Approximation of estimators in the PCA of a stochastic process using B-splines

The objective of this paper is to estimate the principal factors of a continuous time real valued process when we have a collection of independent sample functions which are observed only at discrete time points. We propose to approximate the Principal Component Analysis (PCA) of the process, when the sample functions are regular, by means of the PCA of the natural cubic spline interpolation of the sample curves between the sampling time points. A physical application testing the accuracy of this approach by simulating sample functions of the harmonic oscillator stochastic process is also included. The approximated PCA of this well known process is compared with the exact one and with the classical PCA of the discrete time simulated data.