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Semi-sparse PCA

Author

Listed:
  • Lars Eldén

    (Linköping University)

  • Nickolay Trendafilov

    (The Open University)

Abstract

It is well known that the classical exploratory factor analysis (EFA) of data with more observations than variables has several types of indeterminacy. We study the factor indeterminacy and show some new aspects of this problem by considering EFA as a specific data matrix decomposition. We adopt a new approach to the EFA estimation and achieve a new characterization of the factor indeterminacy problem. A new alternative model is proposed, which gives determinate factors and can be seen as a semi-sparse principal component analysis (PCA). An alternating algorithm is developed, where in each step a Procrustes problem is solved. It is demonstrated that the new model/algorithm can act as a specific sparse PCA and as a low-rank-plus-sparse matrix decomposition. Numerical examples with several large data sets illustrate the versatility of the new model, and the performance and behaviour of its algorithmic implementation.

Suggested Citation

  • Lars Eldén & Nickolay Trendafilov, 2019. "Semi-sparse PCA," Psychometrika, Springer;The Psychometric Society, vol. 84(1), pages 164-185, March.
    • Handle: RePEc:spr:psycho:v:84:y:2019:i:1:d:10.1007_s11336-018-9650-9
    DOI: 10.1007/s11336-018-9650-9
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    References listed on IDEAS

    as
    1. Shen, Haipeng & Huang, Jianhua Z., 2008. "Sparse principal component analysis via regularized low rank matrix approximation," Journal of Multivariate Analysis, Elsevier, vol. 99(6), pages 1015-1034, July.
    2. James Steiger, 1979. "Factor indeterminacy in the 1930's and the 1970's some interesting parallels," Psychometrika, Springer;The Psychometric Society, vol. 44(2), pages 157-167, June.
    3. Unkel, S. & Trendafilov, N.T., 2010. "A majorization algorithm for simultaneous parameter estimation in robust exploratory factor analysis," Computational Statistics & Data Analysis, Elsevier, vol. 54(12), pages 3348-3358, December.
    4. Nickolay T. Trendafilov & Sara Fontanella & Kohei Adachi, 2017. "Sparse Exploratory Factor Analysis," Psychometrika, Springer;The Psychometric Society, vol. 82(3), pages 778-794, September.
    5. JOURNEE, Michel & NESTEROV, Yurii & RICHTARIK, Peter & SEPULCHRE, Rodolphe, 2010. "Generalized power method for sparse principal component analysis," LIDAM Reprints CORE 2232, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. repec:ucp:bkecon:9780226316529 is not listed on IDEAS
    7. Steffen Unkel & Nickolay T. Trendafilov, 2010. "Simultaneous Parameter Estimation in Exploratory Factor Analysis: An Expository Review," International Statistical Review, International Statistical Institute, vol. 78(3), pages 363-382, December.
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