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

Abstract

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 → ∞.

Appendices

Appendix 1 Volumes of k-Spheres and k-Octahedra

The k-volume of a k-dimensional sphere of radius r (cf. Coxeter, 1973, p. 125, Eq. 7.33) is S_k r^k, where S_k is the k-volume of a k-sphere of unit radius, given by:

$${S}_{k}=\frac{{\pi }^{\frac{k}{2}}}{\Gamma \left(1+\frac{k}{2}\right)}$$

This expression is reduced to $S_{k}={\pi}{\frac{k}{2}}(k/2)$! for even values of k, since Γ(n + l) = n! when n is an integer. The (k ‒1)-volume (“surface”) of the sphere is k S_k r^k‒1.

The k-volume of a k-octahedron β_n of edge-length l (Coxeter, 1973, Table 1 (iii)) is 2 ^k/2 l^k / k!, and the (k ‒1)-volume is

$$\frac{{\left({2}^{k+1}k\right)}^\frac{1}{2}{l}^{k-1}}{\left(k-1\right)!}$$

Appendix 2 Limit of [Γ(1 + 1/k)]^k

We find initially the limit of the logarithm by writing z = 1/k. Then,

$$\begin{array}{l}\ln\left[\Gamma\left(1+\frac1k\right)\right]^k=k\;\ln\left[\Gamma\left(1+\frac1k\right)\right]\\=\frac1z\;\mathrm l\mathrm n\;\mathrm\Gamma\left(1+z\right)\\=\frac1z\left[-\ln\left(1+z\right)+z\left(1-\gamma\right)+\sum\nolimits_{n=2}^\infty\frac{\left(-1\right)^n\left[\zeta\left(n\right)-1\right]z^n}n\right]\end{array}$$

where γ is Euler’s constant, i.e. ${\mathrm{lim}}_{m\to \infty }[1+\frac{1}{2}+\frac{1}{3}+\dots +\frac{1}{m}-\mathrm{ln}\ m]\cong 0.5772$, and ζ is the Riemann Zeta Function (Abramowitz & Stegun, 1972, 6.1.33). Thus,

$${\mathrm{ln}[\Gamma (1+\frac{1}{k})]}^{k}=-1+\frac{z}{2}-\frac{{z}^{2}}{3}+\frac{{z}^{3}}{4}-\dots +\left(1-\gamma \right)+\frac{\left[\zeta \left(2\right)-1\right]z}{2}-\frac{\left[\zeta \left(3\right)-1\right]{z}^{2}}{3}+\frac{\left[\zeta \left(4\right)-1\right]{z}^{3}}{4}-\dots \to -\gamma , \mathrm{as}\ {z}\to 0.$$

So [Γ(1 + 1/k)]^k → exp[‒ γ] ≅ 0.5615 as k → ∞.

Appendix 3 Producing a Uniform Random Scatter of Points on the Surface of, or Within the Positive Quadrant of, a k-Sphere

For a uniform random scatter of points on the surface of a k-sphere, A. Abakuks points out that the joint distribution of k independent normal distributions (zero mean, unit variance) forms a spherical cluster centred on the origin so that if x_ij (i = 1, 2,... n; j = 1, 2,... k) are k-variate points normally distributed with zero mean and unit variance, the vector

$${y}_{ij}=\frac{{x}_{ij}}{\sqrt{{\sum }_{j=1}^{k}{x}_{ij}{^{2}}}}$$

is uniformly randomly distributed over the surface of a unit k-sphere. If we then form the product z_ij = |y_ij|[U_i]^1/k, where U_i are uniformly randomly distributed in the interval (0,1), z_ij will be uniformly randomly distributed within the all-positive quadrant of the unit sphere.

Appendix 4 Relations Between F _os, f _s⊂o and f _o⊂s

The following relations hold between F_os, f_s⊂o and f_o⊂s (for definitions, see Table 2):

• (i) F_os = f_s⊂o + f_o⊂s

• (ii) f_s⊂o = a F_os

• (iii) f_o⊂s = (1 – a) F_os, where

  $$\begin{array}{c}a=\frac{\pi -4\theta }{\pi -4\left(\theta -\mathrm{sin}\theta \mathrm{cos}\theta \right)}=0.4252\cdots \\ \theta ={\mathrm{cos}}^{-1}\left(\frac{\sqrt{\pi }}{2}\right)\end{array}$$

that is, the sphere/octahedron angle for λ = 2 (Fig. 1). The first of these relations is the multi-dimensional extension of a simple 2-dimensional relationship. Similar relations hold between F_cs, f_s⊂c  and f_c⊂s.