Data Depth: Center-Outward Ordering of Multivariate Data and Nonparametric Multivariate Statistics

Abstract

This chapter discusses some examples, including depth-based multivariate descriptive statistics, multivariate rank tests, DD-plots for testing in one- and two-sample problems, and multivariate process control. A data depth is a measure of depth or centrality of a given point with respect to a multivariate distribution. It gives rise to a new set of parameters, which can easily quantify the many complex multivariate features of the underlying distribution. These parameters can be expressed by simple one-dimensional graphs that facilitate greatly the visualization and interpretation of multivariate features. Furthermore, a data depth also provides a natural center-outward ordering of data points in a given sample. This ordering leads to a systematic nonparametric multivariate inference scheme and many practical applications. The depth contours shows an almost symmetric nested expansion which reflects the symmetry of the underlying normal distribution. The contours fan out upright and they reflect the asymmetric probabilistic geometry of the underlying exponential distribution. The depth-based descriptive statistics are also elaborated.