title: Optimally weighted L(2) distance for functional data.
abstract: Many techniques of functional data analysis require choosing a measure of distance between functions, with the most common choice being L2 distance. In this article we show that using a weighted L2 distance, with a judiciously chosen weight function, can improve the performance of various statistical methods for functional data, including k-medoids clustering, nonparametric classification, and permutation testing. Assuming a quadratically penalized (e.g., spline) basis representation for the functional data, we consider three nontrivial weight functions: design density weights, inverse-variance weights, and a new weight function that minimizes the coefficient of variation of the resulting squared distance by means of an efficient iterative procedure. The benefits of weighting, in particular with the proposed weight function, are demonstrated both in simulation studies and in applications to the Berkeley growth data and a functional magnetic resonance imaging data set.
authors.affiliation: Department of Child and Adolescent Psychiatry, New York University School of Medicine, New York, New York, U.S.A.
authors.forename: Huaihou
authors.initials: H
authors.lastname: Chen
authors.affiliation: Department of Child and Adolescent Psychiatry, New York University School of Medicine, New York, New York, U.S.A.|Department of Population Health, New York University School of Medicine, New York, New York, U.S.A.|Nathan S. Kline Institute for Psychiatric Research, Orangeburg, New York, U.S.A.
authors.forename: Philip T
authors.initials: PT
authors.lastname: Reiss
authors.affiliation: Department of Mathematics and Statistics, Wright State University, Dayton, Ohio, U.S.A.
authors.forename: Thaddeus
authors.initials: T
authors.lastname: Tarpey
mesh_terms: D000465:Algorithms; D003198:Computer Simulation; D003627:Data Interpretation, Statistical; D004812:Epidemiologic Methods; D015233:Models, Statistical; D018401:Sample Size
keywords: Coefficient of variation; Functional classification; Functional clustering; Penalized splines; Weighted L2 distance