Factor Analysis of Data Matrices: New Theoretical and Computational Aspects With Applications

Abstract

The classical fitting problem in exploratory factor analysis (EFA) is to find estimates for the factor loadings matrix and the matrix of unique factor variances which give the best fit to the sample covariance or correlation matrix with respect to some goodness-of-fit criterion. Predicted factor scores can be obtained as a function of these estimates and the data. In this thesis, the EFA model is considered as a specific data matrix decomposition with fixed unknown matrix parameters. Fitting the EFA model directly to the data yields simultaneous solutions for both loadings and factor scores. Several new algorithms are introduced for the least squares and weighted least squares estimation of all EFA model unknowns. The numerical procedures are based on the singular value decomposition, facilitate the estimation of both common and unique factor scores, and work equally well when the number of variables exceeds the number of available observations. Like EFA, noisy independent component analysis (ICA) is a technique for reduction of the data dimensionality in which the interrelationships among the observed variables are explained in terms of a much smaller number of latent factors. The key difference between EFA and noisy ICA is that in the latter model the common factors are assumed to be both independent and non-normal. In contrast to EFA, there is no rotational indeterminacy in noisy ICA. In this thesis, noisy ICA is viewed as a method of factor rotation in EFA. Starting from an initial EFA solution, an orthogonal rotation matrix is sought that minimizes the dependence between the common factors. The idea of rotating the scores towards independence is also employed in three-mode factor analysis to analyze data sets having a three-way structure. The new theoretical and computational aspects contained in this thesis are illustrated by means of several examples with real and artificial data