Multidimensional medians arising from geodesics on graphs
type: Item
instance of: scholarly article
title: Multidimensional medians arising from geodesics on graphs
zbMATH Open document ID: 0923.62070
author: Christopher G. Small
published in: The Annals of Statistics
publication date: 1999-10-31T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 62H99
Mathematics Subject Classification ID: 05C99
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 05C90
zbMATH DE Number: 1029329
zbMATH Keywords: minimal spanning tree
zbMATH Keywords: medians
zbMATH Keywords: geodesic convexity medians
zbMATH Keywords: permutation
reviewed by: Raúl P. Mentz
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1031833660
OpenAlex ID: W2056388259
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DOI: 10.1214/AOS/1031833660
description: scientific article; zbMATH DE number 1029329