Change Acceleration and Detection

Abstract

A novel sequential change detection problem is proposed, in which the goal is to not only detect but also accelerate the change. Specifically, it is assumed that the sequentially collected observations are responses to treatments selected in real time. The assigned treatments determine the pre-change and post-change distributions of the responses and also influence when the change happens. The goal is to find a treatment assignment rule and a stopping rule that minimize the expected total number of observations subject to a user-specified bound on the false alarm probability. The optimal solution is obtained under a general Markovian change-point model. Moreover, an alternative procedure is proposed, whose applicability is not restricted to Markovian change-point models and whose design requires minimal computation. For a large class of change-point models, the proposed procedure is shown to achieve the optimal performance in an asymptotic sense. Finally, its performance is found in simulation studies to be comparable to the optimal, uniformly with respect to the error probability.

Figures (7)

• Figure 1: An illustration of the proposed setup. The solid arrow is for the treatment assignment model, while the dotted arrows indicate the response model and the change-point model.
• Figure 2: An illustration of the proposed procedure. "LR" is short for the Likelihood Ratio statistic of the pre-change hypothesis against the post-change hypothesis.
• Figure 3: A simulation run of the proposed procedure. The treatments are represented by different shapes, $\Circle, \Delta,\times, \square$. The acceleration block is $\Xi_1 = [\Circle, \Delta, \times]$ and the detection block is $\Xi_2 = [\times, \square]$. The solid line is the logarithm of the posterior odds process and the dashed line is the minus of the logarithm of the likelihood ratio statistic in \ref{['def:detection_testing_rules']}. In an acceleration stage, we use block $\Xi_{1}$, wait until the posterior odds crosses $b_1$, and then switch to a detection stage. In a detection stage, we use block $\Xi_{2}$ and run both the detection rule with parameter $b_2$ and the testing rule with parameter $d$. If the testing rule stops earlier, as in the first detection stage of this figure, we switch back to an acceleration stage; otherwise, we terminate the process, as in the second detection stage above.
• Figure 4: The left two panels show solutions to Equation \ref{['geometric_accleration_opt']} in Appendix \ref{['app:proof_acc_block_fmm']}, while the right two to Equation \ref{['geometric_det_opt']} in Appendix \ref{['app:proof_det_block_fmm']}. Due to space constraints, only a subset of states are shown for the case $K=6,\kappa=3$.
• Figure 5: For both cases, we vary the thresholds of each procedure and plot $|\log_{10}(\text{Err})|$ vs ESS.

Theorems & Definitions (69)

• Remark 2.1
• Remark 2.2
• Lemma 2.1
• Lemma 2.2
• Theorem \oldthetheorem
• Remark 5.1
• Theorem \oldthetheorem
• Remark 5.2
• Corollary 6.1
• Lemma 6.1