A Geometric Approach to Maximum Likelihood Estimation of the Functional Principal Components From Sparse Longitudinal Data

By Jie Peng, Debashis Paul. 2007

128 citations recorded by the source corpus.

In this article, we consider the problem of estimating the eigenvalues and eigenfunctions of the covariance kernel (i.e., the functional principal components) from sparse and irregularly observed longitudinal data. We exploit the smoothness of the eigenfunctions to reduce dimensionality by restricting them to a lower dimensional space of smooth functions. We then approach this problem through a restricted maximum likelihood method. The estimation scheme is based on a Newton–Raphson procedure on the Stiefel manifold using the fact that the basis coefficient matrix for representing the eigenfunctions has orthonormal columns. We also address the selection of the number of basis functions, as well as that of the dimension of the covariance kernel by a second-order approximation to the leave-one-curve-out cross-validation score that is computationally very efficient. The effectiveness of our procedure is demonstrated by simulation studies and an application to a CD4+ counts dataset. In the simulation studies, our method performs well on both estimation and model selection. It also outperforms two existing approaches: one based on a local polynomial smoothing, and another using an EM algorithm. Supplementary materials including technical details, the R package fpca, and data analyzed by this article are available online.

Subfields: Environmental Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/4984b833dddea791eef0852ca1f24ce59cafeca1

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Environmental Engineering

In Statistics and Probability

Citations in the corpus

A general framework for functional regression modelling (2017, background): “…random intercepts and scalar covariates, alternatives exist—some of these for generalized functional responses—that directly estimate the FPCs under orthonormality constraints within one overall model (e.g., James et al., 2000; Van der Linde, 2009; Peng and Paul, 2012; Goldsmith et al., 2015).”
CONSISTENCY OF RESTRICTED MAXIMUM LIKELIHOOD ESTIMATORS OF PRINCIPAL COMPONENTS (2008, methodology): “Throughout this paper, we assume that σ2 is known, even though Peng and Paul (2007) provide estimate of σ2 as well.”
COVARIATE ADJUSTED FUNCTIONAL PRINCIPAL COMPONENTS ANALYSIS FOR LONGITUDINAL DATA (2010, methodology): “…is the uFPCA in Yao, Müller and Wang (2005), the second is the reduced rank approach in James, Hastie and Suger (2000), termed rFPCA with “r” stands for reduced rank, and the third is a geometric approach in Peng and Paul (2009) similar to the reduced rank method but with a different algorithm.”