Biplots: quantitative data

Biplots provide visualizations of two things, usually, but not necessarily, in two dimensions. This paper deals exclusively with biplots for quantitative data X; qualitative data or data in the form of counts will be addressed in a subsequent paper. Data X may represent either (1) a matrix with n rows representing samples/cases and columns representing p quantitative variables or (2) a two-way table whose rows and columns both represent classifying variables. Data sets of both types (1) and (2) are considered. Plotting symbols are usually points (typically for samples and distinguished by shape and/or color) and lines (typically for variables which may be calibrated or treated as arrowed vectors). Furthermore, variables may be nonlinear in both regularity of calibration and/or curvature. Interpretation is through distance, inner-products, and sometimes area. Biplots may be improved by judicious shifts of axes, by scaling and by rotation. Nearly always, biplots give approximations to X and measures, incorporated in the biplot, expressing the degree of approximation are discussed. These aspects are illustrated with reference to examples from principal component analysis, nonlinear biplots, biplots for biadditive models, canonical variate analysis and the analysis of distance between grouped samples. WIREs Comput Stat 2015, 7:42–62. doi: 10.1002/wics.1338 For further resources related to this article, please visit the WIREs website. Conflict of interest: The authors have declared no conflicts of interest for this article.