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Instability of statistical factor analysis

Author: Steven P. Ellis
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 1805-1822
MSC (2000): Primary 62H25; Secondary 65D10
Published electronically: January 7, 2004
MathSciNet review: 2051145
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Abstract: Factor analysis, a popular method for interpreting multivariate data, models the covariance among $p$ variables as being due to a small number ($k$, $1 \leq k < p$) of hidden variables. A factor analysis of $Y$ can be thought of as an ordered or unordered collection, $F(Y)$, of $k$ linearly independent lines in $\mathbb{R} ^{p}$. Let $\mathcal{Y}'$ be the collection of data sets for which $F(Y)$ is defined. The ``singularities'' of $F$ are those data sets, $Y$, in the closure, $\overline{\mathcal{Y}}'$, at which the limit, $\lim_{Y' \to Y, Y' \in \mathcal{Y}'} F(Y')$, does not exist. $F$ is unstable near its singularities.

Let $\Phi(Y)$ be the direct sum of the lines in $F(Y)$. $\Phi$ determines a $k$-plane bundle, $\eta$, over a subset, $\mathcal{X}$, of $\mathcal{Y}$. If $k > 1$ and $\eta$ is rich enough, $F$ ordered or, at least if $k = 2$ or 3, unordered, must have a singularity at some data set in $\mathcal{X}$. The proofs are applications of algebraic topology. Examples are provided.

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Additional Information

Steven P. Ellis
Affiliation: New York State Psychiatric Institute and Columbia University, Unit 42, NYSPI, 1051 Riverside Dr., New York, New York 10032

Keywords: Vector bundle, maximum likelihood, principal components
Received by editor(s): December 3, 2001
Published electronically: January 7, 2004
Additional Notes: This research is supported in part by United States PHS grants MH46745, MH60995, and MH62185.
Communicated by: Richard A. Davis
Article copyright: © Copyright 2004 American Mathematical Society