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Permutation methods for factor analysis and PCA. (English) Zbl 07285315
From the author’s abstract: “Researchers often have datasets measuring features \(x_{ij}\) of samples, such as test scores of students. In factor analysis and PCA, these features are thought to be influenced by unobserved factors, such as skills. Can we determine how many components affect the data? This is an important problem, because decisions made here have a large impact on all downstream data analysis. Consequently, many approaches have been developed. Parallel Analysis is a popular permutation method: it randomly scrambles each feature of the data. It selects components if their singular values are larger than those of the permuted data. Despite widespread use, as well as empirical evidence for its accuracy, it currently has no theoretical justification.”
In this paper, the problem is analyzed under the signal plus noise model. Sufficient conditions on the signal components and on the noise component are established to ensure the consistency of the parallel analysis. A simulation study supports the theoretical results and points out interesting features of the parallel analysis. In particular, the effect of signal strength, the effect of delocalization, and the effect of dimension are studied. Finally, it is shown that strong signals may lead to errors in the detection of weak signal components.
MSC:
62H25 Factor analysis and principal components; correspondence analysis
62H12 Estimation in multivariate analysis
60G35 Signal detection and filtering (aspects of stochastic processes)
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
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