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August 1996 Self-consistency: a fundamental concept in statistics
Bernard Flury, Thaddeus Tarpey
Statist. Sci. 11(3): 229-243 (August 1996). DOI: 10.1214/ss/1032280215


The term "self-consistency" was introduced in 1989 by Hastie and Stuetzle to describe the property that each point on a smooth curve or surface is the mean of all points that project orthogonally onto it. We generalize this concept to self-consistent random vectors: a random vector Y is self-consistent for X if $\mathscr{E}[X|Y] = Y$ almost surely. This allows us to construct a unified theoretical basis for principal components, principal curves and surfaces, principal points, principal variables, principal modes of variation and other statistical methods. We provide some general results on self-consistent random variables, give examples, show relationships between the various methods, discuss a related notion of self-consistent estimators and suggest directions for future research.


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Bernard Flury. Thaddeus Tarpey. "Self-consistency: a fundamental concept in statistics." Statist. Sci. 11 (3) 229 - 243, August 1996.


Published: August 1996
First available in Project Euclid: 17 September 2002

zbMATH: 0955.62540
MathSciNet: MR1436648
Digital Object Identifier: 10.1214/ss/1032280215

Keywords: $k$-means algorithm , elliptical distribution , EM algorithm , mean squared error , principal components , Principal curves , principal modes of variation , principal points , principal variables , regression , self-organizing maps , spherical distribution , Voronoi region

Rights: Copyright © 1996 Institute of Mathematical Statistics

Vol.11 • No. 3 • August 1996
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