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Two frameworks for multivariate functional depth based on multivariate depthsare introduced in this paper. The first framework is multivariate functionalintegrated depth, and the second framework involves multivariate functionalextremal depth, which is an extension of the extremal depth for univariatefunctional data. In each framework, global and local multivariate functionaldepths are proposed. The properties of population multivariate functionaldepths and consistency of finite sample depths to their population versions areestablished. In addition, finite sample depths under irregularly observed timegrids are estimated. As a by-product, the simplified sparse functional boxplotand simplified intensity sparse functional boxplot are proposed forvisualization without data reconstruction. A simulation study demonstrates theadvantages of global multivariate functional depths over local multivariatefunctional depths in outlier detection and running time for big functionaldata. An application of our frameworks to cyclone tracks data demonstrates theexcellent performance of our global multivariate functional depths.

Global Depths for Irregularly Observed Multivariate Functional Data