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Score-based change point detection via tracking the best of infinitely many experts

Anna Markovich, Nikita Puchkin

TL;DR

The paper tackles online, nonparametric change point detection by framing it as sequential prediction with an infinite class of score-based experts. It introduces a score-based test statistic $\widehat{S}_t = \widehat{L}_{1:t}^{EW} - \widehat{L}_{1:t}^{FS}$ derived from competing exponentially weighted average and fixed-share forecasters over an uncountable expert class, exploiting quadratic losses and Fisher-divergence guided score estimates. Theoretical contributions provide non-asymptotic high-probability bounds for the test statistic in both pre-change and post-change regimes, while practical validation on synthetic and real data demonstrates robust detection performance with competitive detection delays and low false-alarm rates. The approach offers a scalable, nonparametric framework for rapid online change detection applicable to streaming data tasks such as activity recognition and sensor monitoring, supported by explicit closed-form updates and efficient recursive recurrences.

Abstract

We propose an algorithm for nonparametric online change point detection based on sequential score function estimation and the tracking the best expert approach. The core of the procedure is a version of the fixed share forecaster tailored to the case of infinite number of experts and quadratic loss functions. The algorithm shows promising results in numerical experiments on artificial and real-world data sets. Its performance is supported by rigorous high-probability bounds describing behaviour of the test statistic in the pre-change and post-change regimes.

Score-based change point detection via tracking the best of infinitely many experts

TL;DR

The paper tackles online, nonparametric change point detection by framing it as sequential prediction with an infinite class of score-based experts. It introduces a score-based test statistic derived from competing exponentially weighted average and fixed-share forecasters over an uncountable expert class, exploiting quadratic losses and Fisher-divergence guided score estimates. Theoretical contributions provide non-asymptotic high-probability bounds for the test statistic in both pre-change and post-change regimes, while practical validation on synthetic and real data demonstrates robust detection performance with competitive detection delays and low false-alarm rates. The approach offers a scalable, nonparametric framework for rapid online change detection applicable to streaming data tasks such as activity recognition and sensor monitoring, supported by explicit closed-form updates and efficient recursive recurrences.

Abstract

We propose an algorithm for nonparametric online change point detection based on sequential score function estimation and the tracking the best expert approach. The core of the procedure is a version of the fixed share forecaster tailored to the case of infinite number of experts and quadratic loss functions. The algorithm shows promising results in numerical experiments on artificial and real-world data sets. Its performance is supported by rigorous high-probability bounds describing behaviour of the test statistic in the pre-change and post-change regimes.
Paper Structure (37 sections, 26 theorems, 284 equations, 4 figures, 6 tables)

This paper contains 37 sections, 26 theorems, 284 equations, 4 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Online Learning with Quadratic Loss
  3. Exponentially Weighted Average Forecaster
  4. Fixed Share Forecaster
  5. Fixed Share Forecaster for the Infinite Class of Experts
  6. Application to Change Point Detection
  7. Numerical Experiments
  8. Activity Detection
  9. Numerical Experiments
  10. Computational details
  11. Synthetic Data Sets
  12. Speech Detection
  13. Room Occupancy Detection
  14. Proofs of the results from Section \ref{['sec:online_learning']}
  15. Proof of Lemma \ref{['lem:ew']}
  16. ...and 22 more sections

Key Result

Lemma 2.1

Assume that the loss function $\ell_t$ is of the form eq:loss for any $t \in \{1, \dots, T\}$. For any $\eta > 0$, let $\widehat{{\boldsymbol \theta}}_{s:t}(\eta)$ be the exponentially weighted average eq:z_st with the Gaussian prior Then it holds that

Figures (4)

  • Figure 1: Activity detection data set (WISDM) after preprocessing. The dotted lines correspond to the annotated change points. Validation and test parts of the series are depicted in darker and lighter colors respectively.
  • Figure 2: Example 4: variance shift detection in a multivariate Gaussian sequence. The top plot shows the generated time series with a single change point $\tau^*$ highlighted with a black dotted line. The coordinates of a multivariate sequence are defined by different colors. The bottom plot demonstrates the behavior of the test statistic for the competing methods: Algorithm \ref{['alg:score-based']} (red), FLH (blue), FALCON (green), KLIEP (magenta), the kernel change point with M-statistic (cyan). Detected change points $\tau$ are marked with bold dots. Values of the test statistics were scaled for a better visualization.
  • Figure 3: Data points of the CENSREC-1-C audio recording with SNR = 20. Dotted pink lines mark the annotated change points.
  • Figure 4: Room occupancy data set after preprocessing. The dotted lines correspond to the annotated change points. Validation and test parts of the series are depicted in darker and lighter colors respectively.

Theorems & Definitions (37)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 3.3
  • Theorem 3.4
  • Proposition B.1
  • Theorem C.1
  • proof
  • Lemma C.2
  • proof
  • ...and 27 more