Shannon index

The Shannon index (often also Shannon-Wiener or Shannon-Weaver index ) is a mathematical quantity that is used in biometrics to describe diversity (cf. biodiversity ). It describes the diversity in the data considered and takes into account both the number of different data categories (e.g. the number of species) and the abundance (number of individuals per species).

definition

The Shannon index of a population made up of individuals in different species, each of which belongs to a species ${\displaystyle H'}$ ${\displaystyle N}$ ${\displaystyle S}$ ${\displaystyle n_{i}}$

${\displaystyle H'=-\sum _{i}{p_{i}\cdot \ln p_{i}}\quad {\text{mit}}\ p_{i}={\frac {n_{i}}{N}}}$.

${\displaystyle p_{i}}$is the proportion of the respective species in the total number , i.e. the relative abundance of the individual species. (Instead of the natural logarithm , the base 2 logarithm,, is also used.) ${\displaystyle i}$ ${\displaystyle N}$ ${\displaystyle \ln }$ ${\displaystyle \log _{2}}$

If the number of species is specified, the Shannon index reaches its maximum when all species are populated with the same frequency. The maximum value of the Shannon index is calculated using the following formula: ${\displaystyle S}$

${\displaystyle H_{\mathrm {max} }=\ln S}$

The quotient of the Shannon index and the maximum Shannon index value is called evenness and is a measure of the distribution of individuals in a population.

Connection with information theory

The Shannon index corresponds to the entropy of a discrete memoryless source (discrete random variable ) over a finite alphabet , which is defined as follows: Each probability of an event is assigned its information content. Then the entropy of a sign is defined as the expected value of the information content ${\displaystyle H}$ ${\displaystyle X}$ ${\displaystyle Z=\{z_{1},z_{2},\dots ,z_{S}\}}$ ${\displaystyle p_{i}}$ ${\displaystyle I(p_{i})=-\log _{2}p_{i}\;}$

${\displaystyle H_{1}=-\sum _{i=1}^{S}p_{i}\cdot \log _{2}p_{i}}$,

where is the probability with which the i th character of the alphabet occurs. ${\displaystyle p_{i}=P(X=z_{i})}$ ${\displaystyle z_{i}}$

The Shannon-Weaver and Shannon-Wiener Debate

Both the designation “Shannon-Weaver-Index” and the designation “Shannon-Wiener-Index” are misleading. Warren Weaver was co-author and popularizer of the bound "A Mathematical Theory of Communication" in which Claude Elwood Shannon published his theory, which had already been laid down in two papers. Norbert Wiener provided the probabilistic calculation methods on which Shannon's work was based. His further research in the context of cybernetics was based on Shannon's information theory . However, the development of the index is solely attributable to Shannon.

See also

• Simpson index

Individual evidence

1. ^ Ian F. Spellerberg, Peter J. Fedor (2003): A tribute to Claude Shannon (1916-2001) and a plea for more rigorous use of species richness, species diversity and the 'Shannon-Wiener' index . In: Global Ecology and Biogeography 12 (3), pp. 177-179, doi: 10.1046 / j.1466-822X.2003.00015.x
2. ^ Charles J. Krebs (1989): Ecological Methodology . HarperCollins, New York.
3. ↑ Nentwig, Wolfgang .: Ecology . 1st edition. Spectrum, Akad. Verl, Heidelberg 2004, ISBN 3-8274-0172-0 , p. 284 .
4. ↑ E. Schramm (2005): Genesis and "Disappearance" of Cybernetics. A literature review. ISOE Discussion Papers No. 25.