Fast DD-classification of functional data

TL;DR

This paper introduces a fast, nonparametric classification method for functional data using a two-step transformation and depth-based classifier, achieving efficiency, robustness, and near-optimal accuracy without smoothing.

Contribution

It presents a novel, computationally efficient classification approach for functional data based on depth functions and hypercube transformations, with theoretical and practical validation.

Findings

Method is robust and computationally efficient.

Achieves Bayes optimality under standard settings.

Validated through simulations and benchmark studies.

Abstract

A fast nonparametric procedure for classifying functional data is introduced. It consists of a two-step transformation of the original data plus a classifier operating on a low-dimensional hypercube. The functional data are first mapped into a finite-dimensional location-slope space and then transformed by a multivariate depth function into the $DD$-plot, which is a subset of the unit hypercube. This transformation yields a new notion of depth for functional data. Three alternative depth functions are employed for this, as well as two rules for the final classification on $[0,1]^q$. The resulting classifier has to be cross-validated over a small range of parameters only, which is restricted by a Vapnik-Cervonenkis bound. The entire methodology does not involve smoothing techniques, is completely nonparametric and allows to achieve Bayes optimality under standard distributional settings. It is robust, efficiently computable, and has been implemented in an R environment. Applicability of the new approach is demonstrated by simulations as well as a benchmark study.