Metric family portraits

By associating a whole distance matrix with a single point in a parameter space, a family of matrices (e.g., all those obeying the triangle inequality) can be shown as a cloud of points. Pictures of the cloud form a family portrait, and its characteristic shape and interrelationship with the portraits of other families can be explored. Critchley (unpublished) used this approach to illustrate, for distances between three points, algebraic results on the nesting relations between various metrics. In this paper, these diagrams are further investigated and then generalized. In the first generalization, projective geometry is used to allow the geometric representation of Additive Mixture, Additive Constant, and Missing Data problems. Then the six-dimensional portraits of four-point distance matrices are studied, revealing differences between the Euclidean, Additive Tree, and General Metric families. The paper concludes with caveats and insights concerning families of generaln-point metric matrices.