Optimal classification of Gaussian processes in homo- and heteroscedastic settings

Abstract

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 to the stochas-

Review on functional data classification

“…J. R. Berrendero et al (2018) provided a formalized theory for binary functional classification, and the system was established by explicitly calculating the Radon‐Nikodym derivatives between Gaussian measures that are absolutely continuous. Torrecilla et al (2020) further extended the homoscedastic setting to a heteroscedastic setting of Gaussian processes, and discussed the optimal discrimination rules more comprehensively. R. J. Berrendero et al (2023) explored the RKHS as an alternative formulation to the ‐based model for functional logistic regression.…”

Optimal Classification for Functional Data

“…. Berrendero et al (2018) and Torrecilla et al (2020) shows that Q * (Z, θ) is well defined and almost surely finite when the probability measures of two classes are equivalent.…”

Optimal Classification for Functional Data

“…is the infinite sequence of mean projection scores and Σ k is a diagonal linear operator from L 2 (T ) to L 2 (T ) satisfying Σ k ψ j = λ (k) j ψ j for j ≥ 1 and k = 1, 2. Given θ, it follows by [4] and [31] that the optimal Bayes classification rule for classifying a new data function Z ∈ L 2 (T ) has an expression…”