Consider a community with \(S\) distinct species and relative abundance vector \(\textbf{p}=(p_1, p_2, ..., p_S)\). Three important diversity indices are reviewed here.
Species Richness
The number of distinct species, \(S\), is clearly an important measure of diversity. This metric ignores, however, species relative abundances.
Shannon Entropy
Randomly select an individual from the community. The Shannon entropy, \(H_{Sh}=-\sum_{i}p_i(1-p_i)\), is a measure of the uncertainty of the species identity of the individual. A higher uncertainty implies a greater diversity.
Gini-Simpson Index
Randomly select two individuals from the community. The Gini-Simpson index, \(H_{\text{GS}}=1-\sum_i p_i^2\), gives the probability that the individuals represent different species. A higher probability implies a greater diversity.
Doubling Property
Let Community A consist of 10 individuals representing species1 and 5 individuals representing species2. Let Community B consist of 10 individuals representing species3 and 5 individuals representing species4. Let Community C be the combination of communities A and B. It seems intuitive that the diversity of Community C–however we measure it–ought to be twice that of Community A.
Important as they are, the indices \(H_{Sh}\) and \(H_{\text{GS}}\) for Community C are not twice that of Community A. They should not, therefore, be regarded as diversities.
Hill Numbers
Diversities ought to obey the doubling property. They should also measure conserved quantities and have the same units. Hill numbers are a class of such diversities. The Hill number order q is defined by \(^qD=\left(\sum_ip_i^q\right)^\frac{1}{1-q}\). (For q =1, see tutorial.)
The diversity indices reviewed above are related to Hill numbers:
\(^0D=S\)
\(^1D=e^{H_{Sh}}\)
\(^2D=\frac{1}{1-H_{\text{GS}}}\)
Your assignment is to complete the tutorial that we began in class and, sometime afterward, review it.