Abstract

Canonical Variate Analysis (CVA) is a well-established multivariate technique for exploring differences between predefined groups. By finding linear combinations of variables that maximise between-group separation relative to within-group variation, CVA provides a supervised low-dimensional representation of multivariate data. When combined with biplot methodology, it offers more than a view of group separation. Observations can be displayed as points alongside representations of the original variables, making it possible to investigate which measurements contribute to the patterns observed.
A fundamental difficulty arises, however, when the number of variables exceeds the number of observations. In these high-dimensional settings, the within-group scatter matrix is singular, and the conventional generalised eigenvalue formulation of CVA cannot be applied directly. Such data are increasingly common across a wide range of disciplines, where large numbers of variables are measured on relatively few observations. Although several approaches have been developed to address singularity in discriminant analysis, the corresponding development of interpretable CVA biplots has received less attention.
This talk presents a CVA biplot construction based on the generalised singular value decomposition (GSVD). Drawing on a GSVD formulation of discriminant analysis, the methodology extends CVA biplots to settings in which the number of variables is not restricted by the sample size. The approach retains the central objective of CVA, namely revealing group separation, while providing a graphical representation that connects the positions of observations to their original measurements. In particular, the biplot representation allows selected variables to be interpreted within the reduced-dimensional display, rather than treating the plot solely as a map of group membership.
The talk will discuss the motivation for using the GSVD, outline the key steps in constructing the resulting biplot, and explain how this formulation relates to conventional CVA when the latter is applicable. An important aspect of the discussion will be the assessment of biplot quality through measures of fit. These measures provide a way to evaluate how well the lower-dimensional representation captures the structure of the original data and how accurately observations and variables are represented. Their interpretation and role in assessing the usefulness of the resulting visualisations will also be discussed. Applications to different multivariate datasets will illustrate the methodology in practice, highlighting both its interpretability and the challenges associated with visualising high-dimensional data.
The talk will also introduce wideRhino, an R package developed to make the methodology accessible and reproducible. The package implements GSVD-based CVA through functions for fitting the analysis and constructing biplots, enabling users to move from grouped multivariate data to an interpretable visual display within R.