aiqbench analysis
This research introduces a finite-sample change detection framework for noisy dynamical systems that decomposes estimation errors into sampling, regularization, and noise components. By leveraging partition-based empirical approximations and Doeblin-type regularization, the method provides rigorous false-alarm guarantees for detecting shifts in stationary distributions within interval-valued processes. The work offers a theoretically grounded approach for monitoring system stability in settings where transition dynamics are obscured by noise.
Empirical Transfer Operators and Finite-Sample Change Detection for Noisy Expanding Interval Maps
arXiv:2606.06785v1 Announce Type: cross Abstract: 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 \(\widehat{\pi}_{0,\rho}\). For each later sliding window, we compute \(\widehat{\pi}_{t,\rho}\) and define the score \[ S_t=\|\widehat{\pi}_{t,\rho}-\widehat{\pi}_{0,\rho}\|_1. \] 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.
Abstract: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 \(\widehat{\pi}_{0,\rho}\). For each later sliding window, we compute \(\widehat{\pi}_{t,\rho}\) and define the score \[ S_t=\|\widehat{\pi}_{t,\rho}-\widehat{\pi}_{0,\rho}\|_1. \] 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.
Machine Learning (stat.ML); Machine Learning (cs.LG); Dynamical Systems (math.DS)
arXiv:2606.06785 [stat.ML]
(or arXiv:2606.06785v1 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2606.06785 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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