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Factor analysis models via I-divergence optimization

Author

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  • Lorenzo Finesso

    (IEIIT - CNR)

  • Peter Spreij

    (Universiteit van Amsterdam)

Abstract

Given a positive definite covariance matrix $$\widehat{\Sigma }$$ Σ ^ of dimension n, we approximate it with a covariance of the form $$HH^\top +D$$ H H ⊤ + D , where H has a prescribed number $$k 0$$ D > 0 is diagonal. The quality of the approximation is gauged by the I-divergence between the zero mean normal laws with covariances $$\widehat{\Sigma }$$ Σ ^ and $$HH^\top +D$$ H H ⊤ + D , respectively. To determine a pair (H, D) that minimizes the I-divergence we construct, by lifting the minimization into a larger space, an iterative alternating minimization algorithm (AML) à la Csiszár–Tusnády. As it turns out, the proper choice of the enlarged space is crucial for optimization. The convergence of the algorithm is studied, with special attention given to the case where D is singular. The theoretical properties of the AML are compared to those of the popular EM algorithm for exploratory factor analysis. Inspired by the ECME (a Newton–Raphson variation on EM), we develop a similar variant of AML, called ACML, and in a few numerical experiments, we compare the performances of the four algorithms.

Suggested Citation

  • Lorenzo Finesso & Peter Spreij, 2016. "Factor analysis models via I-divergence optimization," Psychometrika, Springer;The Psychometric Society, vol. 81(3), pages 702-726, September.
    • Handle: RePEc:spr:psycho:v:81:y:2016:i:3:d:10.1007_s11336-015-9486-5
    DOI: 10.1007/s11336-015-9486-5
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    References listed on IDEAS

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    1. C. Rao, 1955. "Estimation and tests of significance in factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 20(2), pages 93-111, June.
    2. Kohei Adachi, 2013. "Factor Analysis with EM Algorithm Never Gives Improper Solutions when Sample Covariance and Initial Parameter Matrices Are Proper," Psychometrika, Springer;The Psychometric Society, vol. 78(2), pages 380-394, April.
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    5. Zhao, Jianhua & Shi, Lei, 2014. "Automated learning of factor analysis with complete and incomplete data," Computational Statistics & Data Analysis, Elsevier, vol. 72(C), pages 205-218.
    6. Robert Jennrich & Stephen Robinson, 1969. "A Newton-Raphson algorithm for maximum likelihood factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 34(1), pages 111-123, March.
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