Depth-based classification for functional data

Data depth is a modern nonparametric tool for the analysis of multivariate, and recently also functional and general Banach-valued data. The notion of halfspace depth was introduced by Tukey (1975) as a powerful tool for the picturing of multivariate data and exploratory analysis. Two decades later the data depth has start to be developed as a general nonparametric tool for multivariate data. Different depth function were since then introduced; let us mention the simplicial depth, zonoids, L1-depth, projection depth among others. In (Zuo and Serfling, 2000) the statistical depth function was introduced as a function D : SP(S) → R+(→ [0, 1]) satisfying certain general assumptions. Here S is the sample space and P(S) denotes the set of all probability distributions on S. Depth function introduces a linear (semi-)ordering in inward-outward sense to the multivariate data. The higher the depth is the more “inner” with respect to the distribution the point is. Recently the data depth notion was proposed for the functional data as well. One of the considered applications of functional data depth is the classification of functional observations. There are several depth based methods of classification, e.g., the simplest maximum depth classifier or DD-plot based classifier. In our presentation we shall show that even the best classifier will perform poorly if the depth function is not well chosen. In other words, it is depth function what matters. Namely we will compare the depth functions proposed by Fraiman and Muniz (2001] and by López-Pintado and Romo (2007,2009) with the so called K-band depth. Let us review the definitions of the three depth functions: