Abstract
The goal of the paper is to develop a specific application of the convex optimization based hypothesis testing techniques developed in A. Juditsky, A. Nemirovski, Hypothesis testing via affine detectors, Electronic Journal of Statistics 10:2204--2242, 2016. Namely, we consider the Change Detection problem as follows: given an evolving in time noisy observations of outputs of a discrete-time linear dynamical system, we intend to decide, in a sequential fashion, on the null hypothesis stating that the input to the system is a nuisance, vs. the alternative stating that the input is a signal, with both the nuisances and the nontrivial signals modeled as inputs belonging to finite unions of some given convex sets. Assuming the observation noises zero mean sub-Gaussian, we develop computation-friendly sequential decision rules and demonstrate that in our context these rules are provably near-optimal.
Highlights
Quick detection of change-points from data streams is a classic and fundamental problem in signal processing and statistics, with a wide range of applications from cybersecurity [21] to gene mapping [36]
Following the line of research in [8, 16, 17], we allow for rather general structural assumptions on the components of our setup (system (1.1) and descriptions of nuisance and signal inputs) and are looking for computation-friendly inference routine meaning that our easy-to-implement routines and their performance characteristics are given by efficient computation
We consider two different types of detection procedures, those based on affine and on quadratic detectors, each type dealing with its own structure of nuisance and signal inputs
Summary
Quick detection of change-points from data streams is a classic and fundamental problem in signal processing and statistics, with a wide range of applications from cybersecurity [21] to gene mapping [36]. Outstanding contributions include Shewhart’s control chart [33], Page’s CUSUM procedure [29], Shiryaev-Roberts procedure [34], Gordon’s non-parametric procedure [11], and window-limited procedures [19]. High-dimensional change-point detection ( referred to as the multi-sensor change-point detection) is a more recent topic, and various statistical procedures are proposed including [14, 13, 15, 18, 26, 39, 22, 6, 5, 42, 23]. There has been very little research on the computational aspect of change-point detection, especially in the high-dimensional setting
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