[Paper Review] Sketching for Sequential Change-Point Detection

Yang Cao, Andrew Thompson+2 more|arXiv (Cornell University)|May 25, 2015

Distributed Sensor Networks and Detection Algorithms33 references3 citations

TL;DR

This paper proposes a sequential change-point detection method using linear sketches of high-dimensional data based on generalized likelihood ratio (GLR) statistics, enabling efficient real-time detection with reduced computational and storage costs. The key contribution is theoretically accurate approximations for average run length (ARL) and expected detection delay (EDD), showing minimal performance loss when signal strength and sketch dimension M are sufficiently large.

ABSTRACT

We study sequential change-point detection procedures based on linear sketches of high-dimensional signal vectors using generalized likelihood ratio (GLR) statistics. The GLR statistics allow for an unknown post-change mean that represents an anomaly or novelty. We consider both fixed and time-varying projections, derive theoretical approximations to two fundamental performance metrics: the average run length (ARL) and the expected detection delay (EDD); these approximations are shown to be highly accurate by numerical simulations. We further characterize the relative performance measure of the sketching procedure compared to that without sketching and show that there can be little performance loss when the signal strength is sufficiently large, and enough number of sketches are used. Finally, we demonstrate the good performance of sketching procedures using simulation and real-data examples on solar flare detection and failure detection in power networks.

Motivation & Objective

• To address the challenge of real-time change-point detection in high-dimensional streaming data with reduced computational and storage overhead.
• To develop a sequential detection procedure using linear sketches that maintains detection performance while reducing dimensionality from N to M (M ≪ N).
• To derive theoretical approximations for ARL and EDD under fixed and time-varying projections, enabling threshold calibration without extensive simulations.
• To characterize the relative performance of sketching versus full-dimensional detection, identifying conditions under which performance loss is negligible.
• To demonstrate the method’s effectiveness on real-world applications such as solar flare detection and power network failure monitoring.

Proposed method

• Uses linear sketching via an M×N matrix A to project high-dimensional data vectors x_t into lower-dimensional sketches y_t = Ax_t.
• Applies the generalized likelihood ratio (GLR) statistic to detect changes in mean, with the unknown post-change mean estimated via maximum likelihood.
• Derives theoretical approximations for ARL (no change) and EDD (with change) under Gaussian assumptions for pre- and post-change distributions.
• Considers both fixed projection matrices (Gaussian and sparse 0-1 matrices like expander graphs) and time-varying projections A_t.
• Uses Kullback-Leibler divergence and threshold scaling to analyze performance trade-offs between sketching and full-dimensional detection.
• Employs numerical simulations and real-data case studies (solar flares, power networks) to validate theoretical findings and practical performance.

Experimental results

Research questions

• RQ1 Under what conditions does sketching incur minimal performance loss in sequential change-point detection?
• RQ2 How accurately can theoretical approximations predict ARL and EDD for sketch-based detection procedures?
• RQ3 What is the impact of sketch dimension M and signal strength on detection delay and false alarm rate?
• RQ4 How does the choice of sketching matrix (Gaussian vs. expander-based) affect detection performance relative to full-dimensional detection?
• RQ5 Can sketching-based detection achieve near-optimal performance in real-world applications like power network monitoring and solar flare detection?

Key findings

• Theoretical approximations for ARL and EDD are highly accurate, enabling reliable threshold setting without extensive simulations.
• Performance loss due to sketching is minimal when signal strength and sketch dimension M are sufficiently large, especially for Gaussian and expander-based sketching matrices.
• For a signal strength of μ₀ = 4 and M = 100 sketches, the EDD increases by only one sample compared to full-dimensional detection, achieving a 98% dimensionality reduction (100/4941).
• The reduction in Kullback-Leibler divergence from sketching is compensated by a proportional reduction in threshold b, leading to similar EDDs between sketching and full-dimensional methods in high-signal regimes.
• The method achieves near-optimal detection performance in real-world examples, including solar flare detection and power network failure monitoring, with significant computational and storage savings.
• The minimum required sketch dimension M can be determined to bound performance loss under worst-case signals and a target ARL, enabling system design with performance guarantees.