beta Diversity
β-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")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 × p incidence matrix by X = [xij] ∀i ∈ [1, m], j ∈ [1, p] and the p × p corresponding co-occurrence matrix by Y = [yij] ∀i, j ∈ [1, p], with row and column sums:
| 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) |
| βE = exp (βI) − 1 | Routledge (1977) |
| $$ \beta_T = \frac{g(H) + l(H)}{2\alpha} $$ | Wilson & Shmida (1984) |
Where:
- α 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.
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 |
| Measure | Reference |
|---|---|
| $$ C_{BR} = 200 - \sum_{j = 1}^{S} \left| \frac{a_j \times 100}{\sum_{j = 1}^{S} a_j} - \frac{b_j \times 100}{\sum_{j = 1}^{S} b_j} \right|$$ | Brainerd (1951), Robinson (1951) |
| $$ C_N = \frac{2 \sum_{j = 1}^{S} \min(a_j, b_j)}{N_a + N_b} $$ | 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 |
| Measure | Reference |
|---|---|
| $$ C_{Bi} = \frac{o_i - N \times p}{\sqrt{N \times p \times (1 - p)}} $$ | Kintigh (2006) |
Where:
- Sa and Sb denote the total number of taxa observed in samples a and b, respectively,
- Na and Nb denote the total number of individuals in samples a and b, respectively,
- aj and bj denote the number of individuals in the j-th type/taxon, j ∈ [1, S],
- xi and yi denote the number of individuals in the i-th sample/case, i ∈ [1, m],
- oi denotes the number of sample/case common to both type/taxon: $o_i = \sum_{k = 1}^{m} x_k \cap y_k$,
- oj 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))
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.