2007/05/14 by Chen, Dan
Decision Sciences · Mathematics · #Advanced Statistical Methods and Models #Computational Geometry (cs.CG) #FOS: Computer and information sciences #Optimal Experimental Design Methods #Probabilistic and Robust Engineering Design
paper · pdf · doi:10.48550/arxiv.0705.1956
openalex publication_date 2007/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The concept of data depth in non-parametric multivariate descriptive statistics is the generalization of the univariate rank method to multivariate data. Halfspace depth is a measure of data depth. Given a set S of points and a point p, the halfspace depth (or rank) k of p is defined as the minimum number of points of S contained in any closed halfspace with p on its boundary. Computing halfspace depth is NP-hard, and it is equivalent to the Maximum Feasible Subsystem problem. In this thesis a mixed integer program is formulated with the big-M method for the halfspace depth problem. We suggest a branch and cut algorithm. In this algorithm, Chinneck's heuristic algorithm is used to find an upper bound and a related technique based on sensitivity analysis is used for branching. Irreducible Infeasible Subsystem (IIS) hitting set cuts are applied. We also suggest a binary search algorithm which may be more stable numerically. The algorithms are implemented with the BCP framework from the COIN-OR project.