The Mahalanobis distance for functional data with applications to classification

TL;DR

This paper introduces a new Mahalanobis semi-distance for functional data, extending classical concepts to curves generated by stochastic processes, and demonstrates its effectiveness in functional classification tasks.

Contribution

A novel semi-distance for functional data based on a regularized inverse operator, enabling improved classification methods for curve data.

Findings

Enhanced classification accuracy with the Mahalanobis-based methods

Positive results in Monte Carlo simulations

Successful application to real data examples

Abstract

This paper presents a general notion of Mahalanobis distance for functional data that extends the classical multivariate concept to situations where the observed data are points belonging to curves generated by a stochastic process. More precisely, a new semi-distance for functional observations that generalize the usual Mahalanobis distance for multivariate datasets is introduced. For that, the development uses a regularized square root inverse operator in Hilbert spaces. Some of the main characteristics of the functional Mahalanobis semi-distance are shown. Afterwards, new versions of several well known functional classification procedures are developed using the Mahalanobis distance for functional data as a measure of proximity between functional observations. The performance of several well known functional classification procedures are compared with those methods used in conjunction with the Mahalanobis distance for functional data, with positive results, through a Monte Carlo study and the analysis of two real data examples.