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Partially-Observable Sequential Change-Point Detection for Autocorrelated Data via Upper Confidence Region

Haijie Xu, Xiaochen Xian, Chen Zhang, Kaibo Liu

TL;DR

The paper tackles rapid detection of abrupt changes in high-dimensional, autocorrelated data when only a subset of variables can be observed at each time. It frames the problem with a linear state-space model and introduces AUCRSS, which combines a partially observable Kalman filter for state inference, a GLRT-based change-point detector, and an adaptive sampling policy grounded in upper confidence regions and CMAB theory. Key contributions include the design of AUCRSS, its adaptive $oldsymbol\alpha_n$ for exploration-exploitation balance, a tractable approximation E-AUCRSS, and theoretical results on detection-delays and sampling properties under in-control and out-of-control regimes. Extensive simulations and a milling-process case study demonstrate that AUCRSS and its approximation achieve superior detection speed and robustness to autocorrelation, especially with limited sensing budgets. The work advances practical sequential monitoring under resource constraints and links change-point detection with online learning principles for adaptive sensing.

Abstract

Sequential change point detection for multivariate autocorrelated data is a very common problem in practice. However, when the sensing resources are limited, only a subset of variables from the multivariate system can be observed at each sensing time point. This raises the problem of partially observable multi-sensor sequential change point detection. For it, we propose a detection scheme called adaptive upper confidence region with state space model (AUCRSS). It models multivariate time series via a state space model (SSM), and uses an adaptive sampling policy for efficient change point detection and localization. A partially-observable Kalman filter algorithm is developed for online inference of SSM, and accordingly, a change point detection scheme based on a generalized likelihood ratio test is developed. How its detection power relates to the adaptive sampling strategy is analyzed. Meanwhile, by treating the detection power as a reward, its connection with the online combinatorial multi-armed bandit (CMAB) problem is formulated and an adaptive upper confidence region algorithm is proposed for adaptive sampling policy design. Theoretical analysis of the asymptotic average detection delay is performed, and thorough numerical studies with synthetic data and real-world data are conducted to demonstrate the effectiveness of our method.

Partially-Observable Sequential Change-Point Detection for Autocorrelated Data via Upper Confidence Region

TL;DR

The paper tackles rapid detection of abrupt changes in high-dimensional, autocorrelated data when only a subset of variables can be observed at each time. It frames the problem with a linear state-space model and introduces AUCRSS, which combines a partially observable Kalman filter for state inference, a GLRT-based change-point detector, and an adaptive sampling policy grounded in upper confidence regions and CMAB theory. Key contributions include the design of AUCRSS, its adaptive for exploration-exploitation balance, a tractable approximation E-AUCRSS, and theoretical results on detection-delays and sampling properties under in-control and out-of-control regimes. Extensive simulations and a milling-process case study demonstrate that AUCRSS and its approximation achieve superior detection speed and robustness to autocorrelation, especially with limited sensing budgets. The work advances practical sequential monitoring under resource constraints and links change-point detection with online learning principles for adaptive sensing.

Abstract

Sequential change point detection for multivariate autocorrelated data is a very common problem in practice. However, when the sensing resources are limited, only a subset of variables from the multivariate system can be observed at each sensing time point. This raises the problem of partially observable multi-sensor sequential change point detection. For it, we propose a detection scheme called adaptive upper confidence region with state space model (AUCRSS). It models multivariate time series via a state space model (SSM), and uses an adaptive sampling policy for efficient change point detection and localization. A partially-observable Kalman filter algorithm is developed for online inference of SSM, and accordingly, a change point detection scheme based on a generalized likelihood ratio test is developed. How its detection power relates to the adaptive sampling strategy is analyzed. Meanwhile, by treating the detection power as a reward, its connection with the online combinatorial multi-armed bandit (CMAB) problem is formulated and an adaptive upper confidence region algorithm is proposed for adaptive sampling policy design. Theoretical analysis of the asymptotic average detection delay is performed, and thorough numerical studies with synthetic data and real-world data are conducted to demonstrate the effectiveness of our method.
Paper Structure (17 sections, 6 theorems, 28 equations, 8 figures, 3 algorithms)

This paper contains 17 sections, 6 theorems, 28 equations, 8 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Online monitoring for data streams
  4. Partially-observable multivariate data stream change point detection (POMDSCD)
  5. Multi-armed bandit (MAB)
  6. Problem Formulation
  7. Methodology
  8. State inference based on partially observable Kalman filter
  9. Change point detection
  10. Adaptive sampling with upper confidence region
  11. Some properties of AUCRSS
  12. Numerical Study
  13. Influence of $\alpha_{n}$
  14. AUCRSS vs E-AUCRSS
  15. Comparison with benchmarks
  16. ...and 2 more sections

Key Result

Proposition 1

If the null hypothesis is true, the GLRT statistic $\ell(n,k, \hat{\mathbf{f}}(n,k))$ follows a $\chi^{2}$ distribution with $p$ degrees of freedom.

Figures (8)

  • Figure 1: Stream data of six variables (a)-(f) over 500 time points in a milling process
  • Figure 2: Person correlation (a) and autocorrelation function (ACF) (b) of data of milling process
  • Figure 3: Comparison between segment-wise linear $\alpha_n$ and constant $\alpha_n$, for $10$-dimensional data streams with (a) $m = 1$, (b) $m = 2$ and (c) $m = 3$.
  • Figure 4: Comparison between AUCRSS and E-AUCRSS for $10$-dimensional data streams with (a) $m = 2$, (b) $m = 3$ and (c) $m = 4$
  • Figure 5: Online monitoring schematics for IC condition (a) and OC condition (b) and observe dimensions for IC condition (c) and OC condition (d) of E-AUCRSS
  • ...and 3 more figures

Theorems & Definitions (8)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Remark 1
  • Remark 2
  • Theorem 1
  • Lemma 1
  • Theorem 2