Tukey depth: linear programming and applications

Abstract: Determining the representativeness of a point within a data cloud has recently become a desirable task in multivariate analysis. The concept of statistical depth function, which reflects centrality of an arbitrary point, appears to be useful and has been studied intensively during the last decades. Here the issue of exact computation of the classical Tukey data depth is addressed. The paper suggests an algorithm that exploits connection between the Tukey depth and linear separability and is based on iterative application of linear programming. The algorithm further develops the idea of the cone segmentation of the Euclidean space and allows for efficient implementation due to the special search structure. The presentation is complemented by relationship to similar concepts and examples of application.