Area Biplots

Abstract

Classical multivariate analysis techniques such as principal components analysis and correspondence analysis use inner products to approximate data values. Results of these techniques may be visualized by presenting row and column points jointly in a biplot, where the projection of a row point onto a column point vector followed by a multiplication by the length of the column point vector gives the inner product that approximates the corresponding data element. In this article, we propose a new visualization: after a 90○ rotation of the row points, the area spanned by a triangle of a rotated row point, a column point, and the origin approximates the data values. In contrast to the projection biplot, the areas spanned by different row and column points can be compared directly. This property makes the area biplot unique. Therefore, the area biplot is particularly useful for the analysis of a data matrix where all elements can be compared. The area biplot makes it easy to see that, similarly to inner products, higher dimensional area solutions can be represented by summing areas over subsequent pairs of dimensions. Here, the area biplot is developed for principal components analysis, correspondence analysis, and for interaction biplots, but it has general applicability. This article has supplementary material online.