On functional logistic regression: some conceptual issues

Abstract

The main ideas behind the classic multivariate logistic regression model make sense when translated to the functional setting, where the explanatory variable X is a function and the response Y is binary. However, some important technical issues appear (or are aggravated with respect to those of the multivariate case) due to the functional nature of the explanatory variable. First, the mere definition of the model can be questioned: While most approaches so far proposed rely on the L^2-based model, we explore an alternative (in some sense, more general) approach, based on the theory of reproducing kernel Hilbert spaces (RKHS). The validity conditions of such RKHS-based model, and their relation with the L^2-based one, are investigated and made explicit in two formal results. Some relevant particular cases are considered as well. Second, we show that, under very general conditions, the maximum likelihood of the logistic model parameters fails to exist in the functional case, although some restricted versions can be considered. Third, we check (in the framework of binary classification) the practical performance of some RKHS-based procedures, well-suited to our model: They are compared to several competing methods via Monte Carlo experiments and the analysis of real data sets.