Derivative Principal Component Analysis for Representing the Time Dynamics of Longitudinal and Functional Data

• Activity Number: 582 - Statistical Methods for Functional Data
• Type: Contributed
• Date/Time: Wednesday, August 2, 2017 : 2:00 PM to 3:50 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Xiongtao Dai* and Hans-Georg G Müller and Wenwen Tao
• Companies: University of California, Davis and University of California, Davis and Quora
• Keywords: longitudinal data ; principal component analysis ; dimension reduction ; sparse observations ; classification ; spectra

Abstract

We propose a nonparametric method to explicitly model and represent the derivatives of smooth underlying trajectories for longitudinal data. This representation is based on a direct Karhunen--Loeve expansion of the unobserved derivatives and leads to the notion of derivative principal component analysis, which complements functional principal component analysis, one of the most popular tools of functional data analysis. The proposed derivative principal component scores can be obtained for irregularly spaced and sparsely observed longitudinal data, as typically encountered in biomedical studies, as well as for functional data which are densely measured. Novel consistency results and asymptotic convergence rates for the proposed estimates of the derivative principal component scores and other model components are derived under a unified scheme for sparse or dense observations and mild conditions. We demonstrate the derivative principal components are more parsimonious than alternative approaches in terms of underlying derivatives recovery. In an application example, we demonstrate the utility of derivative principal components for the classification of wheat spectra.