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Product formalisms for measures on spaces with binary tree structures: representation, visualization, and multiscale noise

Part of:Measure and integrationStatistics

Published online by Cambridge University Press:  13 November 2020

Devasis Bassu,
Peter W. Jones,
Linda Ness and
David Shallcross
Devasis Bassu
Affiliation:
Blackboard Insurance, Bedminster, NJ, US; E-mail: devasis_bassu@hotmail.com
Peter W. Jones
Affiliation:
Yale University, New Haven, CN, US; E-mail: peterwjones@comcast.net
Linda Ness
Affiliation:
Rutgers University, New Brunswick, NJ, US; E-mail: nesslinda@gmail.com
David Shallcross
Affiliation:
Perspecta Labs, Basking Ridge, NJ, US; E-mail: david.shallcross@perspecta.com

Abstract

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In this paper, we present a theoretical foundation for a representation of a data set as a measure in a very large hierarchically parametrized family of positive measures, whose parameters can be computed explicitly (rather than estimated by optimization), and illustrate its applicability to a wide range of data types. The preprocessing step then consists of representing data sets as simple measures. The theoretical foundation consists of a dyadic product formula representation lemma, and a visualization theorem. We also define an additive multiscale noise model that can be used to sample from dyadic measures and a more general multiplicative multiscale noise model that can be used to perturb continuous functions, Borel measures, and dyadic measures. The first two results are based on theorems in [15, 3, 1]. The representation uses the very simple concept of a dyadic tree and hence is widely applicable, easily understood, and easily computed. Since the data sample is represented as a measure, subsequent analysis can exploit statistical and measure theoretic concepts and theories. Because the representation uses the very simple concept of a dyadic tree defined on the universe of a data set, and the parameters are simply and explicitly computable and easily interpretable and visualizable, we hope that this approach will be broadly useful to mathematicians, statisticians, and computer scientists who are intrigued by or involved in data science, including its mathematical foundations.

Keywords

measuresdyadic setsquasi-conformal mappingsweightsmultiscale noise

MSC classification

Primary:28-02: Research exposition (monographs, survey articles)
Secondary:62-07: Data analysis

Information

Type
Applied Analysis
Information
Forum of Mathematics, Sigma , Volume 8 , 2020, e47
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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