Statistical depth meets computational geometry: a short survey <s> Spatial depth ([Ser02] <s> Abstract An extension of the concept of quantiles in multidimensions that uses the geometry of multivariate data clouds has been considered. The approach is based on blending as well as generalization of the key ideas used in the construction of spatial median and regression quantiles, both of which have been extensively studied in the literature. These geometric quantiles are potentially useful in constructing trimmed multivariate means as well as many other L estimates of multivariate location, and they lead to a directional notion of central and extreme points in a multidimensional setup. Such quantiles can be defined as meaningful and natural objects even in infinite-dimensional Hilbert and Banach spaces, and they yield an effective generalization of quantile regression in multiresponse linear model problems. Desirable equivariance properties are shown to hold for these multivariate quantiles, and issues related to their computation for data in finite-dimensional spaces are discussed. n 1/2 consistenc... <s> BIB001 </s> Statistical depth meets computational geometry: a short survey <s> Spatial depth ([Ser02] <s> The L1-median is a robust estimator of multivariate location with good statistical properties. Several algorithms for computing the L1-median are available. Problem specific algorithms can be used, but also general optimization routines. The aim is to compare different algorithms with respect to their precision and runtime. This is possible because all considered algorithms have been implemented in a standardized manner in the open source environment R. In most situations, the algorithm based on the optimization routine NLM (non-linear minimization) clearly outperforms other approaches. Its low computation time makes applications for large and high-dimensional data feasible. <s> BIB002
). Spatial depth is related to multivariate quantiles proposed in BIB001 : The spatial median is also called the L 1 median (  ). It has breakdown value 1/2, but is not affine equivariant (it is only equivariant with respect to translations, multiplication by a scalar factor, and orthogonal transformations). For a recent survey on the computation of the spatial median see BIB002 . A comparison of the main properties of the different location depth medians is given in Table 1 . 