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Some inequalities contrasting principal component and factor analyses solutions

Adachi, Kohei and Trendafilov, Nickolay T. (2019). Some inequalities contrasting principal component and factor analyses solutions. Japanese Journal of Statistics and Data Science, 2(1) pp. 31–47.

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Principal component analysis (PCA) and factor analysis (FA) are two time-honored dimension reduction methods. In this paper, some inequalities are presented to contrast PCA and FA solutions for the same data set. For this reason, we take advantage of the recently established matrix decomposition (MD) formulation of FA. In summary, the resulting inequalities show that [1] FA gives a better fit to the data than PCA, [2] PCA extracts a larger amount of common “information” than FA, and [3] For each variable, its unique variance in FA is larger than its residual variance in PCA minus the one in FA. The resulting inequalities can be useful to suggest whether PCA or FA should be used for a particular data set. The answers can also be valid for the classic FA formulation not relying on the MD-FA definition, as both “types” FA provide almost equal solutions. Additionally, the inequalities give theoretical explanation of some empirically observed tendencies in PCA and FA solutions, e.g., that the absolute values of PCA loadings tend to be larger than those for FA loadings, and that the unique variances in FA tend to be larger than the residual variances of PCA.

Item Type: Journal Item
Copyright Holders: 2019 Japanese Federation of Statistical Science Associations
ISSN: 2520-8764
Project Funding Details:
Funded Project NameProject IDFunding Body
Not Set(C)-18K11191Japan Society of the Promotion of Sciences
Keywords: Matrix decomposition; Dimension reduction; Common parts; Unique parts; Loadings; Residuals
Academic Unit/School: Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
Faculty of Science, Technology, Engineering and Mathematics (STEM)
Item ID: 58519
Depositing User: Nickolay Trendafilov
Date Deposited: 08 Jan 2019 10:10
Last Modified: 05 Jan 2020 19:51
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