Some Insights About the Small Ball Probability Factorization for Hilbert Random Elements

Abstract

Asymptotic factorizations for the small-ball probability (SmBP) of a Hilbert valued random element X are rigorously established and discussed. In particular, given the first d principal components (PCs) and as the radius ε of the ball tends to zero, the SmBP is asymptotically proportional to (a) the joint density of the first d PCs, (b) the volume of the d-dimensional ball with radius ε, and (c) a correction factor weighting the use of a truncated version of the process expansion. Moreover, under suitable assumptions on the spectrum of the covariance operator of X and as d diverges to infinity when ε vanishes, some simplifications occur. In particular, the SmBP factorizes asymptotically as the product of the joint density of the first d PCs and a pure volume parameter. All the provided factorizations allow to define a surrogate intensity of the SmBP that, in some cases, leads to a genuine intensity. To operationalize the stated results, a non-parametric estimator for the surrogate intensity is introduced and it is proved that the use of estimated PCs, instead of the true ones, does not affect the rate of convergence. Finally, as an illustration, simulations in controlled frameworks are provided.

Classification methods for Hilbert data based on surrogate density

“…In this paper, an unsupervised and a supervised classification method based on the concept of SmBP mixture for Hilbert-valued process have been introduced and analyzed. The novelty lies in the use of the theoretical factorization of the SmBP due to Bongiorno and Goia (2015) and reported in Proposition 1. Such a result introduces a surrogate-density for Hilbert-valued processes that, on the one hand, endorses a "density oriented" clustering approach for detecting the latent structure by incorporating the information on the mixture, and, on the other one, leads to define an optimal Bayes classifier in a supervised classification (discriminant) context.…”

“…The latter expression is the starting point for approaching model-based classification problems: when Y is a latent variable, we deal with an unsupervised classification problem focused on the left-hand side of (1), see Section 2.1; whereas, when Y is observed, the model leads to the construction of a Bayesian classifier whose starting point is the right-hand side of (1), see Section 2.2. In both cases, instead of tackling it directly, we want to simplify it by exploiting an approximation result sketched below (for more details see Bongiorno and Goia, 2015). For the sake of simplicity, it is presented with respect to the process X; however, the same arguments can be applied to (X|Y = g) with g = 1, .…”

“…In fact, from a theoretical point of view, one may wonder if such estimator is consistent for f d and, when this is the case, if it attains the same rate of convergence that holds when Π d is known. A positive answer was provided in Bongiorno and Goia (2015); in particular, consider the special case H n = h 2 n I, and suppose that:…”

“…For a fixed d, consistency properties for estimated modes have been considered, for instance, in Abraham et al (2003), Chen et al (2015) and Sager (1979). Nevertheless, such theoretical issues are little studied in the infinite dimensional framework, since the mathematical concept of the density is still under-developed; some attempts can be found in Gasser et al (1998) and, more recently, in Bongiorno and Goia (2015), Dabo-Niang et al (2007) and Delaigle and Hall (2010).…”

“…by the eigenfunctions of the principal components analysis), and the structure of the pseudo-density (linked to the principal components, i.e. the coefficients of the decomposition), are discussed in Delaigle and Hall (2010) and then in Bongiorno and Goia (2015), where some assumptions are relaxed. One can observe that this approach allows to see the projective approaches in a more general theoretical context.…”