The Weighted Gini-Simpson Index: Revitalizing an Old Index of Biodiversity

The distribution of biodiversity at multiple sites of a region has been traditionally investigated through the additive partitioning of the regional biodiversity into the average within-site biodiversity and the biodiversity among sites. The standard additive partitioning of diversity requires the use of a measure of diversity, which is a concave function of the relative abundance of species, such as the Gini-Simpson index, for instance. Recently, it was noticed that the widely used Gini-Simpson index does not behave well when the number of species is very large. The objective of this paper is to show that the new weighted Gini-Simpson index preserves the qualities of the classic Gini-Simpson index and behaves very well when the number of species is large. The weights allow us to take into account the abundance of species, the phylogenetic distance between species, and the conservation values of species. This measure may also be generalized to pairs of species and, unlike Rao’s index, this measure proves to be a concave function of the joint distribution of the relative abundance of species, being suitable for use in the additive partitioning of biodiversity. The weighted Gini-Simpson index may be easily transformed for use in the multiplicative partitioning of biodiversity as well

Selecting relevant projections onto subsets of coordinates: A minimax dependence-based approach

AbstractThe sum of the entropies of the components of a random vector minus the entropy of the whole vector is known to be a good measure of global interdependence. This measure is used for selecting a relevant pair (or triple) of coordinates based on both the low degree of interdependence exhibited by them and their high degree of dependence on the omitted variables. When the characteristics are Gaussian, this global measure of interdependence has a simple formula, easily calculated from the information offered by MINITAB or SAS. The formalism is applied both to an example involving discrete coordinates and to the known “rabbit's head” diabetes example where the coordinates have continuous range

Diversity Measures and Coarse-graining in Data Analysis with an Application Involving Plant Species on the Galapagos Islands

In a numerical entity-characteristic incidence matrix we can use simple or multiple regression and calculate correlations between pairs of characteristics. However, in order to detect similarities/dissimilarities, interdependence, and multiple probabilistic causality among the characteristics we have to group the entities in classes. The number of uniform classes obtained by coding the given values of these characteristics depends on the balance between the class uncertainty and class ambiguity. The similarity, interdependence, and multiple probabilistic causality among characteristics are analyzed. When a set of entities and the abundance of their components are given, the average within-entity diversity and the average between-entity diversity are studied. The results are applied to the number of endemic and immigrant plant species in the Galapagos Islands

The curvature induced by covariance

The objective of the paper is to construct the signed measure which is the closest one to independence subject to given cavariances between random variables, where closeness is measured by using Pearson's [chi]2 indicator. The difference between this signed measure and the independent, direct product of the marginals gives the curvature induced by the linear dependence between random variables. The signed measure may be extended to the sample space of a time series and used for approximating the conditional mean values, when the joint probability distribution is not known, or for calculating the amount of nonlinear dependence between random variables when the joint probability distribution is known. The integral with respect to this signed measure on the sample space is also analyzed.minimizing [chi]2 the closest signed product measure to independence curvature induced by linear dependence between random variables

The curvature induced by covariance

AbstractThe objective of the paper is to construct the signed measure which is the closest one to independence subject to given cavariances between random variables, where closeness is measured by using Pearson's χ2 indicator. The difference between this signed measure and the independent, direct product of the marginals gives the curvature induced by the linear dependence between random variables. The signed measure may be extended to the sample space of a time series and used for approximating the conditional mean values, when the joint probability distribution is not known, or for calculating the amount of nonlinear dependence between random variables when the joint probability distribution is known. The integral with respect to this signed measure on the sample space is also analyzed