The median in multidimensional spaces

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...

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Bibliographic Details
Published in:Advances in applied mathematics 2004-08, Vol.33 (2), p.366-396
Main Author: Rustad, Alf B
Format: Article
Language:English
Subjects:
Applied sciences
Artificial intelligence
Computer science
control theory
systems
Exact sciences and technology
General topology
Image transformation
Mathematics
Multidimensional median
Pattern recognition. Digital image processing. Computational geometry
Sciences and techniques of general use
Topological measure
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
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Summary: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.
ISSN:0196-8858
1090-2074
DOI:10.1016/j.aam.2003.10.001