Robust template estimation for functional data with phase variability using band depth

Registration, or alignment, of functional observations has been a fundamental problem in functional data analysis. The creation of a template was the key step for alignment of a group of functions. Recent studies have defined the template with the notion of “mean” in the given observations. However, the mean can be sensitive to the, commonly observed, outlier functions in the data. To deal with this problem, a new approach is proposed to adopt the notion of “median” using the time warping functions in the alignment process, based on the recently developed band depth in functional data. A semi-parametric model is provided with an algorithm that yields a consistent estimator for the underlying median template. The robustness of this depth-based registration is illustrated using simulations and two real data sets. In addition, a new depth-based boxplot is proposed for outlier detection in functional data with phase variability.