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 variables as being due to a small number (, ) of hidden variables. A factor analysis of can be thought of as an ordered or unordered collection, , of linearly independent lines in . Let be the collection of data sets for which is defined. The ``singularities'' of are those data sets, , in the closure, , at which the limit, , does not exist. is unstable near its singularities.

Let be the direct sum of the lines in . determines a -plane bundle, , over a subset, , of . If and is rich enough, ordered or, at least if or 3, unordered, must have a singularity at some data set in . 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

Email:
ellis@neuron.cpmc.columbia.edu

DOI:
https://doi.org/10.1090/S0002-9939-04-07272-7

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