Algebraic Factor Analysis: Tetrads, Pentads and Beyond

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.

31 pages, 1 figure

References in corpus (1)

• Higher Lawrence configurations

Cited by in corpus (14)

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• Set-theoretic defining equations of the tangential variety of the Segre variety
• Semidefinite tests for latent causal structures
• Rejoinder: Latent variable graphical model selection via convex optimization
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• Moment Varieties for Mixtures of Products
• Dimensions of Higher Order Factor Analysis Models
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