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Quickest Change Detection Using Mismatched CUSUM

Austin Cooper, Sean Meyn

Abstract

The field of quickest change detection (QCD) concerns design and analysis of algorithms to estimate in real time the time at which an important event takes place and identify properties of the post-change behavior. The goal is to devise a stopping time adapted to the observations that minimizes an $L_1$ loss. Approximately optimal solutions are well known under a variety of assumptions. In the work surveyed here we consider the CUSUM statistic, which is defined as a one-dimensional reflected random walk driven by a functional of the observations. It is known that the optimal functional is a log likelihood ratio subject to special statical assumptions. The paper concerns model free approaches to detection design, considering the following questions: 1. What is the performance for a given functional of the observations? 2. How do the conclusions change when there is dependency between pre- and post-change behavior? 3. How can techniques from statistics and machine learning be adapted to approximate the best functional in a given class?

Quickest Change Detection Using Mismatched CUSUM

Abstract

The field of quickest change detection (QCD) concerns design and analysis of algorithms to estimate in real time the time at which an important event takes place and identify properties of the post-change behavior. The goal is to devise a stopping time adapted to the observations that minimizes an loss. Approximately optimal solutions are well known under a variety of assumptions. In the work surveyed here we consider the CUSUM statistic, which is defined as a one-dimensional reflected random walk driven by a functional of the observations. It is known that the optimal functional is a log likelihood ratio subject to special statical assumptions. The paper concerns model free approaches to detection design, considering the following questions: 1. What is the performance for a given functional of the observations? 2. How do the conclusions change when there is dependency between pre- and post-change behavior? 3. How can techniques from statistics and machine learning be adapted to approximate the best functional in a given class?
Paper Structure (22 sections, 23 theorems, 105 equations, 1 figure)

This paper contains 22 sections, 23 theorems, 105 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Conditionally Independent Model
  3. Examples
  4. Assumptions on CUSUM statistic
  5. Performance approximations
  6. Numerical Experiments
  7. Optimizing CUSUM
  8. In search of stationary points
  9. Linear family
  10. Reconsidering Independence
  11. POMDP model
  12. Meta-stability and approximations
  13. Conclusions
  14. Appendix
  15. Large deviations background
  16. ...and 7 more sections

Key Result

Lemma 2.1

[lemma]t:RelEntRate ${\cal K}(\widecheck{\upmu}^0 \| \upmu^0) = \widecheck{\uppi}^0(F) - \Lambda_0(F)$. $\blacksquare$=0

Figures (1)

  • Figure 1: Two paths: $\{x^{\text{\sf\tiny(H)}}_t : t\ge 0 \}$ approximating the path shown on the left is far more likely than the one shown on the right.

Theorems & Definitions (23)

  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Proposition 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Proposition 3.6
  • Proposition 4.1
  • ...and 13 more