Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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 $p$ 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.


References [Enhancements On Off] (What's this?)

  • 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

Similar Articles

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