Multivariate functional Mahalanobis distance with application to clustering

U. Radojicic${}^a$, J. Oguamalam${}^a$, T.Masak${}^b$ and P. Filzmoser${}^a$

${}^a$TU Wien, Vienna, Austria, ${}^b$WU Wien, Vienna, Austria

The increasing availability of multivariate functional data across diverse scientific domains highlights the importance of precise analyses of such data structures. Unlike univariate functional data, multivariate functional observations not only exhibit temporal dependence but also between-component correlation. A common simplifying assumption, used to make the estimation of the covariance more tractable, is the separability of the covariance operator, which assumes that the correlation across time and between the components can be uncoupled.

Recently, the $\alpha$-Mahalanobis distance was introduced to the univariate functional setting. Building on this idea, this work introduces the regularized multivariate Mahalanobis distance (RMMD) as an extension of this metric to the multivariate functional case under a separable covariance structure. The incorporation of an appropriately chosen regularization operator assures that the RMMD of a multivariate stochastic process can be calculated as the sum of univariate $\alpha$-Mahalanobis distances of its scaled and decorrelated components.

We demonstrate the utility of the RMMD in the context of distance-based clustering of multivariate functional data.
Keywords: multivariate functional data, separable covariance, distance-based clustering