Biplots: the joy of singular value decomposition

The biplot is a generalization of a scatterplot for two variables to the case of many variables. Instead of having samples represented as points with respect to two perpendicular axes, as in a bivariate scatterplot, there are as many axes as variables pointing in different directions. Samples are then perpendicularly projected onto axes to obtain approximate values of the data. The word 'approximate' is important, because it is not possible to represent data on many variables exactly by this procedure, but the biplot arranges the axes to display the data as accurately as possible, usually by least-squares fitting. The 'bi' in biplot refers to the rows and columns of a multivariate data matrix, where the rows are usually cases and the columns are variables. Biplots are almost always displayed in a two-dimensional plot but can just as well be displayed in three-dimensions, with more accurate data representation, using suitable graphical software, for example dynamic rotation or conditioned plots. The usual linear biplot, using least-squares approximation, relies analytically on the singular value decomposition, which in turn can be thought of as a two-sided regression problem. Biplot geometry underlies many classical multivariate procedures, such as principal component analysis, simple and multiple correspondence analysis, discriminant analysis, and other variants of dimension reduction methods such as log-ratio analysis. (C) 2012 Wiley Periodicals, Inc.