sfa_ekc: Empirical Kaiser Criterion for Embedding Similarity Matrices

In semanticfa: Semantic Factor Analysis of Language Model Embeddings

Empirical Kaiser Criterion for Embedding Similarity Matrices

Description

Applies the empirical Kaiser criterion (EKC; Braeken & van Assen, 2017) to an embedding similarity matrix, with the embedding dimension playing the role of the sample size.

Usage

sfa_ekc(sim_matrix, embeddings = NULL, n = NULL)

Arguments

sim_matrix	Numeric similarity matrix (n_items x n_items), or a fitted "sfa" object.
embeddings	Numeric embedding matrix (n_items x embedding_dim), used only for its column count. Optional if n is given.
n	Sample size to use in place of ncol(embeddings): the embedding dimension for similarity matrices, or the respondent count when applying the criterion to a conventional correlation matrix.

Details

The EKC replaces Kaiser's fixed threshold of one with a series of reference eigenvalues. The first reference is the asymptotic maximum sample eigenvalue of a null-model correlation matrix, (1 + \sqrt{\gamma})^2 with \gamma the variables-to-sample-size ratio (Marchenko-Pastur upper edge). Each subsequent reference applies Braeken and van Assen's proportional correction for the variance already absorbed by preceding observed eigenvalues, floored at one. Retention counts the run of leading eigenvalues above their references (the paper's factors-1-to-K rule, operationally the same first-crossing stop as sfa_parallel()). The serial correction addresses the classical parallel-analysis weakness that reference values ignore variance captured by real factors.

Adaptation note: with similarity matrices computed across embedding dimensions (see sfa_similarity()), the sample size is the embedding dimension, so \gamma is items over dimensions. Embedding dimensions are coordinates rather than sampled respondents, so the Marchenko-Pastur bound is a heuristic reference here, not a sampling-theoretic one; treat the result as one voice among the criteria in sfa_nfactors().

Value

A list of class "sfa_ekc" with components:

n_factors

Integer: suggested number of factors.

observed

Numeric vector: observed eigenvalues (descending).

references

Numeric vector: EKC reference eigenvalues.

n

The sample size used.

References

Braeken, J., & van Assen, M. A. L. M. (2017). An empirical Kaiser criterion. Psychological Methods, 22(3), 450–466. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1037/met0000074")}

Examples

data(big5)
sim <- sfa_similarity(big5$embeddings, "mean_centered_pearson")
sfa_ekc(sim, big5$embeddings)

semanticfa documentation built on Sept. 2, 2026, 1:07 a.m.