Using Principal Components from General Hilbert-Space-Valued Covariates for Regression

Abstract: We consider linear regression with covariates that are separable random elements in a general Hilbert space. We first develop a principal component analysis for Hilbert-space-valued covariates based on finite-dimensional projections of the empirical covariance operator and establish asymptotic linearity and joint Gaussian limits for the leading eigenvalues and eigenfunctions under mild moment conditions. We then propose a principal component regression framework that combines Euclidean and Hilbert-space-valued covariates, obtain root-n consistency and asymptotic normality of the regression parameter estimators, and establish the validity of nonparametric and wild bootstrap procedures for inference. Simulation studies with two- and three-dimensional medical imaging predictors demonstrate accurate recovery of eigenstructures, regression coefficients, and bootstrap coverage. The methodology is further illustrated with neuroimaging data.