Generalized biplots for stress-based multidimensionally scaled projections

Abstract

Dimension reduction and visualization are staples of data analytics. Methods such as Principal Component Analysis (PCA) and Multidimensional Scaling (MDS) provide low dimensional (LD) projections of high dimensional (HD) data while preserving an HD relationship between observations. Traditional biplots assign meaning to the LD space of a PCA projection by displaying LD axes for the attributes. These axes, however, are specific to the linear projection used in PCA. Stress-based MDS (s-MDS) projections, which allow for arbitrary stress and dissimilarity functions, require special care when labeling the LD space. An iterative scheme is developed to plot an LD axis for each attribute based on the user-specified stress and dissimilarity metrics. The resulting plot, which contains both the LD projection of observations and attributes, is referred to as the Generalized s-MDS Biplot. The details of the Generalized s-MDS Biplot methodology, its relationship with PCA-derived biplots, and an application to a real dataset are provided.