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

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

Published electronically:
January 7, 2004

MathSciNet review:
2051145

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**1.**Altmann, S. L. (1986)*Rotations, Quaternions, and Double Groups.*Clarendon Press, Oxford. MR**88e:20001****2.**Anderson, T. W. and Rubin, H. (1956) ``Statistical inference in factor analysis,''*Proc. Third Berkeley Sympos. Math. Statist.,*J. Neyman, ed., Univ. of California Press, Berkeley, 111 - 150. MR**18:954f****3.**Boothby, W. M. (1975)*An Introduction to Differentiable Manifolds and Riemannian Geometry.*Academic Press, New York. MR**54:13956****4.**Cayley, A. (1963)*The Collected Mathematical Papers.*Johnson Reprint Corporation, New York.**5.**Dayan, P. and Abbott, L. F. (2001)*Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems.*MIT Press, Cambridge, MA.**6.**Ellis, S. P. (1995) ``Dimension of the singular sets of plane-fitters,''*Annals of Statistics***23**, 490-501. MR**97b:62083****7.**Ellis, S. P. (2002) ``On the instability of factor analysis,'' unpublished manuscript.**8.**Ellis, S. P. (2002) ``Fitting a line to three or four points on a plane,''*SIAM Review***44**, 616-628.**9.**Falconer, K. (1990),*Fractal Geometry: Mathematical Foundations and Applications*. John Wiley & Sons, New York. MR**92j:28008****10.**Golub, G. H. and Van Loan, C. F. (1996)*Matrix Computations, Third Edition.*Johns Hopkins University Press, Baltimore, MD. MR**97g:65006****11.**Greenberg, M. J. and Harper, J. R. (1981)*Algebraic Topology: A First Course.*Addison-Wesley, Reading, MA. MR**83b:55001****12.**Harman, H. H. (1967)*Modern Factor Analysis, Second Edition, Revised.*University of Chicago Press, Chicago. MR**37:4909****13.**Jennrich, R. I. (1973) ``On the stability of rotated factor loadings: The Wexler phenomenon,''*Br. J. Math. Statist. Psychol.***26**, 167-176.**14.**Johnson, R. A. and Wichem, D. W. (1992)*Applied Multivariate Statistical Analysis, Third Edition.*Prentice Hall, Englewood Cliffs, NJ. MR**93c:62103****15.**Lawley, D. N. and Maxwell, A. E. (1971)*Factor Analysis as a Statistical Method.*Butterworth & Co., London. MR**49:8212****16.**Massey, W. S. (1967)*Algebraic Topology: An Introduction.*Harcourt, Brace & World, Inc., New York. MR**35:2271****17.**Milnor, J. W. and Stasheff, J. D. (1974)*Characteristic Classes.*Annals of Mathematics Studies Number 76, Princeton University Press, Princeton, NJ. MR**55:13428****18.**Munkres, J. R. (1984)*Elements of Algebraic Topology.*Benjamin/Cummings, Menlo Park, CA. MR**85m:55001****19.**Psychological Corporation, The (1997)*Wechsler Adult Intelligence Scale - Third Edition, Wechsler Memory Scale - Third Edition: Technical Manual.*Harcourt Brace & Co., New York.**20.**Spanier, E. H. (1966)*Algebraic Topology.*McGraw-Hill, New York. MR**35:1007****21.**Stillwell, J. (2001) ``The story of the 120-cell,''*Notices of Amer. Math. Soc.***48**, 17-24. MR**2001k:52019****22.**Stong, R. E. (1968)*Notes on Cobordism Theory.*Princeton University Press, Princeton, NJ. MR**40:2108**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
62H25,
65D10

Retrieve articles in all journals with MSC (2000): 62H25, 65D10

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