biplots

A biplot is a graphical display used in multivariate analysis to show both the observations (rows) and the variables (columns) of a data matrix in a single two-dimensional plot. It enables simultaneous exploration of relationships among samples and among variables and provides a compact view of the structure revealed by dimensionality reduction methods such as principal component analysis (PCA) or correspondence analysis (CA).

Construction and interpretation commonly rely on a centered (and possibly scaled) data matrix X. A low-dimensional representation is obtained by singular value decomposition or eigen decomposition. In a typical PCA biplot, the first two principal components provide coordinates for both observations and variables: observations appear as points in the component space, while variables appear as arrows or vectors in the same space. The direction and length of a variable vector reflect its contribution to the components and its correlation with them; the angle between two vectors approximates the correlation between the corresponding variables. The position of an observation relative to a variable vector indicates the estimated value of that variable for that observation, given the chosen scaling.

Variants and cautions: several variants exist, including Gabriel's biplot and alternative scaling schemes that balance representation of observations and variables. Biplots are most interpretable when data are standardized and the first two components capture substantial variation. They can reveal clusters, gradients, or group separations, but distort distances and relationships when the data are highly nonlinear or when important variation lies beyond the shown components. Consequently, a biplot should be used as a diagnostic and interpretive aid rather than a precise depiction of all multivariate structure.