# Measuring biodiversity with the Aitchsion distance

NOTE: Jupyter notebook hosted on gitlab.

## Introduction

Biodiversity is widely understood as a way to assess the health of an ecosystem. Although its depletion has become one of the greatest challenge which humanity is facing (REFERENCE UN), its factual expression is still the subject to many debates between ecologists, biologists, statisticians, mathematicians and physicians. MULTIPLE DEFINITIONS OF BIODIVERSITY, RICHNESS, DENSITY, ETC. Biodiversity and richness are often confounded (e.g. REFERENCE). Species richness is the number of counted species in a given region, disregarding their respective proportions. On the other hand, biodiversity is a measure of evenness, i.e. how species are distributed. It is measured by relating counts of individuals as relative amounts of species or phenotype found in a given region, then synthesized to a single index which informs about the state of an ecosystem ($$\alpha$$-diversity) or compares ecosystems across time or space ($$\beta$$​-diversity). Most indices consider species to be unrelated categories. In order to account for similarity between species, some authors proposed to regulate $$\alpha$$ -diversity (REFERENCE) and $$\beta$$ -diversity (REFERENCE) using species pairwise similarities (REFERENCE) or clades (Faith 1992, Forest 2007).

Should the trend continues, there will be more biodiversity indices than species. The meaning and limitations of indices must still be raised to derive convenient pictures of ecological data for wise decision making. The most common indices are the Shannon-Weaver index (Eq. \ref{eqn:shannon}) and the Simpson index (Eq. \ref{eqn:simpson}), which can be slightly modified and inversed (Eq. \ref{eqn:invsimpson}). Many other indices could be presented, such as Margalef (REEFRENCE), McIntosh (REFERENCE), OTHERS etc. These indices are known to return similar trends (Türkmen 2010).

$$\label{eqn:shannon} \label{eqn:shannon}Shannon:H^{\prime}=-\sum_{i=1}^{R}p_{i}\ln p_{i}\\$$
$$\label{eqn:simpson} \label{eqn:simpson}Simpson:D_{1}=1-\sum_{i=1}^{R}p_{i}^{2}\\$$
$$\label{eqn:invsimpson} \label{eqn:invsimpson}Inverse-simpson:D_{2}=\frac{1}{\Sigma_{i=1}^{s}p_{i}^{2}}\\$$

Species eveness is a special class of diversity indices assessing how close proportions are.

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