2 \(\beta\)-diversity
2.1 Turnover
2.1.1 Definitions
The following methods can be used to ascertain 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:
\[\begin{align} x_{i \cdot} = \sum_{j = 1}^{p} x_{ij} && x_{\cdot j} = \sum_{i = 1}^{m} x_{ij} && x_{\cdot \cdot} = \sum_{j = 1}^{p} \sum_{i = 1}^{m} x_{ij} && \forall x_{ij} \in \lbrace 0,1 \rbrace \\ y_{i \cdot} = \sum_{j \geqslant i}^{p} y_{ij} && y_{\cdot j} = \sum_{i \leqslant j}^{p} y_{ij} && y_{\cdot \cdot} = \sum_{i = 1}^{p} \sum_{j \geqslant i}^{p} y_{ij} && \forall y_{ij} \in \lbrace 0,1 \rbrace \end{align}\]
| Measure | Reference |
|---|---|
| \[ \beta_W = \frac{S}{\alpha} - 1 \] | Whittaker (1960) * |
| \[ \beta_C = \frac{g(H) + l(H)}{2} - 1 \] | Cody (1975) * |
| \[ \beta_R = \frac{S^2}{2 y_{\cdot \cdot} + S} - 1 \] | Routledge (1977) * |
| \[ \beta_I = \log x_{\cdot \cdot} - \frac{\sum_{j = 1}^{p} x_{\cdot j} \log x_{\cdot j}}{x_{\cdot \cdot}} - \frac{\sum_{i = 1}^{m} x_{i \cdot} \log x_{i \cdot}}{x_{\cdot \cdot}} \] | Routledge (1977) * |
| \[ \beta_E = \exp(\beta_I) - 1 \] | Routledge (1977) * |
| \[ \beta_T = \frac{g(H) + l(H)}{2\alpha} \] | Wilson and Shmida (1984) * |
Where:
- \(\alpha\) is the mean sample diversity: \(\alpha = \frac{x_{\cdot \cdot}}{m}\),
- \(g(H)\) is the number of taxa gained along the transect,
- \(l(H)\) is the number of taxa lost along the transect.
2.1.2 Usage
2.2 Similarity
2.2.1 Definitions
Similarity between two samples \(a\) and \(b\) or between two types \(x\) and \(y\) 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 \(200\).
| Measure | Reference |
|---|---|
| \[ C_J = \frac{o_j}{S_a + S_b - o_j} \] | Jaccard * |
| \[ C_S = \frac{2 \times o_j}{S_a + S_b} \] | Sorenson * |
| Measure | Reference |
|---|---|
| \[ C_{BR} = 200 - \sum_{j = 1}^{S} \left| \frac{a_j \times 100}{\sum_{j = 1}^{S} a_j} - \frac{b_j \times 100}{\sum_{j = 1}^{S} b_j} \right|\] | (brainerd1951?), (robinson1951?) * |
| \[ C_N = \frac{2 \sum_{j = 1}^{S} \min(a_j, b_j)}{N_a + N_b} \] | Bray and Curtis (1957), Sorenson * |
| \[ C_{MH} = \frac{2 \sum_{j = 1}^{S} a_j \times b_j}{(\frac{\sum_{j = 1}^{S} a_j^2}{N_a^2} + \frac{\sum_{j = 1}^{S} b_j^2}{N_b^2}) \times N_a \times N_b} \] | Morisita-Horn * |
| Measure | Reference |
|---|---|
| \[ C_{Bi} = \frac{o_i - N \times p}{\sqrt{N \times p \times (1 - p)}} \] | (kintigh2006?) * |
Where:
- \(S_a\) and \(S_b\) denote the total number of taxa observed in samples \(a\) and \(b\), respectively,
- \(N_a\) and \(N_b\) denote the total number of individuals in samples \(a\) and \(b\), respectively,
- \(a_j\) and \(b_j\) denote the number of individuals in the \(j\)-th type/taxon, \(j \in \left[ 1,S \right]\),
- \(x_i\) and \(y_i\) denote the number of individuals in the \(i\)-th sample/case, \(i \in \left[ 1,m \right]\),
- \(o_i\) denotes the number of sample/case common to both type/taxon: \(o_i = \sum_{k = 1}^{m} x_k \cap y_k\),
- \(o_j\) denotes the number of type/taxon common to both sample/case: \(o_j = \sum_{k = 1}^{S} a_k \cap b_k\).
2.2.2 Usage
# Brainerd-Robinson (similarity between assemblages)
BR <- similarity(mississippi, method = "brainerd")
plot_spot(BR) +
khroma::scale_colour_YlOrBr()
# Binomial co-occurrence (similarity between types)
BI <- similarity(mississippi, method = "binomial")
plot_spot(BI) +
khroma::scale_colour_PRGn()
