Anytime-Valid Tests for Sparse Anomalies

TL;DR

This paper develops anytime-valid testing procedures for detecting sparse anomalies in multiple data streams, enabling quick detection with adaptive and optimal performance even under unknown parameters.

Contribution

It introduces novel anytime-valid tests for sparse anomalies, addressing the challenge of rapid detection and parameter uncertainty in large-scale streaming data.

Findings

Tests can detect anomalies quickly in large data streams.

Proposed methods are adaptive and perform optimally under unknown parameters.

Numerical results show favorable comparison with oracle and benchmark tests.

Abstract

We consider the problem of detection of sparse anomalies when monitoring a large number of data streams continuously in time. This problem is addressed using anytime-valid tests. In the context of a normal-means model and for a fixed sample, this problem is known to exhibit a nontrivial phase transition that characterizes when anomalies can and cannot be detected. We show, for the anytime-valid version of the problem, testing procedures that can detect the presence of anomalies quickly. Given that the goal is quick detection, existing approaches to anytime-valid testing that study how evidence accumulates for large times through log-optimality criteria is insufficient. This issue is addressed in this context by studying log-optimal procedures for a fixed moment in time, but as the number of streams grows larger. The resulting characterization is related to, but not implied by the existing results for fixed-sample tests. In addition, we also construct and analyze tests that are parameter-adaptive and exhibit optimal performance (in a well defined sense) even when the hypothesized model parameters are unknown. Numerical results illustrate the behavior of the proposed tests in comparison with oracle tests and suitable benchmarks.