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 Free Access

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.**Simon L. Altmann,*Rotations, quaternions, and double groups*, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1986. MR**868858****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.**William M. Boothby,*An introduction to differentiable manifolds and Riemannian geometry*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, No. 63. MR**0426007****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.**Steven P. Ellis,*Dimension of the singular sets of plane-fitters*, Ann. Statist.**23**(1995), no. 2, 490–501. MR**1332578**, https://doi.org/10.1214/aos/1176324532**7.**Ellis, S. P. (2002) ``On the instability of factor analysis,'' unpublished manuscript.**8.**Ellis, S. P. (2002) ``Fitting a line to three or four points on a plane,''*SIAM Review***44**, 616-628.**9.**Kenneth Falconer,*Fractal geometry*, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR**1102677****10.**Gene H. Golub and Charles F. Van Loan,*Matrix computations*, 3rd ed., Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 1996. MR**1417720****11.**Marvin J. Greenberg and John R. Harper,*Algebraic topology*, Mathematics Lecture Note Series, vol. 58, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1981. A first course. MR**643101****12.**Harry H. Harman,*Modern factor analysis*, Second edition, revised, The University of Chicago Press, Chicago, Ill.-London, 1967. MR**0229335****13.**Jennrich, R. I. (1973) ``On the stability of rotated factor loadings: The Wexler phenomenon,''*Br. J. Math. Statist. Psychol.***26**, 167-176.**14.**Richard A. Johnson and Dean W. Wichern,*Applied multivariate statistical analysis*, 3rd ed., Prentice Hall, Inc., Englewood Cliffs, NJ, 1992. MR**1168210****15.**D. N. Lawley and A. E. Maxwell,*Factor analysis as a statistical method*, 2nd ed., American Elsevier Publishing Co., Inc., New York, 1971. MR**0343471****16.**William S. Massey,*Algebraic topology: An introduction*, Harcourt, Brace & World, Inc., New York, 1967. MR**0211390****17.**John W. Milnor and James D. Stasheff,*Characteristic classes*, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 76. MR**0440554****18.**James R. Munkres,*Elements of algebraic topology*, Addison-Wesley Publishing Company, Menlo Park, CA, 1984. MR**755006****19.**Psychological Corporation, The (1997)*Wechsler Adult Intelligence Scale - Third Edition, Wechsler Memory Scale - Third Edition: Technical Manual.*Harcourt Brace & Co., New York.**20.**Edwin H. Spanier,*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210112****21.**John Stillwell,*The story of the 120-cell*, Notices Amer. Math. Soc.**48**(2001), no. 1, 17–24. MR**1798928****22.**Robert E. Stong,*Notes on cobordism theory*, Mathematical notes, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR**0248858**

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