The Shapes of Non-Generic Figures, and Applications to Collinearity Testing

The singular-values decomposition of matrices determines a local representation (Λ, [V]) of the shape space for k labelled points in R^m, which has been used to describe its riemannian structure. Here we use Λ to compute the geodesic distance of a shape from the dimensionally deficient k-ads, which can be used as a statistic for dimensional deficiency. We also characterize, in terms of [V], the `coincidence' set, which can be used to assess whether the deficiencies are `genuine', or merely apparent. These can be used to treat the appropriate analogue, in such curved shape spaces, of the classical multivariate procedure in which one projects the multivariate sample points onto an interesting hyperplane and then regards the foot of the perpendicular as an `estimate' and the length of the perpendicular as an `goodness of fit' statistic. Finally, we discuss some probabilistic properties of such geometrical statistics.