[Paper Review] Quickest Change Detection under Transient Dynamics: Theory and Asymptotic Analysis

Shaofeng Zou, Georgios Fellouris+1 more|arXiv (Cornell University)|Nov 6, 2017

Advanced Statistical Process Monitoring23 references3 citations

TL;DR

This paper proposes adaptive quickest change detection algorithms for systems undergoing transient post-change dynamics, where the shift from pre-change to persistent distribution occurs over unknown-duration phases. It introduces a weighted dynamic CuSum algorithm and proves asymptotic optimality under both Lorden’s and Pollak’s criteria as the average run length to false alarm and transient durations grow, with performance robust to unknown transient phase lengths.

ABSTRACT

The problem of quickest change detection (QCD) under transient dynamics is studied, where the change from the initial distribution to the final persistent distribution does not happen instantaneously, but after a series of transient phases. The observations within the different phases are generated by different distributions. The objective is to detect the change as quickly as possible, while controlling the average run length (ARL) to false alarm, when the durations of the transient phases are completely unknown. Two algorithms are considered, the dynamic Cumulative Sum (CuSum) algorithm, proposed in earlier work, and a newly constructed weighted dynamic CuSum algorithm. Both algorithms admit recursions that facilitate their practical implementation, and they are adaptive to the unknown transient durations. Specifically, their asymptotic optimality is established with respect to both Lorden's and Pollak's criteria as the ARL to false alarm and the durations of the transient phases go to infinity at any relative rate. Numerical results are provided to demonstrate the adaptivity of the proposed algorithms, and to validate the theoretical results.

Motivation & Objective

• Address the challenge of detecting changes in systems that transition through unknown-duration transient phases before reaching a persistent post-change state.
• Develop detection algorithms that are robust to unknown transient phase durations while maintaining low detection delay.
• Establish theoretical performance guarantees under both Lorden’s and Pollak’s criteria for quickest change detection.
• Ensure practical implementability through recursive algorithms that do not require prior knowledge of transient durations.

Proposed method

• Propose a dynamic CuSum (D-CuSum) algorithm adapted from prior work, which recursively computes test statistics based on likelihood ratios under composite hypotheses.
• Introduce a novel weighted dynamic CuSum algorithm that enhances adaptivity by assigning dynamic weights to observations based on phase transitions.
• Formulate the detection problem as a sequential hypothesis test where the null hypothesis assumes no change has occurred, and the alternative includes all possible change-points and transient durations.
• Derive recursive structures for both algorithms to enable real-time implementation without requiring knowledge of transient phase lengths.
• Use generalized likelihood ratio statistics as test statistics, with stopping rules based on thresholds that control the average run length to false alarm.
• Conduct asymptotic analysis under both Lorden’s and Pollak’s criteria, proving optimality as the ARL to false alarm and transient durations tend to infinity at any relative rate.

Experimental results

Research questions

• RQ1 How can quickest change detection be effectively performed when the system transitions through unknown-duration transient phases before reaching a persistent post-change state?
• RQ2 Can detection algorithms be designed to be adaptive to unknown transient phase durations while maintaining theoretical performance guarantees?
• RQ3 What is the asymptotic behavior of detection delay under Lorden’s and Pollak’s criteria when both the average run length to false alarm and transient durations grow?
• RQ4 Can recursive formulations be derived for the detection statistics to ensure practical implementability in real-time systems?
• RQ5 How does the performance of the proposed algorithms compare to classical methods in the presence of transient dynamics?

Key findings

• The proposed weighted dynamic CuSum algorithm achieves asymptotic optimality under both Lorden’s and Pollak’s criteria as the average run length to false alarm and transient phase durations tend to infinity at any relative rate.
• The dynamic CuSum algorithm is shown to be asymptotically optimal under Lorden’s criterion, with detection delay scaling as $ \frac{b}{I_1}(1+o(1)) $ when the transient phase is short.
• For longer transient phases, the detection delay scales as $ b\left(\frac{c_1'}{I_1} + \frac{1-c_1'}{I_2}\right)(1+o(1)) $, where $ c_1' $ represents the relative duration of the first transient phase.
• Theoretical analysis confirms that both algorithms maintain low false alarm rates and adapt to unknown transient durations through recursive computation.
• Numerical results validate the theoretical findings, demonstrating the robustness and adaptivity of the algorithms across various transient phase configurations.
• The recursive structure of the algorithms ensures practical feasibility, enabling real-time implementation without requiring prior knowledge of transient phase lengths.