# beta Diversity

## N. Frerebeau

### 2025-04-02

\(\beta\)-diversity measures how different local systems are from one another (Moreno and Rodríguez 2010).

    ## Install extra packages (if needed)
    # install.packages("folio") # Datasets

    ## Load packages
    library(tabula)

    ## Ceramic data from Lipo et al. 2015
    data("mississippi", package = "folio")

# 1 Turnover

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 & Shmida (1984)
Table: Turnover measures. |

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 Similarity

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
Table: Qualitative similarity measures (between samples). |

Measure | Reference
--- | ---
$$ C_{BR} = 200 - \sum_{j = 1}^{S} \left | \frac{a_j \times 100}{\sum_{j = 1}^{S} a_j} - \frac{b_j \times 100}{\sum_{j = 1}^{S} b_j} \right
$$ C_N = \frac{2 \sum_{j = 1}^{S} \min(a_j, b_j)}{N_a + N_b} $$ | Bray & Curtis (1957), Sorenson
$$ 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
Table: Quantitative similarity measures (between samples). |

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: \(o_j = \sum_{k = 1}^{S} a_k \cap b_k\).

    ## Brainerd-Robinson (similarity between assemblages)
    BR <- similarity(mississippi, method = "brainerd")
    plot_spot(BR, col = khroma::colour("YlOrBr")(12))

plot of chunk similarity

# 3 References

Brainerd, G. W. 1951. The Place of Chronological Ordering in Archaeological Analysis. American Antiquity, 16(4), 301-313. DOI: 10.2307/276979.

Bray, J. R. & Curtis, J. T. (1957). An Ordination of the Upland Forest Communities of Southern Wisconsin. Ecological Monographs, 27(4), 325-349. DOI: 10.2307/1942268.

Cody, M. L. (1975). Towards a Theory of Continental Species Diversity: Bird Distributions Over Mediterranean Habitat Gradients. In M. L. Cody & J. M. Diamond (Eds.), Ecology and Evolution of Communities, 214-257. Cambridge, MA: Harvard University Press.

Moreno, C. E. & Rodríguez, P. (2010). A Consistent Terminology for Quantifying Species Diversity? Oecologia, 163(2), 279-782. DOI: 10.1007/s00442-010-1591-7.

Robinson, W. S. (1951). A Method for Chronologically Ordering Archaeological Deposits. American Antiquity, 16(4), 293-301. DOI: 10.2307/276978.

Routledge, R. D. (1977). On Whittaker’s Components of Diversity. Ecology, 58(5), 1120-1127. DOI: 10.2307/1936932.

Whittaker, R. H. (1960). Vegetation of the Siskiyou Mountains, Oregon and California. Ecological Monographs, 30(3), 279-338. DOI: 10.2307/1943563..

Wilson, M. V. & Shmida, A. (1984). Measuring Beta Diversity with Presence-Absence Data. The Journal of Ecology, 72(3), 1055-1064. DOI: 10.2307/2259551.