Functional Data Analysis: Smooth Covariance Estimation and Principal Components Analysis

The collection of functional data has significantly increased due to the rapid advancements in wearable devices, remote sensing, and the Internet of Things. Unlike traditional data forms, functional data are inherently infinite-dimensional, consisting of curves, surfaces, or any data viewed over a continuum. This thesis focuses on enhancing Functional Data Analysis through three novel methodologies addressing covariance function estimation, functional principal component analysis, and multivariate functional principal component analysis, with a particular emphasis on challenges presented by sparsely observed data typical in longitudinal studies. The first contribution introduces two novel methods for estimating the smooth covariance function. The first method provides a low-rank approximation of the covariance matrix, facilitating dimension reduction and computational efficiency. The second method ensures a positive definite covariance matrix by employing a Cholesky decomposition, offering stable and interpretable results. Both methods approximate the covariance using a basis function expansion and employ penalized regression to estimate the basis function coefficients, ensuring smoothness in the resulting covariance estimates. The second contribution proposes a novel method for extracting functional principal components from sparse functional data, and it details a technique for selecting the optimal number of basis functions and principal components. Employing conditional estimation, we effectively estimate the principal scores and recover the underlying trajectory across the entire domain from the sparse functional data. Our approach maintains orthogonality in the eigenfunctions and ensures positive eigenvalues, thereby guaranteeing a low-rank positive semi-definite covariance function.

The third contribution introduces a novel method to extract multivariate functional principal components from multivariate sparse functional data, capturing correlations across multiple observations through an eigen-decomposition of the combined univariate principal component scores. Utilizing the relationship between the univariate and multivariate functional principal components, we propose a simple estimation strategy for obtaining multivariate functional principal components based on their univariate counterparts.

Through simulation studies, we comprehensively compare our methods to existing approaches, demonstrating significant improvements. To demonstrate the practical applicability of these methodologies, the thesis applies them to data from the VistaMilk research center, which includes both densely sampled and sparsely observed functional data related to the milking process of dairy cows.