Multiscale Change-point Segmentation: Beyond Step Functions

Abstract

Modern multiscale type segmentation methods are known to detect multiple change-points with hii gh statistical accuracy, while allowing for fast computation. Underpinning theory has been developed mainly for models which assume the signal as an unkk nown piecewise constant function. In this paper this will be extended to certain function classes beyond step functions in a nonparametric regression see tting, revealing certain multiscale segmentation methods as robust to deviation from such piecewise constant functions. Although these methods are desigg ned for step functions, our main finding is its adaptation over such function classes for a universal thresholding. On the one hand, this includes nearll y optimal convergence rates for step functions with increasing number of jumps. On the other hand, for models which are characterized by certain approxii mation spaces, we obtain nearly optimal rates as well. This includes bounded variation functions, and (piecewise) Hölder functions of smoothness orr der 0<α≤1. All results are formulated in terms of Lp-loss (0<p<∞) both almost surely and in expectation. Theoretical findings are examined by various nuu merical simulations.