Robust depth-based estimation in the time warping model

In functional data analysis, the time warping model aims at representing a set of curves exhibiting phase and amplitude variation with respect to a common continuous process. Many biological processes, when observed across the time among different individuals, fit into this concept. The observed curves are modeled as the composition of an "amplitude process," which governs the common behavior, and a "warping process" that induces time distortion among the individuals. We aim at characterizing the first one. Because of the phase variation present among the curves, classical sample statistics computed on the observed sample provide poor representations of the amplitude process. Existing methods to estimate the mean behavior of the amplitude process consist on aligning the curves, that is, eliminating time variation, before estimation. However, since they rely on the use of sample means, they are very sensitive to the presence of outliers. In this article, we propose the use of a functional depth-based median as a robust estimator of the central behavior of the amplitude process. We investigate its properties in the time warping model, and we evaluate its performance in different simulation studies where we compare it to existing estimators, and we show its robustness against atypical observations. Finally, we illustrate its use with real data on a yeast time course microarray data set.