Shrinkage Estimation for Functional Principal Component Scores with Application to the Population Kinetics of Plasma Folate

Abstract

We present the application of a nonparametric method to perform functional principal components analysis for functional curve data that consist of measurements of a random trajectory for a sample of subjects. This design typically consists of an irregular grid of time points on which repeated measurements are taken for a number of subjects. We introduce shrinkage estimates for the functional principal component scores that serve as the random effects in the model. Scatterplot smoothing methods are used to estimate the mean function and covariance surface of this model. We propose improved estimation in the neighborhood of and at the diagonal of the covariance surface, where the measurement errors are reflected. The presence of additive measurement errors motivates shrinkage estimates for the functional principal components scores. Shrinkage estimates are developed through best linear prediction and in a generalized version, aiming at minimizing one-curve-leave-out prediction error. The estimation of individual trajectories combines data obtained from that individual as well as all other individuals. We apply our methods to new data regarding the analysis of the level of 14 C -folate in plasma as a function of time since dosing healthy adults with a small tracer dose of 14 C -folic acid. A time transformation was incorporated to handle design irregularity concerning the time points on which the measurements were taken.The proposed methodology incorporating shrinkage and data-adaptive features is seen to be well suited for describing population kinetics of 14 C -folate specific activity and random effects, and can also be applied to other functional data analysis problems.

Longitudinal Functional Principal Component Analysis

“…Our approach can be seen as an extension of functional principal component analysis for multilevel functional data [7]. Our methods apply to longitudinal data where each observation is functional, and should thus not be confused with nonparametric methods for the longitudinal profiles of scalar variables [17,30,31,37,41,48,50,51]. For good introductions to functional data analysis in general, please see [10,34].…”

Estimating Mixture of Gaussian Processes by Kernel Smoothing

“…As depicted in Figure 1, the collected sample of random curves are typically not smooth in practice. Following Yao et al (2003), it is assumed that the observed curve {y i (t), t = t ij , j = 1, ···, N i } is where ε i (t) is additive measurement error, and it is assumed that ε i (t ij ), for all i and j, are independent and identically distributed as N(0, σ 2 ). Denote y ij = y i (t ij ) and ε ij = ε i (t ij ).…”

“…Among many approaches, functional principal component (FPC) analysis serves as a key technique in functional data analysis. Rice and Silverman (1991) and James et al (2000) studied the spline smoothing methods in FPC analysis; Staniswalis and Lee (1998) and Yao et al (2003) applied kernel-based smoothing methods for FPC analysis in irregular and sparse longitudinal data. The asymptotic properties of principal component functions are investigated in Yao et al (2005) and Hall et al (2006).…”

Functional data analysis for volatility

“…Positive definiteness of the corresponding covariance surface can be guaranteed by a projection of the initial estimateĜ on a positive definite versionG, as described in Yao et al (2003). In a last step, the PART algorithm yields estimates of the individual FPC scores.…”