Given a times series Y in , with a piecewise constant mean and independent components, the twin problems of change-point detection and change-point localization, respectively amount to detecting the existence of times where the mean varies and estimating the positions of those change-points. In this work, we tightly characterize optimal rates for both problems and uncover the phase transition phenomenon from a global testing problem to a local estimation problem. Introducing a suitable definition of the energy of a change-point, we first establish in the single change-point setting that the optimal detection threshold is . When the energy is just above the detection threshold, then the problem of localizing the change-point becomes purely parametric: it only depends on the difference in means and not on the position of the change-point anymore. Interestingly, for most change-point positions, including all those away from the endpoints of the time series, it is possible to detect and localize them at a much smaller energy level. In the multiple change-point setting, we establish the energy detection threshold and show similarly that the optimal localization error of a specific change-point becomes purely parametric. Along the way, tight minimax rates for Hausdorff and estimation losses of the vector of all change-points positions are also established. Two procedures achieving these optimal rates are introduced. The first one is a least-squares estimator with a new multiscale penalty that favours well spread change-points. The second one is a two-step multiscale post-processing procedure whose computational complexity can be as low as . Notably, these two procedures accommodate with the presence of possibly many low-energy and therefore undetectable change-points and are still able to detect and localize high-energy change-points even with the presence of those nuisance parameters.
The work of N. Verzelen has been partially supported by ANR-21-CE23-0035 (ASCAI).
The work of Patricia Reynaud-Bouret was supported by the French government, through the and 3IA Côte d’Azur Investissements d’Avenir managed by the National Research Agency (ANR-15- IDEX-01 and ANR-19-P3IA-0002), by the interdisciplinary Institute for Modeling in Neuroscience and Cognition (NeuroMod) of the Université Côte d’Azur and directly by the National Research Agency (ANR-19-CE40-0024) with the ChaMaNe project.
We are grateful to Guillem Rigaill for many stimulating discussions. We would also would like to thank two referees for suggestions that improved the presentation. Nicolas Verzelen is also affiliated to Institut Agro, Montpellier and Univ. Montpellier. Patricia Reynaud-Bouret is also affiliated with the CNRS.
"Optimal change-point detection and localization." Ann. Statist. 51 (4) 1586 - 1610, August 2023. https://doi.org/10.1214/23-AOS2297