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In ecology, a diversity index is a statistic which is intended to measure the biodiversity of an ecosystem. More generally, diversity indices can be used to assess the diversity of any population in which each member belongs to a unique species. Estimators for diversity indices are likely to be biased, so caution is advisable when comparing similar values. The word ecology is often used in common parlance as a synonym for the natural environment or environmentalism. ...
Rainforests are the most biodiverse ecosystems on earth Biodiversity or biological diversity is the diversity of life. ...
In ecology, the word ecosystem is an abbreviation of the term, ecological system. ...
In statistics, an estimator is a function of the known data that is used to estimate an unknown parameter; an estimate is the result from the actual application of the function to a particular set of data. ...
For other uses, see Bias (disambiguation). ...
Species richness The species richness S is simply the number of species present in an ecosystem. This index makes no use of relative abundances. Species Richness is a measure of biodiversity. ...
Simpson's diversity index If pi is the fraction of all organisms which belong to the i-th species, then Simpson's diversity index is most commonly defined as the statistic Simpsons Diversity Index is a measure of diversity. ...
 This quantity was introduced by Edward Hugh Simpson.
If ni is the number of individuals of species i which are counted, and N is the total number of all individuals counted, then
 is an estimator for Simpson's index for sampling without replacement.
Note that  , with values near zero corresponding to highly diverse or heterogeneous ecosystems and values near one corresponding to more homogeneous ecosystems. Biologists who find this confusing sometimes use 1 / D instead; confusingly, this reciprocal quantity is also called Simpson's index. A more sensible response is to redefine Simpson's index as
 (called by statisticians the index of diversity), since The index of diversity (also referred to as the Index of Variability) is a commonly used measure, in demographic research, to determine the variation in categorical data. ...
- this quantity has a simple intuitive interpretation: it represents the probability that if we randomly choose two individuals, that they will belong to distinct species,
- this quantity is comparable with the so-called Shannon-Weaver diversity index, which has an even better theoretical justification as a measure of statistical inhomogeneity.
Shannon's diversity index Shannon's diversity index is simply the ecologist's name for the communication entropy introduced by Claude Shannon: The Shannon–Weaver Index (also called the Shannon Index or the Shannon–Wiener Index) is one of several diversity indices used to measure biodiversity. ...
Claude Elwood Shannon (April 30, 1916 _ February 24, 2001) has been called the father of information theory, and was the founder of practical digital circuit design theory. ...
 where pi is the fraction of individuals belonging to the i-th species. This is by far the most widely used diversity index. The intuitive significance of this index can be described as follows. Suppose we devise binary codewords for each species in our ecosystem, with short codewords used for the most abundant species, and longer codewords for rare species. As we walk around and observe individual organisms, we call out the corresponding codeword. This gives a binary sequence. If we have used an efficient code, we will be able to save some breath by calling out a shorter sequence than would otherwise be the case. If so, the average codeword length we call out as we wander around will be close to the Shannon diversity index. In telecommunication, the term code word has the following meanings: A cryptonym used to identify sensitive intelligence data. ...
It is possible to write down estimators which attempt to correct for bias in finite sample sizes, but this would be misleading since communication entropy does not really fit expectations based upon parametric statistics. Differences arising from using two different estimators are likely to be overwhelmed by errors arising from other sources. Current best practice tends to use bootstrapping procedures to estimate communication entropy. Bootstrapping alludes to a German legend about Baron Münchhausen, who was able to lift himself out of a swamp by pulling himself up by his own hair. ...
Shannon himself showed that his communication entropy enjoys some powerful formal properties, and furthermore, it is the unique quantity which does so. These observations are the foundation of its interpretation as a measure of statistical diversity (or "surprise", in the arena of communications). The applications of this quantity go far beyond the one discussed here; see the textbook cited below for an elementary survey of the extraordinary richness of modern information theory.
Berger-Parker index The Berger-Parker diversity index is simply  This is an example of an index which uses only partial information about the relative abundances of the various species in its definition.
Renyi entropy The Species richness, the Shannon index, Simpson's index, and the Berger-Parker index can all be identified as particular examples of quantities bearing a simple relation to the Renyi entropy, In information theory, the Rényi entropy, a generalisation of Shannon entropy, is one of a family of functionals for quantifying the diversity, uncertainty or randomness of a system. ...
 for α approaching  respectively.
Unfortunately, the powerful formal properties of communication entropy do not generalize to Renyi's entropy, which largely explains the much greater power and popularity of Shannon's index with respect to its competitors.
External links
References - Colinvaux, Paul A. (1973). Introduction to Ecology. Wiley. ISBN 0-471-16498-4.
- Cover, Thomas M.; and Thomas, Joy A. (1991). Elements of Information Theory. Wiley. See chapter 5 for an elaboration of coding procedures described informally above.
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