The Mahalanobis distance for functional data with applications to classification
This work addresses classification challenges for functional data, offering an incremental extension of a classical multivariate concept to this domain.
The paper tackles the problem of extending the Mahalanobis distance to functional data, introducing a new semi-distance using a regularized square root inverse operator in Hilbert spaces, and shows positive results in classification through Monte Carlo studies and real data examples.
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.