Biplots for the Correlation Matrix

The visualization of the correlation matrix by means of biplots is considered. The classical centering operations, either by the overall mean, the column means, or row and column means are shown to be problematic for the visualisation of the correlation matrix, and sub-optimal in terms of goodness-of-fit. More flexible adjustments are possible by using a single scalar adjustment, a set of column scalars or both row-and-column scalars using a weighted alternating least squares algorithm. Recently, correlation biplots with a single scalar adjustment have been advocated and outperform the usual correlation biplots made by principal component analysis. This article presents an iterative algorithm for a column adjustment of the correlation matrix with the goal of improving the goodness-of-fit over the the use of a single scalar adjustment and studies its usefulness in practical data analysis. The resulting biplots are harder to read but can be made more interpretable by using correlation tally sticks. Correlation tally sticks are advocated for improving the visualization of correlation structure. The weighted root mean squared error is used to compare low-dimensional approximations to the correlation matrix across methods.