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.
Publié le : 1996-09-14
Classification:
Elliptical distribution,
EM algorithm,
$k$-means algorithm,
mean squared error,
principal components,
principal curves,
principal modes of variation,
principal points,
principal variables,
regression,
self-organizing maps,
spherical distribution,
Voronoi region
@article{1032280215,
author = {Tarpey, Thaddeus and Flury, Bernard},
title = {Self-consistency: a fundamental concept in statistics},
journal = {Statist. Sci.},
volume = {11},
number = {1},
year = {1996},
pages = { 229-243},
language = {en},
url = {http://dml.mathdoc.fr/item/1032280215}
}
Tarpey, Thaddeus; Flury, Bernard. Self-consistency: a fundamental concept in statistics. Statist. Sci., Tome 11 (1996) no. 1, pp. 229-243. http://gdmltest.u-ga.fr/item/1032280215/