PCA - Based Dimension Reduction for Splines

PCA in a reproducing kernel Hilbert space is analysed as probabilistically optimal procedure of dimension reduction given a covariance structure by the reproducing kernel. It provides a unifying framework for various seemingly disparate and special techniques of dimension reduction applied to splines, in geostatistical “kriging” or in interpolation of data resulting from computer experiments. Regarding the covariance as de...ning a prior for Bayesian analyses in a Hilbert function space several suggestions are derived for data analyses involving functions particularly on multivariate domains, including the choice of a parsimonious interpolation spline as regression function in generalized models, the use of a Demmler-Reinsch-like basis in kriging or interpolation and derivation of principal modes of variation in collections of surfaces.