Multiscale quantile segmentation

Abstract

We introduce a new methodology for analyzing serial data by quantile regression assuming that the underlying quantile function consists of constant segments. The procedure does not rely on any distributional assumption besides serial independence. It is based on a multiscale statistic, which allows to control the (finite sample) probability for selecting the correct number of segments S at a given error level, which serves as a tuning parameter. For a proper choice of this parameter, this tends exponentially fast to the true S, as sample size increases. We further show that the location and size of segments are estimated at minimax optimal rate (compared to a Gaussian setting) up to a log-factor. Thereby, our approach leads to (asymptotically) uniform confidence bands for the entire quantile regression function in a fully nonparametric setup. The procedure is efficiently implemented using dynamic programming techniques with double heap structures, and software is provided. Simulations and data examples from genetic sequencing and ion channel recordings confirm the robustness of the proposed procedure, which at the same hand reliably detects changes in quantiles from arbitrary distributions with precise statistical guarantees.

A review on minimax rates in change point detection and localisation

“…Finally, we mention a list of papers working on different aspects in this area: Hawkins and Deng (2010), Haynes et al (2017b), Itoh and Kurths (2010), Matteson and James (2014), Vanegas et al (2019), Zou et al (2014), Harchaoui and Cappé (2007), Garreau and Arlot (2018), Celisse et al (2018) and Arlot et al (2019), among others.…”

“…Quite the contrary, the study of the fundamental limits of the testing problem is lagged behind. The literature on different aspects of testing includes Yao and Au (1989), Frick et al (2014), Enikeeva et al (2019), Vanegas et al (2019), Dette and Kutta (2019), Dette et al (2018a), Akashi et al (2018), Dette et al (2018c), Aue et al (2018), Aue and Horváth (2013), Robbins et al (2011), Liu et al (2019, Stoehr et al (2020), Kirch et al (2015), Jewell et al (2019), Chen (2019a), Jirak (2015), Chu and Chen (2019) and Verzelen et al (2020), among others.…”

Optimistic search: Change point estimation for large-scale data via adaptive logarithmic queries

“…For instance, for detection of changes in quantiles (cf. Vanegas et al, 2020), one might consider to store the data in a tree structure, e.g. AVL tree (Adelson-Velskiȋ and Landis, 1962).…”