Quickest Change Detection for Multiple Data Streams Using the James-Stein Estimator

TL;DR

This work tackles quickest change detection in multiple Gaussian data streams with an unknown mean shift. It leverages the James-Stein shrinkage estimator to form two tests: JS-WL-CuSum and JS-SRRS, achieving uniform reductions in detection delay across post-change parameters and ARL constraints, especially for large $K$. The authors provide non-asymptotic upper bounds and second-order asymptotic minimax results, showing that shrinkage improves performance without sacrificing optimality, and they validate the theory with simulations showing notable gains over ML-based and GLR methods. The results highlight the practical impact of integrating prior structural information via shrinkage into online detection, offering scalable and robust performance in high-dimensional settings.

Abstract

The problem of quickest change detection is studied in the context of detecting an arbitrary unknown mean-shift in multiple independent Gaussian data streams. The James-Stein estimator is used in constructing detection schemes that exhibit strong detection performance both asymptotically and non-asymptotically. Our results indicate that utilizing the James-Stein estimator in the recently developed window-limited CuSum test constitutes a uniform improvement over its typical maximum likelihood variant. That is, the proposed James-Stein version achieves a smaller detection delay simultaneously for all possible post-change parameter values and every false alarm rate constraint, as long as the number of parallel data streams is greater than three. Additionally, an alternative detection procedure that utilizes the James-Stein estimator is shown to have asymptotic detection delay properties that compare favorably to existing tests. The second-order asymptotic detection delay term is reduced in a predefined low-dimensional subspace of the parameter space, while second-order asymptotic minimaxity is preserved. The results are verified in simulations, where the proposed schemes are shown to achieve smaller detection delays compared to existing alternatives, especially when the number of data streams is large.

Figures (6)

• Figure 1: Mean squared-errors (MSE) of the maximum likelihood, James-Stein, and positive-part James-Stein estimators (with shrinkage towards $\bm 0$) when $K = 10$.
• Figure 2: The upper bounds of Theorem \ref{['thm:WL_UB']} (dotted lines) and approximation in eq. \ref{['eq:simple_approx']} compared against the true detection delay for a range of values of $b$. Both the upper bound and the approximation are smaller when using the JS estimator (blue) compared to the ML estimator (yellow). It is also seen that in particular when using the JS estimator, the analytical results accurately reflect the true detection delay (solid lines).
• Figure 3: Tradeoff between the average run length to false alarm (ARL) and detection delay (ADD) for the studied tests with $K=5$ (left) and $K=10$ (right). James-Stein variants (blue) uniformly improve on the ML versions (gold). For $K=30$, the proposed tests are also significantly better than the GLR test LAI_1998 in terms of reduced detection delay.
• Figure 4: Detection delay against the dimension $K$, when $\lVert{\boldsymbol{\theta} }\rVert = 1$ and $\gamma = 2000$. It is observed, that the higher the dimension, the larger the gap in performance in favor of the James-Stein tests. Hence, JS-based test scales well for larger number of streams and sensors.
• Figure 5: Average detection delay when the change affects a varying number of sensors out of 20, with total signal magnitude $\lVert{\boldsymbol{\theta} }\rVert$ fixed at 1. JS-based tests are uniformly better than ML-alternatives and better than the GLR test when more than 40% of the streams are affected.

Theorems & Definitions (11)

• Lemma 1: Information lower bound
• Remark 1
• Remark 2
• Theorem 1
• proof
• Remark 3
• Theorem 2
• proof
• Corollary 1
• proof