Robust Quickest Change Detection in Non-Stationary Processes

TL;DR

This work develops robust quickest change detection for non-stationary post-change processes where the post-change law is unknown. By introducing a Least Favorable Law (LFL) and designing Shiryaev and CUSUM statistics against it, the authors obtain exact or asymptotic robust optimality under Bayesian and minimax formulations, respectively. They establish general conditions for robustness, provide concrete LFL constructions for Gaussian and Poisson families, and validate the approach with simulations and real data (Pittsburgh flight arrivals and COVID-19 infection rates). The results yield recursive, implementable detectors with strong performance guarantees in practical non-stationary settings, enabling robust anomaly detection across aerospace, defense, and public health domains.

Abstract

Optimal algorithms are developed for robust detection of changes in non-stationary processes. These are processes in which the distribution of the data after change varies with time. The decision-maker does not have access to precise information on the post-change distribution. It is shown that if the post-change non-stationary family has a distribution that is least favorable in a well-defined sense, then the algorithms designed using the least favorable distributions are robust and optimal. Non-stationary processes are encountered in public health monitoring and space and military applications. The robust algorithms are applied to real and simulated data to show their effectiveness.

Figures (8)

• Figure 1: Distance measurements and corresponding signals extracted from datasets on aircraft trajectories collected from aircraft around the Pittsburgh-Butler Regional Airport (Patrikar2021).
• Figure 2: Daily infection rates for Allegheny and St. Louis counties in the first $200$ days starting 2020/1/22. The number of infections over time (even beyond the dates shown here) has multiple cycles of high values and low values.
• Figure 3: [Left] Comparison of robust CUSUM algorithm designed using pre-change density $f=\mathcal{N}(0,1)$ and post-change density $\mathcal{N}(0.5, 1)$ with a non-robust test designed using pre-change density $f=\mathcal{N}(0,1)$ and post-change density $\mathcal{N}(1.5, 1)$. The data samples are generated after change using $\mathcal{N}(0.5, 1)$. [Right] A similar plot for Poisson laws, with $f=\text{Pois}(0.5)$, LFL as $\text{Pois}(0.8)$, non-robust post-change as $\text{Pois}(1.5)$, and data samples are generated from $\text{Pois}(0.8)$. Here $\text{EDD}$ is used to denote $\mathsf{E}_1^{\bar{G}}[\tau_c^*-1]$ and MFA $= \mathsf{E}_{\infty}[\tau]$ is the mean time to a false alarm, where change time $\nu = \infty$ (no change).
• Figure 4: Comparisons for the robust Shiryaev test. These two figures were generated using the same setup used for generating Fig. \ref{['fig:CUSUM']}. In addition, the change point is assumed to be a geometric random variable with parameter $0.01$. Here EDD represents $\mathsf{E}[(\tau^*-\nu)^+]$ and PFA $= \mathsf{P}^{\pi}(\tau < \nu)$ is the probability of false alarm.
• Figure 5: [Left] Flight signals for the last $100$ seconds of $35$ randomly chosen aircraft arriving at the Pittsburgh-Butler Regional Airport (Patrikar2021). Signals are padded with zeros. The first $100$ seconds can be thought of as time before the aircraft appears in the sensor system. [Right] Flight signal corrupted by Gaussian noise, $\mathcal{N}(0,1)$.

Theorems & Definitions (14)

• Definition 2.1: Stochastic Boundedness in Non-Stationary Processes; Least Favorable Law (LFL)
• Lemma 2.2: unni-etal-ieeeit-2011
• Theorem 2.3
• Remark 1
• Theorem 3.1
• Corollary 4.1
• Corollary 4.2
• Theorem 5.1: brucks2023modeling,lai-ieeetit-1998
• Example 5.2: I.P.I.D. Process
• Example 5.3: MLR Order Processes