Instability of statistical factor analysis

Ellis, Steven

doi:10.1090/S0002-9939-04-07272-7

Instability of statistical factor analysis
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by Steven P. Ellis PDF

Proc. Amer. Math. Soc. 132 (2004), 1805-1822 Request permission

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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