Improved Approximation and Visualization of the Correlation Matrix

The graphical representation of the correlation matrix by means of different multivariate statistical methods is reviewed, a comparison of the different procedures is presented with the use of an example dataset, and an improved representation with better fit is proposed. Principal component analysis is widely used for making pictures of correlation structure, though as shown a weighted alternating least squares approach that avoids the fitting of the diagonal of the correlation matrix outperforms both principal component analysis and principal factor analysis in approximating a correlation matrix. Weighted alternating least squares is a very strong competitor for principal component analysis, in particular if the correlation matrix is the focus of the study, because it improves the representation of the correlation matrix, often at the expense of only a minor percentage of explained variance for the original data matrix, if the latter is mapped onto the correlation biplot by regression. In this article, we propose to combine weighted alternating least squares with an additive adjustment of the correlation matrix, and this is seen to lead to further improved approximation of the correlation matrix.