Premium
Optimally weighted L 2 distance for functional data
Author(s) -
Chen Huaihou,
Reiss Philip T.,
Tarpey Thaddeus
Publication year - 2014
Publication title -
biometrics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.298
H-Index - 130
eISSN - 1541-0420
pISSN - 0006-341X
DOI - 10.1111/biom.12161
Subject(s) - weight function , mathematics , weighting , nonparametric statistics , functional data analysis , cluster analysis , permutation (music) , measure (data warehouse) , algorithm , function (biology) , medoid , statistics , computer science , data mining , medicine , physics , evolutionary biology , biology , acoustics , radiology
Summary Many techniques of functional data analysis require choosing a measure of distance between functions, with the most common choice being L 2 distance. In this article we show that using a weighted L 2 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.