Data depth and optimization

In this short article, we consider the notion of data depth which generalizes the me-ian to higher dimensions. Our main objective is to present a snapshot of the data depth, several closely related notions, associated optimization problems and algorithms. In particular, we briefly touch on our recent approaches to compute the data depth using linear and integer optimization programming. Although the problem is NP-hard, there are ways to compute nontrivial lower and upper bounds of the depth. The notion of data depth has been studied independently in statistics, discrete geometry, political science and optimization. The motivation and the necessity in statistics to generalize the median and the rank is very natural, as the mean is not considered to be a robust measure of central location. It is enough to place one outlier to change the mean. In contrast, the median in one dimension is very robust as half of the observations need to be changed to corrupt the value of the median. In nonparametric statistics, several data depth measures were introduced as multivariate generalizations of ranks to complement classical multivariate analysis, first by Tuke