Exploratory Factor Analysis of Data Matrices With More Variables Than Observations

Abstract

A new approach for exploratory factor analysis (EFA) of data matrices with more variables p than observations n is presented. First, the classic EFA model (n > p) is considered as a specific data matrix decomposition with fixed unknown matrix parameters. Then, it is generalized to a new model, called for short GEFA, which covers both cases of data, with either n > p or p ≥ n. An alternating least squares algorithm GEFALS is proposed for simultaneous estimation of all GEFA model parameters. Like principal component analysis (PCA), GEFALS is based on singular value decomposition, which makes GEFA an attractive alternative to PCA for descriptive data analysis and dimensionality reduction. The existence and uniqueness of the GEFA parameter estimation is studied and the convergence properties of GEFALS are established. Finally, the new approach is applied to both artificial (Thurstone’s 26-variable box data) and real high-dimensional data, while the performance of GEFALS is illustrated with simulation experiment. Some codes and data are available online as supplemental materials.

Keywords:  :

• Factor analysis
• p > n
• Procrustes problem
• Singular value decomposition