Thereafter, we denote by:
\[ D_{Mg} = \frac{S - 1}{\ln N} \]
\[ D_{Mn} = \frac{S}{\sqrt{N}} \]
Hurlbert (1971) unbiaised estimate of Sander (1968) rarefaction:
\[ E(S) = \sum_{i = 1}^{S} 1 - \frac{{N - N_i} \choose n}{N \choose n} \]
Diversity:
\[ H' = - \sum_{i = 1}^{S} p_i \ln p_i \]
Evenness:
\[ E = \frac{H'}{H'_{max}} = \frac{H'}{\ln S} = - \sum_{i = 1}^{S} p_i \log_S p_i \]
When \(p_i\) is unknown in the population, an estimate is given by \(\hat{p}_i =\frac{n_i}{N}\) (maximum likelihood estimator - MLE). As the use of \(\hat{p}_i\) results in a biased estimate, Hutcheson (1970) and Bowman et al. (1971) suggest the use of:
\[ \hat{H}' = - \sum_{i = 1}^{S} \hat{p}_i \ln \hat{p}_i - \frac{S - 1}{N} + \frac{1 - \sum_{i = 1}^{S} \hat{p}_i^{-1}}{12N^2} + \frac{\sum_{i = 1}^{S} (\hat{p}_i^{-1} - \hat{p}_i^{-2})}{12N^3} + \cdots \]
This error is rarely significant (Peet 1974), so the unbiaised form is not implemented here (for now).
Diversity:
\[ HB = \frac{\ln (N!) - \sum_{i = 1}^{S} \ln (n_i!)}{N} \]
Evenness:
\[ E = \frac{HB}{HB_{max}} \]
with:
\[ HB_{max} = \frac{1}{N} \ln \frac{N!}{\left( \lfloor \frac{N}{S} \rfloor! \right)^{S - r} \left[ \left( \lfloor \frac{N}{S} \rfloor + 1 \right)! \right]^{r}} \]
where: \(r = N - S \lfloor \frac{N}{S} \rfloor\).
The following methods return a dominance index, not the reciprocal or inverse form usually adopted, so that an increase in the value of the index accompanies a decrease in diversity.
Dominance for an infinite sample:
\[ D = \sum_{i = 1}^{S} p_i^2 \]
Dominance for a finite sample:
\[ \lambda = \sum_{i = 1}^{S} \frac{n_i \left( n_i - 1 \right)}{N \left( N - 1 \right)} \]
Dominance:
\[ D = \frac{N - U}{N - \sqrt{N}} \]
Evenness:
\[ E = \frac{N - U}{N - \frac{N}{\sqrt{S}}} \]
where \(U\) is the distance of the sample from the origin in an \(S\) dimensional hypervolume:
\[U = \sqrt{\sum_{i = 1}^{S} n_i^2}\]
Dominance:
\[ d = \frac{n_{max}}{N} \]
The following methods can be used to acertain the degree of turnover in taxa composition along a gradient on qualitative (presence/absence) data. This assumes that the order of the matrix rows (from 1 to \(m\)) follows the progression along the gradient/transect.
We denote the \(m \times p\) incidence matrix by \(X = \left[ x_{ij} \right] ~\forall i \in \left[ 1,m \right], j \in \left[ 1,p \right]\) and the \(p \times p\) corresponding co-occurrence matrix by \(Y = \left[ y_{ij} \right] ~\forall i,j \in \left[ 1,p \right]\), with row and column sums:
\[\begin{align} x_{i \cdot} = \sum_{j = 1}^{p} x_{ij} && x_{\cdot j} = \sum_{i = 1}^{m} x_{ij} && x_{\cdot \cdot} = \sum_{j = 1}^{p} \sum_{i = 1}^{m} x_{ij} && \forall x_{ij} \in \lbrace 0,1 \rbrace \\ y_{i \cdot} = \sum_{j \geqslant i}^{p} y_{ij} && y_{\cdot j} = \sum_{i \leqslant j}^{p} y_{ij} && y_{\cdot \cdot} = \sum_{i = 1}^{p} \sum_{j \geqslant i}^{p} y_{ij} && \forall y_{ij} \in \lbrace 0,1 \rbrace \end{align}\]\[ \beta_W = \frac{S}{\alpha} - 1 \]
where \(\alpha\) is the mean sample diversity: \(\alpha = \frac{x_{\cdot \cdot}}{m}\).
\[ \beta_C = \frac{g(H) + l(H)}{2} - 1 \]
where \(g(H)\) is the number of taxa gained along the transect and \(l(H)\) the number of taxa lost.
\[ \beta_T = \frac{g(H) + l(H)}{2\alpha} \]
where \(g(H)\) is the number of taxa gained along the transect, \(l(H)\) the number of taxa lost and \(\alpha\) the mean sample diversity, \(\alpha = \frac{x_{\cdot \cdot}}{m}\).
Similarity between two samples \(a\) and \(b\) can be measured as follow. These indices provide a scale of similarity from \(0\)-\(1\) where \(1\) is perfect similarity and \(0\) is no similarity (with the exception of the Brainerd-Robinson index which is scaled between \(0\) and \(200\))
\(a_i\) and \(b_i\) denote the number of individuals in the \(i\)-th taxon, \(i \in \left[ 1,n \right]\). \(j\) denotes the number of taxa common to both samples: \(j = \sum_{k = 1}^{n} a_k \cap b_k\).
\[ C_J = \frac{j}{S_a + S_b - j} \]
\[ C_S = \frac{2j}{S_a + S_b} \]
\[ C_{BR} = 200 - \sum_{i = 1}^{S} \left| \frac{a_i \times 100}{\sum_{i = 1}^{S} a_i} - \frac{b_i \times 100}{\sum_{i = 1}^{S} b_i} \right|\]
Bray and Curtis (1957) modified version of Sorenson’s index:
\[ C_N = \frac{2 \sum_{i = 1}^{S} \min(a_i, b_i)}{N_a + N_b} \]
\[ C_{MH} = \frac{2 \sum_{i = 1}^{S} a_i \times b_i}{(\frac{\sum_{i = 1}^{S} a_i^2}{N_a^2} + \frac{\sum_{i = 1}^{S} b_i^2}{N_b^2}) \times N_a \times N_b} \]
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