Corrected Confidence Bands for Functional Data Using Principal Components

• Activity Number: 120
• Type: Topic Contributed
• Date/Time: Monday, July 30, 2012 : 8:30 AM to 10:20 AM
• Sponsor: Biometrics Section
• Author(s): Jeff Goldsmith*+ and Sonja Greven and Ciprian Crainiceanu
• Companies: The Johns Hopkins University and Ludwig-Maxilians-Universität München and The Johns Hopkins University
• Keywords: Bootstrap ; Functional Principal Components Analysis ; Iterated Expectation and Variance ; Simultaneous Bands

Abstract

Functional principal components (FPC) analysis is widely used to decompose and express functional observations. Curve estimates implicitly condition on basis functions and other quantities derived from FPC decompositions; however these objects are unknown in practice. In this paper, we propose a method for obtaining correct curve estimates by accounting for uncertainty in FPC decompositions. Additionally, pointwise and simultaneous confidence intervals that account for both model-based and decomposition-based variability are constructed. Standard mixed-model representations of functional expansions are used to construct curve estimates and variances conditional on a specific decomposition. A bootstrap procedure is implemented to understand the uncertainty in principal component decomposition quantities. Iterated expectation and variance formulas combine both sources of uncertainty by combining model-based conditional estimates across the distribution of decompositions. Our method compares favorably to competing approaches in simulation studies and real data examples that include both densely- and sparsely-observed functions.