High-Dimensional Sequential Change Detection

TL;DR

This paper extends quickest change detection to high-dimensional multivariate normal data, introducing the NHDKL measure to analyze detection delays and proposing estimators that optimize this divergence for improved performance.

Contribution

It introduces the NHDKL divergence as a key tool for high-dimensional change detection and develops estimators that asymptotically minimize this divergence, enhancing detection efficiency.

Findings

Detection delay inversely proportional to NHDKL differences.

Perfect estimation reduces delay to divergence between post- and pre-change distributions.

Proposed estimators outperform standard fixed-dimension methods.

Abstract

We address the problem of detecting a change in the distribution of a high-dimensional multivariate normal time series. Assuming that the post-change parameters are unknown and estimated using a window of historical data, we extend the framework of quickest change detection (QCD) to the highdimensional setting in which the number of variables increases proportionally with the size of the window used to estimate the post-change parameters. Our analysis reveals that an information theoretic quantity, which we call the Normalized High- Dimensional Kullback-Leibler divergence (NHDKL), governs the high-dimensional asymptotic performance of QCD procedures. Specifically, we show that the detection delay is asymptotically inversely proportional to the difference between the NHDKL of the true post-change versus pre-change distributions and the NHDKL of the true versus estimated post-change distributions. In cases of perfect estimation, where the latter NHDKL is zero, the delay is inversely proportional to the NHDKL between the post-change and pre-change distributions alone. Thus, our analysis is a direct generalization of the traditional fixed-dimension, large-sample asymptotic framework, where the standard KL divergence is asymptotically inversely proportional to detection delay. Finally, we identify parameter estimators that asymptotically minimize the NHDKL between the true versus estimated post-change distributions, resulting in a QCD method that is guaranteed to outperform standard approaches based on fixed-dimension asymptotics.