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Sequential monitoring for distributional changepoint using degenerate U-statistics

Cooper Boniece, Lajos Horvath, Lorenzo Trapani

TL;DR

The paper develops online changepoint detection for distributional changes in a sequence $\{\mathbf{X}_i\}$ using degenerate U-statistics, introducing weighted CUSUM and Page-CUSUM detectors and a novel repurposing scheme that incorporates past monitoring data into the training sample. It delivers a thorough asymptotic theory under $H_0$ and $H_A$, requiring only square-summability $\sum_{\ell} \lambda_{\ell}^2<\infty$ of kernel eigenvalues, and provides Monte Carlo methods to approximate critical values and to quantify detection delays for early and late changes. The work demonstrates strong empirical performance, particularly for multivariate data, and offers practical kernel choices (including energy distance and Grothendieck divergences) along with training-sample stability testing and moving-window extensions. Overall, it broadens online changepoint detection by relaxing kernel-spectral assumptions, enabling robust, distribution-wide monitoring with interpretable delay metrics and adaptable horizon strategies.

Abstract

We investigate the online detection of changepoints in the distribution of a sequence of observations using degenerate U-statistic-type processes. We study weighted versions of: an ordinary, CUSUM-type scheme, a Page-CUSUM-type scheme, and an entirely novel approach based on recycling past observations into the training sample. With an emphasis on completeness, we consider open-ended and closed-ended schemes, in the latter case considering both short- and long-running monitoring schemes. We study the asymptotics under the null in all cases, also proposing a consistent, Monte-Carlo based approximation of critical values; and we derive the limiting distribution of the detection delays under early and late occurring changes under the alternative, thus enabling to quantify the expected delay associated with each procedure. As a crucial technical contribution, we derive all our asymptotics under the assumption that the kernels associated with our U-statistics are square summable, instead of requiring the typical absolute summability, which makes our assumption naturally easier to check. Our simulations show that our procedures work well in all cases considered, having excellent power versus several types of distributional changes, and appearing to be particularly suited to the analysis of multivariate data.

Sequential monitoring for distributional changepoint using degenerate U-statistics

TL;DR

The paper develops online changepoint detection for distributional changes in a sequence using degenerate U-statistics, introducing weighted CUSUM and Page-CUSUM detectors and a novel repurposing scheme that incorporates past monitoring data into the training sample. It delivers a thorough asymptotic theory under and , requiring only square-summability of kernel eigenvalues, and provides Monte Carlo methods to approximate critical values and to quantify detection delays for early and late changes. The work demonstrates strong empirical performance, particularly for multivariate data, and offers practical kernel choices (including energy distance and Grothendieck divergences) along with training-sample stability testing and moving-window extensions. Overall, it broadens online changepoint detection by relaxing kernel-spectral assumptions, enabling robust, distribution-wide monitoring with interpretable delay metrics and adaptable horizon strategies.

Abstract

We investigate the online detection of changepoints in the distribution of a sequence of observations using degenerate U-statistic-type processes. We study weighted versions of: an ordinary, CUSUM-type scheme, a Page-CUSUM-type scheme, and an entirely novel approach based on recycling past observations into the training sample. With an emphasis on completeness, we consider open-ended and closed-ended schemes, in the latter case considering both short- and long-running monitoring schemes. We study the asymptotics under the null in all cases, also proposing a consistent, Monte-Carlo based approximation of critical values; and we derive the limiting distribution of the detection delays under early and late occurring changes under the alternative, thus enabling to quantify the expected delay associated with each procedure. As a crucial technical contribution, we derive all our asymptotics under the assumption that the kernels associated with our U-statistics are square summable, instead of requiring the typical absolute summability, which makes our assumption naturally easier to check. Our simulations show that our procedures work well in all cases considered, having excellent power versus several types of distributional changes, and appearing to be particularly suited to the analysis of multivariate data.
Paper Structure (21 sections, 22 theorems, 357 equations, 5 figures, 13 tables)

This paper contains 21 sections, 22 theorems, 357 equations, 5 figures, 13 tables.

Table of Contents

  1. Introduction
  2. Assumptions and monitoring schemes
  3. Main results
  4. Monitoring under $H_{0}$
  5. Monitoring under the alternative
  6. Complements and extensions
  7. A “ repurposing” approach
  8. Testing for the stability of the training sample
  9. Examples of kernel functions
  10. On implementation
  11. Simulations and applications
  12. Simulation study
  13. Empirical illustration
  14. Discussion and conclusions
  15. Further Monte Carlo evidence
  16. ...and 6 more sections

Key Result

Theorem 3.1

Assume $H_{0}$ holds, and consider the detector $\mathcal{D}_{m}(k)=\mathcal{D}_{m}^{(1)}(k)$. Let $g_{m}$ be as in e:boundaryfxn. As $m\rightarrow \infty$, Suppose $M=M_{m}\rightarrow \infty$ such that $M/m\rightarrow a_{0}\in (0,\infty ]$, and let $u_{0}=a_{0}/(1+a_{0})$. Then Suppose $M=M_{m}\rightarrow \infty$ such that $M/m\rightarrow 0$, and let the boundary function $g_{m}$ be given by $g

Figures (5)

  • Figure A.1: Detection delays - density under strong breaks
  • Figure A.2: Detection delays - density under weak breaks
  • Figure B.1: Logs of beats per minute - training sample, from $01:37:59$ till $02:29:43$
  • Figure B.2: Logs of beats per minute - training and monitoring sample, from $01:37:59$ till $02:51:19$
  • Figure B.3: Logs of beats per minute - training and monitoring sample, from $01:37:59$ to $03:55:59$

Theorems & Definitions (49)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 4.1
  • Theorem 4.2
  • Example 4.1
  • Example 4.2
  • Example 4.3
  • ...and 39 more