Univariate mean change point detection: Penalization, CUSUM and optimality

Abstract

The problem of univariate mean change point detection and localization based on a sequence of n independent observations with piecewise constant means has been intensively studied for more than half century, and serves as a blueprint for change point problems in more complex settings. We provide a complete characterization of this classical problem in a general framework in which the upper bound σ 2 on the noise variance, the minimal spacing ∆ between two consecutive change points and the minimal magnitude κ of the changes, are allowed to vary with n. We first show that consistent localization of the change points, when the signal-to-noise ratio κdiverges with n at the rate of at least log(n), we demonstrate that two computationally-efficient change point estimators, one based on the solution to an ℓ 0 -penalized least squares problem and the other on the popular wild binary segmentation algorithm, are both consistent and achieve a localization rate of the order σ 2 κ 2 log(n). We further show that such rate is minimax optimal, up to a log(n) term.

Multiscale change point detection for dependent data

“…Remark In a recent article Wang et al (2019) investigated optimality properties for mean change point detection and localization in model () with a piecewise constant function and independent identically distributed sub‐Gaussian errors. In particular they derived necessary and sufficient conditions under which consistent localization of the change points is possible and derived the minimax optimal rate.…”

“…In particular they derived necessary and sufficient conditions under which consistent localization of the change points is possible and derived the minimax optimal rate. Wang et al (2019) also demonstrated that localization based on an ℓ 0 ‐penalized least squares approach or wild binary segmentation achieves a rate that is minimax optimal up to a logarithmic factor. It follows from theorem 7 in Frick et al (2014) that in the case of independent identically normal distributed errors SMUCE also attains the minimax optimal rate up to a logarithmic factor.…”

“…We mention exemplarily Zou, Yin, Feng, and Wang (2014), who developed a maximum likelihood approach for multiple change point detection in a nonparametric setting and the recent work of Anastasiou and Fryzlewicz (2019), who presented a change point detection procedure based on an isolation technique. Wang, Yu, and Rinaldo (2019) and Padilla, Yu, Wang, and Rinaldo (2019) derived minimax optimal rates for the localization of the changes for independent identically distributed sub-Gaussian error processes. Theoretical results on detecting multiple changes in dependent data are published less frequently in the literature and we refer to the work of Eichinger and Kirch (2018), Fryzlewicz (2018a) and Baranowski, Chen, and Fryzlewicz (2019a) among others.…”

Discussion of ‘Detecting possibly frequent change-points: Wild Binary Segmentation 2 and steepest-drop model selection’

“…In a frequent jump regime i is small (related to outlier detection) and necessarily corresponding jumps f ′ i need to be large to be detectable. In both situations, an adaptation of Lemma 1 of Wang et al (2018) shows that no consistent estimator of the locations of change point exists when −2 min 1≤i≤N ( i (f � i ) 2 ) < log(T). While WBS2.SDLL is shown to perform well in both regimes numerically, the paper does not provide a theoretical underpinning of this good behaviour, in the sense that only a linear-time change point setting with T ∶= min i i being of the same order as the sample size T is considered: Such an assumption is not necessary for consistent change point detection and, moreover, it excludes models such as extreme.teeth (ET) and extreme.extreme.teeth (EET), which are reasonably considered as belonging to the frequent jump regime with T ≤ 5 .…”

“…In addition, the best currently available results for the localisation rate attained by WBS as well as the requirement on the magnitude of changes for their detection, are sub-optimal when T ∕T → 0 (see Appendix A of Cho and Kirch 2019). Baranowski et al (2019) and Wang et al (2018) suggest modifications of WBS that alleviate the sub-optimality at the cost of introducing additional tuning parameters such as a threshold or an upper bound on the length of random intervals. However, even in these papers, the assumptions are formulated in terms of…”