Multiple Change-Point Estimation With a Total Variation Penalty

• Published in: Journal of the American Statistical Association Vol. 105; no. 492; pp. 1480 - 1493
• Main Authors: Harchaoui, Z., Lévy-Leduc, C.
• Format: Journal Article
• Language: English
• Published: Alexandria, VA Taylor & Francis 01.12.2010; American Statistical Association; Taylor & Francis Ltd
• Subjects: Algorithms; Applications; Bayesian method; Change-point estimation; Computer Science; Computer Vision and Pattern Recognition; Consistent estimators; Data corruption; Datasets; Dynamic models; Dynamic programming; Estimating techniques; Estimation; Evolutionary noise; Exact sciences and technology; General topics; Infinity; LARS; LASSO; Low noise; Mathematics; Methodology; Parametric inference; Point estimators; Probability and statistics; Probability theory and stochastic processes; Random variables; Regression analysis; Sciences and techniques of general use; Sparsity; Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications); Statistical methods; Statistics; Theory and Methods; type penalty; Change point; Algorithm; Implementation; Penalty method; type penalty; Statistical method; Location estimation; ℓ; One-dimensional calculations; Sparsity; Consistent estimator; Least squares method; LARS; Point estimation; Change-point estimation; LASSO; White noise; Asymptotic approximation; Dynamic programming; Variable selection; ℓ1-type penalty
• ISSN: 0162-1459; 1537-274X
• DOI: 10.1198/jasa.2010.tm09181

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.