In the statistical literature on factor analysis many ingenious graphical and analytical procedures have been developed for transforming arbitrary factor loading matrices into meaningful ones while preserving factor scores orthogonality. All of them aim at obtaining an interpretable set of factors, according to Thurstons simple-structure principle, by focusing on and suitably modifying the factor loading matrix. Recently, starting from the signal processing community, the literature has witnessed a growing interest towards modeling a set of observed non gaussian variables through linear combinations of non gaussian independent latent ones. In this paper we show that if such a model holds, it can be easily estimated by suitably rotating the ordinary factor analysis solution (obtained by a distribution free method, as the problem is outside the classic gaussian context in which independence and uncorrelatedness coincide). This involves shifting the attention from the structure of the factor loading matrix to the fac- tor scores relationship, as independence does not necessarily impose a simple structure on the factor loading matrix. The question is then: how can one transform a set of sphered variables (the factor scores which are assumed unit variance and uncorrelated) into a set of as many independent ones? An answer is o®ered by Independent Component Analysis proposed by Comon (1994) in order to model p given variables as linear mixtures of p unknown independent ones: as the factor scores are sphered, performing ICA on them simply amounts to look for the orthogonal rotation which leads to the least gaussian projections of the original factor scores. We illustrate the performances of the proposed method both on real and simulated data and also highlight the links of the method with another recently proposed model also allowing for independent latent variables, the independent factor analysis model (Attias, 1999; Montanari and Viroli, 2004).

Looking for independent factors: a new factor rotation method

MONTANARI, ANGELA;VIROLI, CINZIA
2004

Abstract

In the statistical literature on factor analysis many ingenious graphical and analytical procedures have been developed for transforming arbitrary factor loading matrices into meaningful ones while preserving factor scores orthogonality. All of them aim at obtaining an interpretable set of factors, according to Thurstons simple-structure principle, by focusing on and suitably modifying the factor loading matrix. Recently, starting from the signal processing community, the literature has witnessed a growing interest towards modeling a set of observed non gaussian variables through linear combinations of non gaussian independent latent ones. In this paper we show that if such a model holds, it can be easily estimated by suitably rotating the ordinary factor analysis solution (obtained by a distribution free method, as the problem is outside the classic gaussian context in which independence and uncorrelatedness coincide). This involves shifting the attention from the structure of the factor loading matrix to the fac- tor scores relationship, as independence does not necessarily impose a simple structure on the factor loading matrix. The question is then: how can one transform a set of sphered variables (the factor scores which are assumed unit variance and uncorrelated) into a set of as many independent ones? An answer is o®ered by Independent Component Analysis proposed by Comon (1994) in order to model p given variables as linear mixtures of p unknown independent ones: as the factor scores are sphered, performing ICA on them simply amounts to look for the orthogonal rotation which leads to the least gaussian projections of the original factor scores. We illustrate the performances of the proposed method both on real and simulated data and also highlight the links of the method with another recently proposed model also allowing for independent latent variables, the independent factor analysis model (Attias, 1999; Montanari and Viroli, 2004).
Proceedings in Computational Statistics, Compstat 2004, 16th Symposium of IASC Praga
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Montanari A; Viroli C
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/111090
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