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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 | 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.


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  • 1. Simon L. Altmann, Rotations, quaternions, and double groups, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1986. MR 868858
  • 2. T. W. Anderson and Herman Rubin, Statistical inference in factor analysis, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. V, University of California Press, Berkeley and Los Angeles, 1956, pp. 111–150. MR 0084943
  • 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, 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

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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