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Change Detection of Markov Kernels with Unknown Pre and Post Change Kernel

Hao Chen, Jiacheng Tang, Abhishek Gupta

TL;DR

A new change detection algorithm for detecting a change in the Markov kernel over a metric space in which the post-change kernel is unknown is developed and an upper bound and lower bound are derived on the mean delay and mean time between false alarms.

Abstract

In this paper, we develop a new change detection algorithm for detecting a change in the Markov kernel over a metric space in which the post-change kernel is unknown. Under the assumption that the pre- and post-change Markov kernel is uniformly ergodic, we derive an upper bound on the mean delay and a lower bound on the mean time between false alarms. A numerical simulation is provided to demonstrate the effectiveness of our method.

Change Detection of Markov Kernels with Unknown Pre and Post Change Kernel

TL;DR

A new change detection algorithm for detecting a change in the Markov kernel over a metric space in which the post-change kernel is unknown is developed and an upper bound and lower bound are derived on the mean delay and mean time between false alarms.

Abstract

In this paper, we develop a new change detection algorithm for detecting a change in the Markov kernel over a metric space in which the post-change kernel is unknown. Under the assumption that the pre- and post-change Markov kernel is uniformly ergodic, we derive an upper bound on the mean delay and a lower bound on the mean time between false alarms. A numerical simulation is provided to demonstrate the effectiveness of our method.
Paper Structure (14 sections, 32 equations, 2 figures)

This paper contains 14 sections, 32 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Formulation
  4. RKHS Embedding of Markov Kernels
  5. MMD Estimation for Dependent Observations
  6. Motivating Examples
  7. Main Results
  8. Numerical Simulations
  9. Conclusion
  10. Appendix
  11. Proof of Lemma \ref{['lem: RKHS for MC']}
  12. Proof of Lemma \ref{['lem: UE for extended space']}
  13. Proof of Theorem \ref{['thm: MTBFA']}
  14. Proof of Theorem \ref{['thm: MD']}

Figures (2)

  • Figure 1: A change in the variance of $\omega_n$'s distribution is applied at the red breaking line ($\tau=1000$). The left plot shows the 2-norm of the system state. The middle plot shows the update term $s(B_t^r)$ at each step. The right plot shows the kernel CUSUM statistic $\hat{S}_n$.
  • Figure 2: A change in the mean of $\omega_n$'s distribution is applied at the red breaking line ($\tau=1000$).

Theorems & Definitions (3)

  • Definition 1
  • Example 1
  • Example 2