A deconvolution approach to estimation of a common shape in a shifted curves model

TL;DR

This paper introduces an adaptive wavelet thresholding method for estimating a common shape in shifted curve models, linking curve registration to linear inverse problems and providing new shift estimation techniques.

Contribution

It proposes a novel wavelet-based adaptive estimator for mean pattern recovery in shifted curves, connecting curve registration with inverse problem analysis.

Findings

Estimator achieves near-minimax convergence rates

Method effectively estimates unknown random shifts

Numerical experiments demonstrate competitive performance

Abstract

This paper considers the problem of adaptive estimation of a mean pattern in a randomly shifted curve model. We show that this problem can be transformed into a linear inverse problem, where the density of the random shifts plays the role of a convolution operator. An adaptive estimator of the mean pattern, based on wavelet thresholding is proposed. We study its consistency for the quadratic risk as the number of observed curves tends to infinity, and this estimator is shown to achieve a near-minimax rate of convergence over a large class of Besov balls. This rate depends both on the smoothness of the common shape of the curves and on the decay of the Fourier coefficients of the density of the random shifts. Hence, this paper makes a connection between mean pattern estimation and the statistical analysis of linear inverse problems, which is a new point of view on curve registration and image warping problems. We also provide a new method to estimate the unknown random shifts between curves. Some numerical experiments are given to illustrate the performances of our approach and to compare them with another algorithm existing in the literature.