Nonlinear biplots.

The classical biplot (Gabriel, 1971) of a multivariate sample X of n units by p variables is generalized for any Euclidean imbeddable metric dij. Sample-units are represented in the usual way by points in an ordination, most conveniently using principal coordinates. Variables are represented by a set of p concurrent nonlinear trajectories whose lengths and dispositions relative to the sample points aid interpretation. When dij2 is defined to have independent contributions from each variable, the centroid of any p points, one on each trajectory, may be used to interpolate and interpret sample points in the ordination. The basic idea may be exploited further for use with any form of metric or nonmetric scaling and for structured multivariate samples.