Multiple Change-Point Estimation With a Total Variation Penalty

Abstract

Abstract. We propose a new approach for dealing with the estimation of the location of change-points in one-dimensional piecewise constant signals observed in white noise.Our approach consists in reframing this task in a variable selection context. We use a penalized least-square criterion with a ℓ 1 -type penalty for this purpose. We explain how to implement this method in practice by using the LARS/LASSO algorithm. We then prove that, in an appropriate asymptotic framework, this method provides consistent estimators of the change-points with an almost optimal rate. We finally provide an improved practical version of this method by combining it with a reduced version of the dynamic programming algorithm and we successfully compare it with classical methods.

Fast Detection of Block Boundaries in Block-Wise Constant Matrices

“…The proofs of Proposition 1, Lemmas 1 and 2 can be seen as a natural extension of the results of [5].…”

“…Such a complexity is however prohibitive for dealing with very large data sets. In this situation, [5] proposed to rephrase the change-point estimation issue as a variable selection problem. This approach has also been extended by [15] to find shared change-points between several signals.…”

Change-Point Estimation in the Multivariate Model Taking into Account the Dependence: Application to the Vegetative Development of Oilseed Rape

“…Then, after a vectorization of the data, the Lasso criterion proposed by (Tibshirani 1996) is applied for nding the positions of the non null coecients after having applied a preconditioning described in (Jia & Rohe 2015) to the whitened observations. Our approach can thus be seen as an extension of the methodology proposed by (Harchaoui & Lévy-Leduc 2010) and (Tibshirani 2014 (Harchaoui & Lévy-Leduc 2010).…”

“…Thanks to this modeling, nding the positions of the non-null coecients in B allows us to nd the thermal times at which the changes occur in the slope of the function having piecewise constant rst derivatives. Note that, as in (Harchaoui & Lévy-Leduc 2010), we modeled the change-point detection problem as an estimation issue in a sparse linear model. However, there are some dierences.…”

“…This complexity has recently been improved by (Killick, Fearnhead & Eckley 2012), (Rigaill 2015) and (Maidstone, Hocking, Rigaill & Fearnhead 2016) in some specic cases. A dierent route to reducing the computational complexity of the multiple change-point detection problem in the univariate case is considered in (Harchaoui & Lévy-Leduc 2010) who consider the least-squares criterion with a total variation penalty, which enables them to use the LARS algorithm of (Efron, Hastie, Johnstone & Tibshirani 2004). Other penalties are proposed in (Ng, Lee & Lee 2018).…”