, particularly when the mean structure is complex, such as being piecewise constant with frequent shifts. In this work, we propose a new inferential framework for autocorrelation in time series data under frequent mean shifts. In particular, we introduce a Shift-Immune Portmanteau (SIP) test that reliably tests for autocorrelation and is robust against mean shifts. We illustrate an application of our method to nanopore sequencing data.

Autocorrelation Test under Frequent Mean Shifts

TL;DR

This work tackles autocorrelation testing in nonstationary time series with frequent mean shifts by developing the Shift-Immune Portmanteau (SIP) framework, which uses mean-robust quadratic forms built from circulant Toeplitz matrices. The SIP statistic, supported by asymptotic normality and chi-square limits, provides reliable inference under piecewise-constant means and offers two practical variants (SIP1, SIP2) with different estimator choices for the long-run variance. A Shift-Immune ACF plot is proposed to visualize dependence patterns obscured by mean shifts, and the method is demonstrated via simulations and a nanopore sequencing dataset, where it uncovers autocorrelation not detectable by classical tools. Overall, SIP delivers robust type I error control and competitive power in settings where frequent mean shifts would otherwise invalidate standard diagnostics, with direct applicability to high-throughput sequencing data and other nonstationary time series.

Abstract

Testing for the presence of autocorrelation is a fundamental problem in time series analysis. Classical methods such as the Box-Pierce test rely on the assumption of stationarity, necessitating the removal of non-stationary components such as trends or shifts in the mean prior to application. However, this is not always practical, particularly when the mean structure is complex, such as being piecewise constant with frequent shifts. In this work, we propose a new inferential framework for autocorrelation in time series data under frequent mean shifts. In particular, we introduce a Shift-Immune Portmanteau (SIP) test that reliably tests for autocorrelation and is robust against mean shifts. We illustrate an application of our method to nanopore sequencing data.
Paper Structure (16 sections, 12 theorems, 74 equations, 2 figures, 5 tables)

This paper contains 16 sections, 12 theorems, 74 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Separation of the serial correlation and mean shifts
  3. Model descriptions
  4. Quadratic forms
  5. Shift-Immune Portmanteau Test
  6. Construction of the Test Statistic
  7. Choice of Basis
  8. What is being tested?
  9. Asymptotic theory
  10. Test procedures
  11. Shift-Immune ACF plot
  12. Numerical Studies
  13. Simulated data examples
  14. Type I error control
  15. Power Analysis
  16. ...and 1 more sections

Key Result

Proposition 1

Let ${\boldsymbol{A}}$ be a symmetric Toeplitz matrix of the form V6. For ${\boldsymbol{X}}$ generated from model V1 where the noise is mean zero and stationary with a finite variance, we have

Figures (2)

  • Figure 1: An illustration of nanopore sequencing data from wang2024adapting. Top: 5000 data points from sequence id=33; bottom: 5000 data points from sequence id=39. Each of these sequences contains numerous mean shifts.
  • Figure 2: Left: the standard ACF plots for nanopore sequences with id$=33$ and 39; right: shift-immune ACF plots for nanopore sequences with id=33 and 39.

Theorems & Definitions (12)

  • Proposition 1
  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Theorem 2
  • Theorem 3
  • Proposition 7
  • ...and 2 more