On data depth in infinite dimensional spaces

The concept of data depth leads to a center-outward ordering of multivariate data, and it has been effectively used for developing various data analytic tools. While different notions of depth were originally developed for finite dimensional data, there have been some recent attempts to develop dept...

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Bibliographic Details
Published in:Annals of the Institute of Statistical Mathematics Vol. 66; no. 2; pp. 303 - 324
Main Authors: Chakraborty, Anirvan, Chaudhuri, Probal
Format: Journal Article
Language:English
Published: Tokyo Springer Japan 01.04.2014
Springer Nature B.V
Subjects:
Brownian motion
Economics
Finance
Gene expression
Insurance
Management
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Order disorder
Probability distribution
Random variables
Regression analysis
Statistical analysis
Statistics
Statistics for Business
Stochasticity
Studies
Mixing sequences
Band depth
Half-region depth
Projection depth
Half-space depth
Spatial depth
Fractional Brownian motions
Geometric Brownian motions
Integrated data depth
ISSN:0020-3157, 1572-9052
Online Access:Get full text
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Summary:The concept of data depth leads to a center-outward ordering of multivariate data, and it has been effectively used for developing various data analytic tools. While different notions of depth were originally developed for finite dimensional data, there have been some recent attempts to develop depth functions for data in infinite dimensional spaces. In this paper, we consider some notions of depth in infinite dimensional spaces and study their properties under various stochastic models. Our analysis shows that some of the depth functions available in the literature have degenerate behaviour for some commonly used probability distributions in infinite dimensional spaces of sequences and functions. As a consequence, they are not very useful for the analysis of data satisfying such infinite dimensional probability models. However, some modified versions of those depth functions as well as an infinite dimensional extension of the spatial depth do not suffer from such degeneracy and can be conveniently used for analyzing infinite dimensional data.
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ISSN:0020-3157
1572-9052
DOI:10.1007/s10463-013-0416-y