# beta Diversity

### 2024-05-16

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

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

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:

mk\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}mk

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} $$Morisita-Horn Table: Quantitative similarity measures (between samples). Measure Reference$$ C_{Bi} = \frac{o_i - N \times p}{\sqrt{N \times p \times (1 - p)}}  Kintigh (2006)
Table: Quantitative similarity measures (between types).

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))


## Binomial co-occurrence (similarity between types)
BI <- similarity(mississippi, method = "binomial")
plot_spot(BI, col = khroma::colour("PRGn")(12))


# 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.

Kintigh, K. (2006). Ceramic Dating and Type Associations. In J. Hantman & R. Most (Eds.), Managing Archaeological Data: Essays in Honor of Sylvia W. Gaines, 17–26. Anthropological Research Paper 57. Tempe, AZ: Arizona State University. DOI: 10.6067/XCV8J38QSS.

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.