Unified biplot geometry

Gower, John (2003). Unified biplot geometry. metodološki zvezki (Develpments in Applied Statistics), 19 pp. 3–22.

URL: http://mrvar.fdv.uni-lj.si/pub/mz/mz19/gower.pdf


The fundamental geometry is outlined that underlies all biplots of a data-matrix X of n cases and p variables. Cases are represented by n points and variables by a reference system. The reference system for quantitative variables may be orthogonal Cartesian axes, other linear axes or nonlinear trajectories. The reference system for categorical variables is a set of category-level-points (CLPs) one for each category-level; CLPs for ordered categories are collinear. Axes are labelled by a set of graduated numerical markers; CLPs are labelled by the names of their category levels. The point representing a case is nearer the markers that give the values of its variables, than to any other markers. This high dimensional representation is approximated in few (often two) dimensions in such a way that the approximated reference system gives optimal approximations to the values of X. Furthermore, new cases may be interpolated into the approximation space. Special cases within this general framework are illustrated by several examples of biplots.

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