Multidimensional medians arising from geodesics on graphs
In dimension 1 the median is characterized by a variety of equivalent definitions. However, in higher dimensions, these definitions cease to be equivalent. The characterization of the median in dimension 1 as the deepest point in the distribution was generalized by \textit{J. W. Tukey} [Proc. Int. Congr. Math., Vancouver 1974, Vol. 2, 523-531 (1975)] to what is usually now called the Tukey depth median. In this paper the author introduces a family of medians applicable to distributions and data on graphs. They are called geodesic convexity medians (GCM), and are based on a geodesic convexity depth. This median will be applicable to data sets in \(\mathbb{R}^p\), and to data and distributions which lie naturally on a graph. As a special case of this, permutation medians for the analysis of rank or permutation data are considered. The graph median is seen to be an extension of the univariate median, which arises as the linear graph associated with the ordering of the data.
scientific article; zbMATH DE number 1029329
scientific article