Multiscale change-point segmentation: beyond step functions

Abstract

Modern multiscale type segmentation methods are known to detect multiple changepoints with high statistical accuracy, while allowing for fast computation. Underpinning theory has been developed mainly for models that assume the signal as a piecewise constant function. In this paper this will be extended to certain function classes beyond such step functions in a nonparametric regression setting, revealing certain multiscale segmentation methods as robust to deviation from such piecewise constant functions. Our main finding is the adaptation over such function classes for a universal thresholding, which includes bounded variation functions, and (piecewise) Hölder functions of smoothness order 0 < α ≤ 1 as special cases. From this we derive statistical guarantees on feature detection in terms of jumps and modes. Another key finding is that these multiscale segmentation methods perform nearly (up to a logfactor) as well as the oracle piecewise constant segmentation estimator (with known jump locations), and the best piecewise constant approximants of the (unknown) true signal. Theoretical findings are examined by various numerical simulations.MSC 2010 subject classifications: 62G08, 62G20, 62G35.

Univariate mean change point detection: Penalization, CUSUM and optimality

“…In term of localization, Theorem 2.8 in Frick et al (2014) has also provided a localization error rate that match the minimax rate we derive in this paper. Similar results can also be found in other papers including Dümbgen and Spokoiny (2001), Dümbgen and Walther (2008), Li et al (2017), Jeng et al (2012), Enikeeva et al (2018), to name but a few.…”

A Variational View on Statistical Multiscale Estimation

“…The proper choices of w a and s a depend on the type of signal preferably to be reconstructed and the model itself, see e.g. Schmidt-Hieber et al (2013) for convolution models, and Spokoiny (2009), Frick et al (2014), Pein et al (2017) and Li et al (2019) for change point regression. The constraint in ( 9) forces the residuals Y − Xβ to satisfy…”

“…Frick et al (2014) introduced a remedy, SMUCE (simultaneous multiscale change point estimator) following the MIND idea, by combining variational estimation with multiple tests on residuals over different scales. More generally, multiscale change point segmentation (MCPS; Li et al, 2019) is defined as any solution to the constrained non-convex optimization problem…”

Multiscale change point detection for dependent data

“…Li, Munk, and Sieling (2016) argue that in situations with low signal to noise ratio or with many change‐points compared with the number of observations SMUCE necessarily leads to a conservative estimate and propose to control the false discovery instead of the family wise error rate. More recently, Li, Guo, and Munk (2019) extend the procedure to certain function classes beyond step functions.…”

“…Li, Munk, and Sieling (2016) argue that in situations with low signal to noise ratio or with many change-points compared with the number of observations SMUCE necessarily leads to a conservative estimate and propose to control the false discovery instead of the family wise error rate. More recently, Li, Guo, and Munk (2019) extend the procedure to certain function classes beyond step functions. Even though (1) appears to be relatively simple at first glance, inferring information about the signal * is an important problem in statistics and (1) can be considered as a prototype model to understand the fundamental properties of the multiple change point problem.…”