---
title: "beta Diversity"
author: "N. Frerebeau"
date: "`r Sys.Date()`"
output:
  markdown::html_format:
    options:
      toc: true
      number_sections: true
bibliography: bibliography.bib
vignette: >
  %\VignetteIndexEntry{beta Diversity}
  %\VignetteEngine{knitr::knitr}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

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

```{r intro}
## 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 $m \times p$ 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.

# 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              |
|:--------------------------------------------|:-----------------------|
| $$ C_{BR} = 200 - \sum_{j = 1}^{S} \left| \frac{a_j \times 100}{\sum_{j = 1}^{S} a_j} - \frac{b_j \times 100}{\sum_{j = 1}^{S} b_j} \right|$$ | 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 |
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: $o_j = \sum_{k = 1}^{S} a_k \cap b_k$.

```{r similarity, fig.width=7, fig.height=5, fig.align="center"}
## 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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/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](https://doi.org/10.2307/276978).

Routledge, R. D. (1977). On Whittaker's Components of Diversity. *Ecology*, 58(5), 1120-1127. DOI: [10.2307/1936932](https://doi.org/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.](https://doi.org/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](https://doi.org/10.2307/2259551).