A clustering method for Hilbert functional data based on the Small Ball Probability

In the present work, motivated by the definition of a clustering method for functional data, the small-ball probability (SmBP) of a Hilbert valued process is considered. In particular, asymptotic factorizations for the SmBP are rigorously established exploiting the Karhunen--Lo\'eve expansion whose basis turns out to be the optimal one in controlling the approximation errors. In fact, as the radius of the ball tends to zero, the SmBP is asymptotically proportional to the joint density of an increasing number (with the radius) of principal components (PCs) evaluated at the center of the ball up to a factor depending only on the radius. As a consequence, the joint distribution of the first PCs provides a surrogate density of the process and, hence, in a very natural way, becomes the core in defining a density based unsupervised classification algorithm. To implement the latter, a non parametric estimator for such joint density is introduced and it is proved that used estimated PCs does not affect the rate of convergence. Finally, after a discussion on the proposed clustering algorithm, as an illustration, an application to a real dataset is provided.