Quickest Change Detection with Cost-Constrained Experiment Design

TL;DR

This work extends Quickest Change Detection (QCD) to settings with multiple, costed experiments, introducing performance constraints via $ARLFA$ and $POR_i$ while aiming to minimize the worst-case detection delay $WADD$. It develops the 2E-CUSUM and its truncated variant as foundational building blocks, then generalizes to 3E-CUSUM, and further to the $m$E-CUSUM framework for arbitrary numbers of experiments, with rigorous asymptotic optimality guarantees. The authors also incorporate data-efficiency by allowing a no-experiment option and derive corresponding algorithms (DE2E-CUSUM, DE3E-CUSUM, etc.) that retain constant-gap optimality relative to the high-quality CUSUM benchmark. Across both classic and data-efficient regimes, the methods leverage undershoot scaling and test truncation, and are supported by renewal-reward analyses to guarantee ARLFA feasibility and tunable PORs, enabling practical deployment in energy- or cost-constrained sensing networks.

Abstract

In the classical quickest change detection problem, an observer performs a single experiment to monitor a stochastic process. The goal in the classical problem is to detect a change in the statistical properties of the process, with the minimum possible delay, subject to a constraint on the rate of false alarms. This paper considers the case where, at each observation time, the decision-maker must choose between multiple experiments with varying information qualities and costs. The change can be detected using any of the experiments. The goal here is to detect the change with the minimum delay, subject to constraints on the rate of false alarms and the fraction of time each experiment is performed before the time of change. The constraint on the fraction of time can be used to control the overall cost of using the system of experiments. An algorithm called the two-experiment cumulative sum (2E-CUSUM) algorithm is first proposed to solve the problem when there are only two experiments. The algorithm for the case of multiple experiments, starting with three experiments, is then designed iteratively using the 2E-CUSUM algorithm. Two key ideas used in the design are the scaling of undershoots and the truncation of tests. The multiple-experiment algorithm can be designed to satisfy the constraints and can achieve the delay performance of the experiment with the highest quality within a constant. The important concept of data efficiency, where the observer has the choice of not performing any experiment, is explored as well.

Figures (10)

• Figure 1: An evolution of the statistic $D_n$ of the 2E-CUSUM algorithm for $f_0^X = f_0^Y = \mathcal{N}(0, 1)$, $f_1^X = \mathcal{N}(0.75, 1)$, $f_1^Y = \mathcal{N}(1, 1)$, $\nu = 50$, and $\gamma = 1000$, with $a_Y = 1.0$ and $N_X = 2.0$. The decision-maker performs experiment $Y$ whenever $D_n \geq 0$, and experiment $X$ when $D_n < 0$.
• Figure 2: $\textup{POR}_Y$ of the 2E-CUSUM algorithm as a function of the scaling factor $a_Y$ for $f_0^X = f_0^Y = \mathcal{N}(0, 1)$, $f_1^X = \mathcal{N}(0.75, 1)$, and $f_1^Y = \mathcal{N}(1, 1)$ with truncation $N_X \in \{ 0.1, 1, 10, 100 \}$. $\textup{POR}_Y$ exponentially decreases as a function of $a_Y$.
• Figure 3: The 2E-CUSUM Algorithm. Experiment $Y$ is performed first. When the statistic $D_n$ falls below 0, the undershoot is scaled by a factor of $a_Y$, and experiment $X$ is performed at most $N_X$ times. When the statistic crosses 0 from below, experiment $Y$ is performed again. Using pre-change distributions of $\{ X_n \}$ and $\{ Y_n \}$, $\sigma^Y$ is the length of time experiment $Y$ is performed while $\tau_{\textup{ C}}^X (\vert a_Y D_{\sigma^Y} \vert, N_X)$ is the length of time experiment $X$ is performed before change occurs at time $\nu$. The algorithm stops and declares change at time $\tau_{\textup{ 2E}}$ when the statistic crosses threshold $A$.
• Figure 4: WADD vs $\log (\textup{ARLFA})$ graphs of the 2E-CUSUM algorithm for $f_0^X = f_0^Y = \mathcal{N}(0, 1)$, $f_1^X = \mathcal{N}(0.75, 1)$, and $f_1^Y = \mathcal{N}(1, 1)$ and different $\textup{POR}_Y$ levels. The 2E-CUSUM algorithm performs better at a higher $\textup{POR}_Y$ level.
• Figure 5: WADD vs $\log (\textup{ARLFA})$ graphs of the 2E-CUSUM, CUSUM, and RSS algorithms for $f_0^X = f_0^Y = \mathcal{N}(0, 1)$, $f_1^X = \mathcal{N}(0.75, 1)$, and $f_1^Y = \mathcal{N}(1, 1)$. For the 2E-CUSUM algorithm, we set the parameters at $a_Y = 1.0$ and $N_X = 2.0$, corresponding to $\textup{POR}_Y = \textup{POR}_X = 0.50$. For the RSS algorithm, we set the probability of performing experiment $Y$ at $p_Y = 0.50$. The CUSUM algorithm has the lowest detection delay. The 2E-CUSUM algorithm outperforms the RSS algorithm.

Theorems & Definitions (14)

• Remark 1
• Remark 2
• Remark 3
• Theorem 1
• proof
• Corollary 1
• proof
• Theorem 2
• proof
• Corollary 2