Description Length and Dimensionality Reduction in Functional Data Analysis

In this paper we investigate the use of description length principles to select an appropriate number of basis functions for functional data. We provide a flexible definition of the dimension of a random function that is constructed directly from the Karhunen-Loève expansion of the observed process. Our results show that although the classical, principle component variance decomposition technique will behave in a coherent manner, in general, the dimension chosen by this technique will not be consistent. We describe two description length criteria, and prove that they are consistent and that in low noise settings they will identify the true finite dimension of a signal that is embedded in noise. Two examples, one from mass-spectroscopy and the one from climatology, are used to illustrate our ideas. We also explore the application of different forms of the bootstrap for functional data and use these to demonstrate the workings of our theoretical results.