The problem of quickest change detection (QCD) in autoregressive (AR) models is investigated. A system is being monitored with sequentially observed samples. At some unknown time, a disturbance signal occurs and changes the distribution of the observations. The disturbance signal follows an AR model, which is dependent over time. Before the change, observations only consist of measurement noise, and are independent and identically distributed (i.i.d.). After the change, observations consist of the disturbance signal and the measurement noise, are dependent over time, which essentially follow a continuous-state hidden Markov model (HMM). The goal is to design a stopping time to detect the disturbance signal as quickly as possible subject to false alarm constraints. Existing approaches for general non-i.i.d. settings and discrete-state HMMs cannot be applied due to their high computational complexity and memory consumption, and they usually assume some asymptotic stability condition. In this paper, the asymptotic stability condition is firstly theoretically proved for the AR model by a novel design of forward variable and auxiliary Markov chain. A computationally efficient Ergodic CuSum algorithm that can be updated recursively is then constructed and is further shown to be asymptotically optimal. The data-driven setting where the disturbance signal parameters are unknown is further investigated, and an online and computationally efficient gradient ascent CuSum algorithm is designed. The algorithm is constructed by iteratively updating the estimate of the unknown parameters based on the maximum likelihood principle and the gradient ascent approach. The lower bound on its average running length to false alarm is also derived for practical false alarm control. Simulation results are provided to demonstrate the performance of the proposed algorithms.

翻译：研究自回归（AR）模型中的最速变化检测（QCD）问题。系统通过顺序观测样本进行监测，在未知时刻，扰动信号发生并改变观测数据的分布。该扰动信号遵循随时间具有依赖性的AR模型。变化前，观测值仅由测量噪声构成，服从独立同分布（i.i.d.）；变化后，观测值包含扰动信号与测量噪声，呈现时序依赖性，本质上服从连续状态隐马尔可夫模型（HMM）。目标是在虚警约束下，设计停时以最快速度检测到扰动信号。现有针对一般非独立同分布场景及离散状态HMM的方法因计算复杂度和内存消耗过高而无法直接应用，且通常需要假设某种渐近稳定性条件。本文首次通过前向变量与辅助马尔可夫链的创新设计，从理论上证明了AR模型的渐近稳定性条件；进而构建了可递归更新的高效遍历CuSum算法，并证明其渐近最优性。进一步研究了扰动信号参数未知的数据驱动场景，设计了在线且计算高效的自适应梯度上升CuSum算法：该算法基于最大似然原理与梯度上升方法迭代更新未知参数估计。同时推导了其虚警平均运行长度下界，以支持实际虚警控制。仿真结果验证了所提算法的性能。