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International Journal of Ecology

Volume 2012 (2012), Article ID 478728, 10 pages

http://dx.doi.org/10.1155/2012/478728

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

^{1}Environmental and Health Studies Program, Department of Multidisciplinary Studies, Glendon College, York University, 2275 Bayview Avenue, Toronto, ON, Canada M4N 3M6^{2}Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, Canada M3J 1P3

Received 19 September 2011; Revised 22 November 2011; Accepted 6 December 2011

Academic Editor: Jean-Guy Godin

Copyright © 2012 Radu Cornel Guiasu and Silviu Guiasu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

Measuring biodiversity is a major, and much debated, topic in ecology and conservation biology. The simplest measure of biodiversity is the number of species from a given community, habitat, or site. Obviously, this ignores how many individuals each species has. The best known measures of biodiversity, that also take into account the relative abundance of species, are the Gini-Simpson index and the Shannon entropy . Both measures have been imported into biology from other fields Thus, Gini [1] introduced his formula in statistics, in 1912. Much later, after 37 years, Simpson [2] pleaded convincingly in favour of using Gini’s formula as a measure of biodiversity. Shannon was an engineer who introduced his discrete entropy in information theory [3], in 1948, as a measure of uncertainty, inspired by Boltzmann’s continuous entropy from classical statistical mechanics [4], defined half a century earlier. Shannon’s formula was adopted by biologists about 17 years later [5–8], as a measure of specific diversity. This import of mathematical formulas has continued. Rényi, a probabilist, introduced his own entropy [9], in order to unify several generalizations of the Shannon entropy. He was a pure mathematician without any interest in applications, but later, Hill [10] claimed that by taking the exponential of Rényi’s entropy we obtain a class of suitable measures of biodiversity, called Hill’s numbers, which were praised by Jost [11–13] as being the “true” measures of biodiversity. In 1982, Rao [14], a statistician, introduced the so-called quadratic entropy , which in fact has nothing to do with the proper entropy and depends not only on the relative abundance of species but also on the phylogenetic distance between species. This function has also been quickly adopted by biologists as a measure of dissimilarity between the pairs of species. In the last 20 years, a lot of other measures of diversity have been proposed. According to Ricotta [15], there is currently a “jungle” of biological measures of diversity. However, as mentioned by S. Hoffmann and A. Hoffmann [16], there is no unique “true” measure of diversity.

Starting with MacArthur [5], MacArthur and Wilson [7], and Whittaker [8], the distribution of biodiversity at multiple sites of a region has been traditionally investigated through the partitioning of the regional or total biodiversity, called *γ*-diversity, into the average within-site biodiversity, called *α*-diversity, and the between-site biodiversity or diversity turnover, called *β*-diversity. All these diversities, namely, *α*-diversity, *β*-diversity, and *γ*-diversity, should be nonnegative numbers. Unlike *α*-diversity and *γ*-diversity, there is no consensus about how to interpret and calculate *β*-diversity. According to Whittaker [8], who introduced the terminology, *β*-diversity is the ratio between *γ*-diversity and *α*-diversity. This is the multiplicative partitioning of diversity. According to MacArthur [5], Lande [17], and, more recently, Veech at al. [18], *β*-diversity is the difference between *γ*-diversity and *α*-diversity. This is the additive partitioning of diversity.

