Adv Data Anal Classif (2014) 8:321–338 DOI 10.1007/s11634-014-0166-6 REGULAR ARTICLE Simplicial band depth for multivariate functional data Sara López-Pintado · Ying Sun · Juan K. Lin · Marc G. Genton Received: 26 April 2013 / Revised: 26 January 2014 / Accepted: 10 February 2014 / Published online: 5 March 2014 © Springer-Verlag Berlin Heidelberg 2014 Abstract We propose notions of simplicial band depth for multivariate functional data that extend the univariate functional band depth. The proposed simplicial band depths provide simple and natural criteria to measure the centrality of a trajectory within a sample of curves. Based on these depths, a sample of multivariate curves can be ordered from the center outward and order statistics can be defined. Properties of the proposed depths, such as invariance and consistency, can be established. A simulation study shows the robustness of this new definition of depth and the advantages of using a multivariate depth versus the marginal depths for detecting outliers. Real data examples from growth curves and signature data are used to illustrate the performance and usefulness of the proposed depths. Keywords Band depth · Functional boxplot · Functional and image data · Modified band depth · Multivariate · Simplicial Mathematics Subject Classification 62F07 · 62M10 S. López-Pintado Department of Biostatistics, Columbia University, New York, NY 10032, USA e-mail: sl2929@columbia.edu Y. Sun Department of Statistics, The Ohio State University, Columbus, OH 43210, USA e-mail: sunwards@stat.osu.edu J. K. Lin SearchForce, Inc., San Mateo, CA 94403, USA e-mail: juan.k.lin@gmail.com M. G. Genton (B ) CEMSE Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia e-mail: marc.genton@kaust.edu.sa 123