arXiv subjects

Jan Graffelman

Publications and source records attributed to Jan Graffelman.

3 recordsLinked to original sources

arXiv2026

On the approximation of the between-set correlation matrix by canonical correlation analysis

Canonical correlation analysis is a classic well-known multivariate statistical method focusing on the relationships between two sets of variables. The visualisation of those relationships can be achieved by means of a biplot of the between-set correlation matrix. The canonical analysis provides a low-rank approximation to the between-set correlation matrix that is optimal in a generalised least squares sense. This article proposes to adjust the between-set correlation matrix using either a single scalar effect, or column and/or row effects. An alternating generalised least squares algorithm is proposed to obtain optimal adjustments and low-rank factorisations. The adjustment leads to a better approximation of the between-set correlation matrix that achieves a lower root mean squared error in comparison with the classic canonical analysis. The results of the adjusted analysis can be efficiently visualised using biplots, with a minimal change in interpretation rules that only affects the biplot origin. Biplot calibration is used to enhance the visualisation of the results of the adjusted analysis. Some examples with publicly available data sets from social science, geochemistry and medical science illustrate the proposed improvement. Software for carrying out the adjusted canonical analysis in the R environment is provided.

stat.ME↗

arXiv2024

On the visualisation of the correlation matrix

Extensions of earlier algorithms and enhanced visualization techniques for approximating a correlation matrix are presented. The visualization problems that result from using column or colum--and--row adjusted correlation matrices, which give numerically a better fit, are addressed. For visualization of a correlation matrix a weighted alternating least squares algorithm is used, with either a single scalar adjustment, or a column-only adjustment with symmetric factorization; these choices form a compromise between the numerical accuracy of the approximation and the comprehensibility of the obtained correlation biplots. Some illustrative examples are discussed.

stat.CO↗

arXiv2022

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 data set, 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.

stat.CO↗