The Mahalanobis distance for functional data with applications to classification

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 regu-larized 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 Ma-halanobis distance for functional data as a measure of proximity between functional observations. The performance of several well known functional classification proce-dures 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