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Modeling Minimal Spanning Trees with Beta Vectors Open Access

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We examine methods of generating beta random vectors to model the normalized interpoint distances on the minimal spanning tree (MST). Using properties of the univariate beta distribution, we propose three methods for generating multivariate beta vectors. We use overlapping sums of the components of a Dirichlet distribution to construct beta vectors. We investigate the products of beta variables that follow an ordered Dirichlet distribution. The geometric mean of beta variables is explored to produce a multivariate beta distribution. We define a multivariate Gini index for the normalized distances on the MST to measure the amount of scatter in a multivariate sample and the inequity among the interpoint distances. An example shows the MST of 11 European languages with respect to the first 10 numerals. A simulation study compares the parametric bootstrap of the Gini index, the maximum and the range of the interpoint distances with results from modeling the distances on the MST.

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