Characteristics of Distance Matrices Based on Euclidean, Manhattan and Hausdorff Coefficients

From n-size samples of k-variate points, we construct n × n distance-matrices based on the widely used Euclidean, Manhattan and Hausdorff coefficients and study (individually and in pairs) their properties P, R and ρ using theoretical analysis and both computer-generated and empirical data. The concordance P_EM is shown by analysis of uniformly-distributed data to decrease asymptotically as k → ∞ to exp [‒ exp [‒ γ]] ≅  0.5704, and P_EH and P_MH to decrease to zero, as also in generated N(0,1) and empirical data. In geological data, P_EM is higher than predicted for 10 < k < 50. The robustness R of single matrices increases asymptotically as k → ∞ to 1.0, but the benefit to phenetic analysis is offset by the opposite response in P. For paired matrices, the distance-correlation coefficient ρ_E²_M is > 0.9 and independent of k for both U(0,1) and N(0,1); ρ_EH  and ρ_EM decline to zero as k → ∞.