We study finite-sample change detection for one-dimensional noisy dynamical systems using partition-based empirical approximations of stationary behaviour. Given observations from an interval-valued process, we partition the state space, estimate a finite transition matrix from observed transitions between partition elements, and apply a small Doeblin-type regularisation to ensure a unique stationary distribution. From an initial reference segment, we compute a baseline empirical stationary distribution \(\widehatπ_{0,ρ}\). For each later sliding window, we compute \(\widehatπ_{t,ρ}\) and define the score \[ S_t=\|\widehatπ_{t,ρ}-\widehatπ_{0,ρ}\|_1. \] Large values of \(S_t\) indicate a change in stationary behaviour relative to the baseline. The statistic detects changes in invariant density or stationary law, but not all possible changes in transition dynamics. Under explicit assumptions on empirical transition concentration, finite-state stationary distribution stability, partition approximation, regularisation bias, and noise stability, we derive a finite-sample bound for the empirical stationary density. The bound separates sampling error, regularisation bias, partition approximation error, and noise bias. We then obtain a single-window false-alarm guarantee and a sufficient detection condition when the invariant density changes by more than the estimation error. We illustrate the method on synthetic noisy beta-map change-point experiments.

翻译：我们研究基于分区的经验逼近稳态行为的一维噪声动力系统有限样本变化检测问题。给定区间值过程的观测数据后，我们对状态空间进行分区，从分区元素间的观测转移中估计有限转移矩阵，并采用小型Doeblin型正则化确保唯一稳态分布。基于初始参考片段，计算基线经验稳态分布\(\widehatπ_{0,ρ}\)。针对后续每个滑动窗口，计算\(\widehatπ_{t,ρ}\)并定义评分\[ S_t=\|\widehatπ_{t,ρ}-\widehatπ_{0,ρ}\|_1. \]当\(S_t\)取较大值时，表明相对于基线存在稳态行为变化。该统计量可检测不变测度或稳态分布的变化，但无法检测所有可能的转移动力学变化。在经验转移集中性、有限状态稳态分布稳定性、分区逼近、正则化偏差及噪声稳定性的显式假设下，我们推导出经验稳态密度的有限样本界。该界分离了采样误差、正则化偏差、分区逼近误差与噪声偏差。进而获得单窗口虚警保证，以及当不变密度变化超过估计误差时的充分检测条件。我们通过含噪贝塔映射变点合成实验验证该方法。