High-Dimensional, Multiscale Online Changepoint Detection

Abstract

We introduce a new method for high-dimensional, online changepoint detection in settings where a p-variate Gaussian data stream may undergo a change in mean.The procedure works by performing likelihood ratio tests against simple alternatives of different scales in each coordinate, and then aggregating test statistics across scales and coordinates. The algorithm is online in the sense that both its storage requirements and worstcase computational complexity per new observation are independent of the number of previous observations; in practice, it may even be significantly faster than this. We prove that the patience, or average run length under the null, of our procedure is at least at the desired nominal level, and provide guarantees on its response delay under the alternative that depend on the sparsity of the vector of mean change. Simulations confirm the practical effectiveness of our proposal, which is implemented in the R package ocd, and we also demonstrate its utility on a seismology data set. K E Y W O R D S average run length, detection delay, high-dimensional changepoint detection, online algorithm, sequential method | 235 CHEN et al. 2 A random variable τ taking values in ℕ ∪ {∞} is an extended stopping time with respect to the filtration ( n ) n∈ℕ , if { = n} ∈  n for all n ∈ ℕ.𝔼 wc (N) : = sup z ∈ ℕ ∪ {0} ess sup 𝔼 z, (N − z) ∨ 0|X 1 , … , X z .

Inference in high-dimensional online changepoint detection

“…For i ∈ N and j ∈ [p], let X j i denote the jth coordinate of X i . The ocd algorithm of Chen et al (2021), which forms part of Algorithm 1, relies on a lower bound β > 0 for the 2 -norm of the vector of mean change and sets of signed scales B and B 0 defined in terms of β. From our perspective, the key aspects of this multiscale algorithm are that, in addition to returning a stopping time N as output, it produces a matrix of residual tail lengths (t j N,b ) j∈[p],b∈B∪B 0 with t j N,b := sargmax 0≤h≤N N i=N −h+1 (X j i − b/2) (similarly to (1)), an 'anchor' coordinate ĵ ∈ [p], a signed anchor scale b ∈ B and a tail partial sum vector A…”

“…There are two main differences between the algorithms. First, in ocd CI, the base changepoint detection procedure is ocd, while in ocd CI , we use the ocd procedure of Chen et al (2021) instead. This latter algorithm is designed to avoid difficulties caused by adversarial pre-change observations that may lead to lengthy response delays for the ocd procedure.…”

“…Nevertheless, the related problem where one observes data sequentially and seeks to declare changes as soon as possible after they have occurred, is nowadays receiving increasing attention (e.g. Kirch and Stoehr, 2019;Dette and Gösmann, 2020;Gösmann et al, 2020;Chen et al, 2021;Yu et al, 2021b). Although the focus of our review here has been on recent developments, including finite-sample results in multivariate and high-dimensional settings, we also mention that changepoint analysis has a long history (e.g.…”

“…In terms of the base sequential changepoint detection procedures, we focus on the ocd algorithm (short for online changepoint detection) introduced by Chen et al (2021), as well as its variant ocd , which provides guarantees on both the average and worst-case detection delays, subject to a guarantee on the patience, or average false alarm rate under the null hypothesis of no change. Crucially, these are both online algorithms.…”

High-dimensional, multiscale online changepoint detection

“…For simplicity of exposition, we have not presented this computational speed-up in Algorithm 1, and it appears to be difficult to provide theoretical guarantees on |T |. Nevertheless we have implemented the algorithm in this form in the R package ocd (Chen, Wang and Samworth, 2020), and have found it to provide substantial computational savings in practice.…”