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

Abstract

In this paper, 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 approach this problem through a maximum likelihood method assuming that the covariance kernel is smooth and finite dimensional. We exploit the smoothness of the eigenfunctions to reduce dimensionality by restricting them to a lower dimensional space of smooth functions. The estimation scheme is developed based on a Newton-Raphson procedure using the fact that the basis coefficients representing the eigenfunctions lie on a Stiefel manifold. We also address the selection of the right 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 data set. 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 of the empirical covariances, and another using an EM algorithm.

Archetypoid analysis for sports analytics

“…We use the geometric approach to obtain the maximum likelihood estimation of the functional principal components from sparse functional data proposed by Peng and Paul (2009), which outperforms other estimation procedures (James et al, 2000;Yao and Müller, 2005) and which also incorporates information from all the curves. In Peng and Paul (2009), a model selection procedure based on the minimization of 8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64an approximate cross-validation (CV) score was also proposed for choosing both m and the number of basis functions M . These M functions are cubic B-splines with equally spaced knots, and are used in the model to represent the eigenfunctions.…”

“…Following Peng and Paul (2009), we have chosen to use m = 4 eigenfunctions with M = 5 basis functions for representing the eigenfunctions, as it gives the smallest approximate CV score. The estimated eigenvalues are: 3.127180e + 02, 1.989077e + 02, 8.946427e + 01 and 3.68e − 13.…”

“…Figures 3 and 4 show the mean function and the first four eigenfunctions. Although we do not present functional variance in the plots, it can be computed using Equation 6 in Peng and Paul (2009), which gives the projected covariance kernel and which is implemented in the accompanying software. ADA is applied to the matrix of functional principal component scores.…”

On the Pareto-Optimal Beam Structure and Design for Multi-User MIMO Interference Channels

“…By exploiting the product structure of the parameter space, the optimization problem (24) can be solved by an alternating technique again in a similar way to the technique in [31]. That is, first we fix and update by the steepest descent or Newton method on the Stiefel manifold [21], [26].…”

“…the elements of , i.e., . The gradient of at is given by (31) Hessian: [21] For a general Riemannian manifold , the Hessian operator of a smooth function at a point is defined as a linear operator: with for all , where is the LeviCivita connection on . Just as in the Euclidean case, a smooth function on admits Taylor expansion.…”