Computational considerations in functional principal component analysis

Computing estimates in functional principal component analysis (FPCA) from discrete data is usually based on the approximation of sample curves in terms of a basis (splines, wavelets, trigonometric functions, etc.) and a geometrical structure in the data space (L ² spaces, Sobolev spaces, etc.). Until now, the computational efforts have been focused in developing ad hoc algorithms to approximate those estimates by previously selecting an efficient approximating technique and a convenient geometrical structure. The main goal of this paper consists of establishing a procedure to formulate the algorithm for computing estimates of FPCA under general settings. The resulting algorithm is based on the classic multivariate PCA of a certain random vector and can thus be implemented in the majority of statistical packages. In fact, it is derived from the analysis of the effects of modifying the norm in the space of coordinates. Finally, an application on real data will be developed to illustrate the so derived theoretic results. This research has been supported by Project MTM2004-5992 from Dirección General de Investigación, Ministerio de Ciencia y Tecnología.