Exact fast computation of band depth for large functional datasets: How quickly can one million curves be ranked?

Band depth is an important nonparametric measure that generalizes order statistics and makes univariate methods based on order statistics possible for functional data. However, the computational burden of band depth limits its applicability when large functional or image datasets are considered. This paper proposes an exact fast method to speed up the band depth computation when bands are defined by two curves. Remarkable computational gains are demonstrated through simulation studies comparing our proposal with the original computation and one existing approximate method. For example, we report an experiment where our method can rank one million curves, evaluated at fifty time points each, in 12.4 seconds with Matlab.