A Comparison of Band-based Approaches to Functional Depth
摘要
Statistical depth functions have become essential tools for analyzing multivariate data, as they provide center-outward ordering of the data and thus measure centrality. Classical depth notions, such as Tukey depth and simplicial depth, are widely applied in various statistical tasks, including outlier detection and data visualization. With the growing demand for analyzing complex data structures, depth functions have been extended to functional data, where observations are treated as functions rather than finite-dimensional vectors. This article focuses on functional depths defined using bands formed by random functions. We study four such depth functions: (i) band depth, (ii) modified band depth, (iii) spherical depth, and (iv) modified spherical depth, the last of which is introduced here for the first time. We compare these notions with an emphasis on theoretical properties and evaluate their practical performance through an application to real-world data.