Consistent change-point detection with kernels

Damien Garreau (1, 2) , Sylvain Arlot (3, 4)

1 DI-ENS - Département d'informatique - ENS Paris
2 SIERRA - Statistical Machine Learning and Parsimony
3 LMO - Laboratoire de Mathématiques d'Orsay
4 SELECT - Model selection in statistical learning

In this paper we study the kernel change-point algorithm (KCP) proposed by Arlot, Celisse and Harchaoui (2012), which aims at locating an unknown number of change-points in the distribution of a sequence of independent data taking values in an arbitrary set. The change-points are selected by model selection with a penalized kernel empirical criterion. We provide a non-asymptotic result showing that, with high probability, the KCP procedure retrieves the correct number of change-points, provided that the constant in the penalty is well-chosen; in addition, KCP estimates the change-points location at the optimal rate. As a consequence, when using a characteristic kernel, KCP detects all kinds of change in the distribution (not only changes in the mean or the variance), and it is able to do so for complex structured data (not necessarily in $\mathbb{R}^d$). Most of the analysis is conducted assuming that the kernel is bounded; part of the results can be extended when we only assume a finite second-order moment.

Keywords: Kernel methods; Change-point detection; Penalized least-squares

Domains: Statistics Theory [stat.TH]

Damien Garreau, Sylvain Arlot. Consistent change-point detection with kernels. Electronic Journal of Statistics , 2018, 12 (2), pp.4440-4486. ⟨hal-01416704v2⟩