Multivariate Functional Halfspace Depth

Abstract

This article defines and studies a depth for multivariate functional data. By the multivariate nature and by including a weight function, it acknowledges important characteristics of functional data, namely differences in the amount of local amplitude, shape, and phase variation. We study both population and finite sample versions. The multivariate sample of curves may include warping functions, derivatives, and integrals of the original curves for a better overall representation of the functional data via the depth. We present a simulation study and data examples that confirm the good performance of this depth function. Supplementary materials for this article are available online.

KEY WORDS:

• Functional data
• Multivariate data
• Statistical depth
• Time warping

Additional information

Funding

The authors acknowledge the support of KU Leuven grant GOA/12/14 and of the IAP Research Network P7/06 of the Belgian Science Policy. The authors thank B. De Ketelaere for providing the data, and the reviewers for their comments.