Let us assume that in a certain region there are species, *m* sites, and is the distribution of the relative abundance of species at site . Let be an arbitrary parameter assigned to site , such that . These parameters may be used to make adjustments for differences (in size, altitude, etc.) between the sites. If no adjustment is made, we take these parameters to be equal, that is, , for every . If *μ* is a nonnegative measure of diversity, which assigns a nonnegative number to each distribution of the relative abundance of the species, then the corresponding *γ*-diversity is and the *α*-diversity is . The *β*-diversity is taken to be , in the additive partitioning of diversity, and , in the multiplicative partitioning of diversity. In general, a measure of biodiversity ought to be nonnegative, in which case the corresponding *α*-diversity and *γ*-diversity calculated by using such a measure are also nonnegative, as they should be. From a systemic point of view, the *β*-diversity shows to what extent the total, or regional diversity differs from the average diversity of the communities/habitats/sites taken together, as a system, reflecting the dissimilarity, or differentiation between communities/habitats/sites of the region with respect to the individual species. If the measure of biodiversity is a concave function of the distribution of the relative abundance of species , then the corresponding -diversity is , in the additive partitioning of diversity, and , in the multiplicative partitioning of diversity, for arbitrary parameters . If a measure of diversity is not a concave function of the distribution of the relative abundance of species , then the corresponding *β*-diversity could be negative, in the additive partitioning of biodiversity, or less than 1, in the multiplicative partitioning of biodiversity, for some parameters satisfying , which is absurd. As discussed by Jost [11, 12], if a measure of diversity is not a concave function of the distribution of the relative abundance of species , we can still attempt the partitioning of biodiversity into -, -, and -diversity if a new kind of *α*-diversity may be introduced. This new type of -diversity would be based on a different way of averaging the diversities of the individual communities/habitats/sites instead of the simple, golden mean value from statistics, which works so well for the concave measures of diversity. However, finding such an unorthodox, nonstandard *α*-diversity when the measure of biodiversity is not concave is not easy. It is also difficult to find a mathematical interpretation for such a new kind of *α*-diversity. In spite of the passage of time, the most popular measures of biodiversity are still , , and . Both and are concave functions of the distribution of the relative abundance of species and therefore can be used for doing the additive partitioning of biodiversity. The first two Hill’s numbers are mathematical transformations of and , namely, and , and are used in the multiplicative partitioning of biodiversity. Recently, however, it was noted [13, 19] that both Shannon’s entropy and the Gini-Simpson index do not behave well when the number of species is very large. On the other hand, when a distance between species, such as the phylogenetic distance for instance, is also taken into account along with the relative abundance of species, Rao’s index [14] is a widely used measure of dissimilarity. But, unfortunately, Rao’s index is not a concave function of the distribution of the relative abundance of species, for an arbitrary distance matrix between species. Consequently, it proves to be suitable for use in the standard additive partitioning of diversity only in some special cases, but not in general. The objective of this paper is to show that the weighted Gini-Simpson quadratic index, a generalization of the classic Gini-Simpson index of biodiversity, offers a solution to both of the drawbacks just mentioned. Unlike Shannon’s entropy and the classic Gini-Simpson index, this new weighted measure of biodiversity behaves very well even if the number of species is very large. The weights allow us to measure biodiversity when a distance between species and/or conservation values of the species are taken into account, along with the abundance of species. When the phylogenetic distance between species is taken as the weight, the corresponding weighted Gini-Simpson index, unlike Rao’s index, is a concave function of the distribution of the relative abundance of the pairs of species, being suitable for use in the additive partitioning of biodiversity. A simple algebraic transformation makes the weighted Gini-Simpson index suitable for use in the multiplicative partitioning of biodiversity as well.

In Methodology, the weighted Gini-Simpson quadratic index is defined both for individual species and for pairs of species. This new measure of biodiversity is used for calculating the average within-site biodiversity (*α*-diversity), the intersite biodiversity (*β*-diversity), and the regional or total biodiversity (*γ*-diversity). It is also shown that the weighted Gini-Simpson quadratic index may be easily modified, by a simple algebraic transformation, to get a measure of biodiversity suitable for use in the multiplicative partitioning of biodiversity as well. In Section 3, a numerical example is presented, which illustrates how the mathematical formalism should be applied from a practical standpoint.

#### 2. Methodology

##### 2.1. The Weighted Measure of Diversity with Respect to Individual Species

Let us assume that there are species in a certain community/habitat/site and let be the relative abundance of species (the number of individuals of species divided by the total number of individuals in that community/habitat/site). We have diversity if species is present at that location but other species are found there as well. The probability that the species is present *and* there are other species present as well is . If we take all possible values of from the unit interval into account, the wave function corresponding to the species is a nonnegative, symmetric, bell-shaped, concave function, reaching its maximum value at . If we sum up these wave functions, for all species, we obtain the classic Gini-Simpson index corresponding to the given distribution of the relative abundance of the species . Since 1949, this has been considered to be a very good measure of biodiversity. In order to generalize it, we may assign an amplitude to the wave function of the species , and the resulting new wave function continues to be a nonnegative, symmetric, bell-shaped, concave function of , but this time its maximum value is . Summing up these wave functions for all the species, we get the weighted Gini-Simpson index:
which depends both on the distribution of the relative abundance of species and on the nonnegative weights . The concavity of was proven in [20, 21]. The weight could be anything which contributes to the increase in the diversity induced by the species . However, the weights may not depend on the relative abundance of species. If , for each , then (1) becomes the so-called Rich-Gini-Simpson index introduced in [22], which is essentially dependent on the species richness of the respective community/habitat/site. If there are some conservation values assigned to the species , which are positive numbers on a certain scale of values, and the weights are , the corresponding weighted Gini-Simpson index is denoted by . Obviously, if , for each species , then (1) is the classic Gini-Simpson index . An upper bound for , which depends only on the maximum weight and the number of species, is

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