In this paper, we consider sampling an Ornstein-Uhlenbeck (OU) process through a channel for remote estimation. The goal is to minimize the mean square error (MSE) at the estimator under a sampling frequency constraint when the channel delay statistics is unknown. Sampling for MSE minimization is reformulated into an optimal stopping problem. By revisiting the threshold structure of the optimal stopping policy when the delay statistics is known, we propose an online sampling algorithm to learn the optimum threshold using stochastic approximation algorithm and the virtual queue method. We prove that with probability 1, the MSE of the proposed online algorithm converges to the minimum MSE that is achieved when the channel delay statistics is known. The cumulative MSE gap of our proposed algorithm compared with the minimum MSE up to the (k + 1)th sample grows with rate at most O(In k). Our proposed online algorithm can satisfy the sampling frequency constraint theoretically. Finally, simulation results are provided to demonstrate the performance of the proposed algorithm.

中文翻译：

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通过具有未知延迟统计的通道对 Ornstein-Uhlenbeck 过程进行远程估计的采样

在本文中，我们考虑通过通道对 Ornstein-Uhlenbeck (OU) 过程进行采样以进行远程估计。目标是当信道延迟统计数据未知时，在采样频率约束下最小化估计器的均方误差 (MSE)。MSE 最小化的采样被重新表述为最优停止问题。通过重新审视延迟统计已知时最优停止策略的阈值结构，我们提出了一种在线采样算法，使用随机逼近算法和虚拟队列方法来学习最优阈值。我们证明，以概率 1，所提出的在线算法的 MSE 收敛到信道延迟统计已知时实现的最小 MSE。与第 (k + 1) 个样本的最小 MSE 相比，我们提出的算法的累积 MSE 差距以最多 O(In k) 的速率增长。我们提出的在线算法理论上可以满足采样频率约束。最后，提供仿真结果来证明所提出算法的性能。