Update references

README.md

@@ -166,7 +166,7 @@ l’information*. Nouvelle bibliothèque scientifique. Paris: Flammarion.
 
 <div id="ref-desachy2004" class="csl-entry">
 
-Desachy, Bruno. 2004. “Le sériographe EPPM: un outil informatisé de
+Desachy, B. 2004. “Le sériographe EPPM: un outil informatisé de
 sériation graphique pour tableaux de comptages.” *Revue archéologique de
 Picardie* 3 (1): 39–56. <https://doi.org/10.3406/pica.2004.2396>.
 

pkgdown/_pkgdown.yml

@@ -20,13 +20,3 @@ reference:
 - title: Datasets
   contents:
   - has_concept("datasets")
-
-navbar:
-  components:
-    articles:
-      text: Articles
-      menu:
-      - text: Diversity Measures
-        href: articles/diversity.html
-      - text: Bibliography
-        href: articles/bibliography.html

vignettes/beta.Rmd

@@ -0,0 +1,132 @@
+---
+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 packages}
+## Install extra packages (if needed)
+# install.packages("folio") # Datasets
+
+## Load packages
+library(tabula)
+```
+
+```{r data}
+## 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. *: implemented.
+
+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). *: implemented.
+
+| 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). *: implemented.
+
+| 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). *: implemented.
+
+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).