Optimal classification of Gaussian processes in homo- and heteroscedastic settings

A procedure to derive optimal discrimination rules is formulated for binary functional classification problems in which the instances available for induction are characterized by random trajectories sampled from different Gaussian processes, depending on the class label. Specifically, these optimal rules are derived as the asymptotic form of the quadratic discriminant for the discretely monitored trajectories in the limit that the set of monitoring points becomes dense in the interval on which the processes are defined. The main goal of this work is to provide a detailed analysis of such optimal rules in the dense monitoring limit, with a particular focus on elucidating the mechanisms by which near-perfect classification arises. In the general case, the quadratic discriminant includes terms that are singular in this limit. If such singularities do not cancel out, one obtains near-perfect classification, which means that the error approaches zero asymptotically, for infinite sample sizes. This singular limit is a consequence of the orthogonality of the probability measures associated with the stochastic processes from which the trajectories are sampled. As a further novel result of this analysis, we formulate rules to determine whether two Gaussian processes are equivalent or mutually singular (orthogonal).