In the problem of quickest change detection, a change occurs at some unknown time in the distribution of a sequence of random vectors that are monitored in real time, and the goal is to detect this change as quickly as possible subject to a certain false alarm constraint. In this work we consider this problem in the presence of parametric uncertainty in the post-change regime and controlled sensing. That is, the post-change distribution contains an unknown parameter, and the distribution of each observation, before and after the change, is affected by a control action. In this context, in addition to a stopping rule that determines the time at which it is declared that the change has occurred, one also needs to determine a sequential control policy, which chooses the control action at each time based on the already collected observations. We formulate this problem mathematically using Lorden's minimax criterion, and assuming that there are finitely many possible actions and post-change parameter values. We then propose a specific procedure for this problem that employs an adaptive CuSum statistic in which (i) the estimate of the parameter is based on a fixed number of the more recent observations, and (ii) each action is selected to maximize the Kullback-Leibler divergence of the next observation based on the current parameter estimate, apart from a small number of exploration times. We show that this procedure, which we call the Windowed Chernoff-CuSum (WCC), is first-order asymptotically optimal under Lorden's minimax criterion, for every possible possible value of the unknown post-change parameter, as the mean time to false alarm goes to infinity. We also provide simulation results to illustrate the performance of the WCC procedure.

翻译：在快速变化检测问题中，某个未知时刻，实时监测的随机向量序列的分布发生改变，目标是在满足一定虚警约束条件下尽可能快地检测到这种变化。本文考虑在变化后存在参数不确定性以及受控感知情况下的该问题。即，变化后分布包含未知参数，且变化前后每次观测的分布均受控制动作影响。在此背景下，除需确定宣告变化发生的停止规则外，还需确定序贯控制策略，该策略基于已收集的观测值在每个时刻选择控制动作。我们采用Lorden的极小化极大准则对该问题进行数学建模，并假设可用动作和变化后参数值均为有限个。针对该问题，我们提出一种具体方法，采用自适应CuSum统计量，其中：(i) 参数估计基于固定数量的最近观测值；(ii) 除少量探索时刻外，每次动作选择旨在基于当前参数估计最大化下一次观测的Kullback-Leibler散度。我们证明，该方法（称为窗口Chernoff-CuSum，WCC）在Lorden的极小化极大准则下，对于未知变化后参数的每个可能取值，当平均虚警时间趋于无穷时，具有一阶渐近最优性。我们还提供仿真结果以说明WCC方法的性能。