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Algebraic factor analysis: tetrads, pentads and beyond
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  • Published: 15 November 2006

Algebraic factor analysis: tetrads, pentads and beyond

  • Mathias Drton1,
  • Bernd Sturmfels2 &
  • Seth Sullivant3 

Probability Theory and Related Fields volume 138, pages 463–493 (2007)Cite this article

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  • 47 Citations

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Abstract

Factor analysis refers to a statistical model in which observed variables are conditionally independent given fewer hidden variables, known as factors, and all the random variables follow a multivariate normal distribution. The parameter space of a factor analysis model is a subset of the cone of positive definite matrices. This parameter space is studied from the perspective of computational algebraic geometry. Gröbner bases and resultants are applied to compute the ideal of all polynomial functions that vanish on the parameter space. These polynomials, known as model invariants, arise from rank conditions on a symmetric matrix under elimination of the diagonal entries of the matrix. Besides revealing the geometry of the factor analysis model, the model invariants also furnish useful statistics for testing goodness-of-fit.

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Author information

Authors and Affiliations

  1. Department of Statistics, University of Chicago, Chicago, IL, 60637, USA

    Mathias Drton

  2. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    Bernd Sturmfels

  3. Society of Fellows and Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Seth Sullivant

Authors
  1. Mathias Drton
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  2. Bernd Sturmfels
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  3. Seth Sullivant
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Corresponding author

Correspondence to Mathias Drton.

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Drton, M., Sturmfels, B. & Sullivant, S. Algebraic factor analysis: tetrads, pentads and beyond. Probab. Theory Relat. Fields 138, 463–493 (2007). https://doi.org/10.1007/s00440-006-0033-2

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  • Received: 20 September 2005

  • Revised: 07 October 2006

  • Published: 15 November 2006

  • Issue Date: July 2007

  • DOI: https://doi.org/10.1007/s00440-006-0033-2

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Keywords

  • Covariance Matrix
  • Pentad
  • Model Invariant
  • Diagonal Entry
  • Hide Variable
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