The median in multidimensional spaces

Abstract

The multidimensional sample median is defined with respect to a finite collection of variables. The multidimensional median is defined for regular Borel measures. Both concepts are shown to yield topological measures. They are shown to be the only possible definitions being invariant under monotone maps. The construction is done by an image transformation. The non-linear behaviour of various medians and sample medians subject to multivariate observations is investigated through the corresponding topological measures. Continuity properties of the medians are presented. Several applications are introduced.  2004 Elsevier Inc. All rights reserved.

Approximation of quasi-states on manifolds

“…Quasi-states had long remained a largely theoretical topic, but this is starting to change. For instance, in [22] this concept is applied to the definition of a multidimensional median. See also the references therein.…”

Topological measures, image transformations and self-similar sets

“…We have the following definition and proposition from [13]: Definition 9. Let X 1 and X 2 be compact Hausdorff spaces.…”