Functional independent component analysis by choice of norm: a framework for near-perfect classification

Abstract

We develop a theory for functional independent component analysis in an infinite dimensional framework using Sobolev spaces that accommodate smoother functions. The notion of penalized kurtosis is introduced motivated by Silverman’s method for smoothing principal components. This approach allows for a classical definition of in dependent components obtained via projection onto the eigenfunctions of a smoothed kurtosis operator mapping a whitened functional random variable. We discuss the theoretical properties of this operator in relation to a generalized Fisher discriminant function and the relationship it entails with the Feldman-Hajek dichotomy for ´ Gaussian measures, both of which are critical to the principles of functional classification. The proposed estimators are a particularly competitive alternative in binary classification of functional data and can eventually achieve the so-called near-perfect classification, which is a genuine phenomenon of high-dimensional data. Our methods are illustrated through simulations, various real datasets, and used to model electroencephalographic biomarkers for the diagnosis of depressive disorder.