# renyi: Renyi and Hill Diversities and Corresponding Accumulation... In vegan: Community Ecology Package

## Description

Function `renyi` find Rényi diversities with any scale or the corresponding Hill number (Hill 1973). Function `renyiaccum` finds these statistics with accumulating sites.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ```renyi(x, scales = c(0, 0.25, 0.5, 1, 2, 4, 8, 16, 32, 64, Inf), hill = FALSE) ## S3 method for class 'renyi' plot(x, ...) renyiaccum(x, scales = c(0, 0.5, 1, 2, 4, Inf), permutations = 100, raw = FALSE, collector = FALSE, subset, ...) ## S3 method for class 'renyiaccum' plot(x, what = c("Collector", "mean", "Qnt 0.025", "Qnt 0.975"), type = "l", ...) ## S3 method for class 'renyiaccum' persp(x, theta = 220, col = heat.colors(100), zlim, ...) ```

## Arguments

 `x` Community data matrix or plotting object. `scales` Scales of Rényi diversity. `hill` Calculate Hill numbers. `permutations` Usually an integer giving the number permutations, but can also be a list of control values for the permutations as returned by the function `how`, or a permutation matrix where each row gives the permuted indices. `raw` if `FALSE` then return summary statistics of permutations, and if `TRUE` then returns the individual permutations. `collector` Accumulate the diversities in the order the sites are in the data set, and the collector curve can be plotted against summary of permutations. The argument is ignored if `raw = TRUE`. `subset` logical expression indicating sites (rows) to keep: missing values are taken as `FALSE`. `what` Items to be plotted. `type` Type of plot, where `type = "l"` means lines. `theta` Angle defining the viewing direction (azimuthal) in `persp`. `col` Colours used for surface. Single colour will be passed on, and vector colours will be selected by the midpoint of a rectangle in `persp`. `zlim` Limits of vertical axis. `...` Other arguments which are passed to `renyi` and to graphical functions.

## Details

Common `diversity` indices are special cases of Rényi diversity

H.a = 1/(1-a) log sum(p^a)

where a is a scale parameter, and Hill (1975) suggested to use so-called ‘Hill numbers’ defined as N.a = exp(H.a). Some Hill numbers are the number of species with a = 0, exp(H') or the exponent of Shannon diversity with a = 1, inverse Simpson with a = 2 and 1/max(p) with a = Inf. According to the theory of diversity ordering, one community can be regarded as more diverse than another only if its Rényi diversities are all higher (Tóthmérész 1995).

The `plot` method for `renyi` uses lattice graphics, and displays the diversity values against each scale in separate panel for each site together with minimum, maximum and median values in the complete data.

Function `renyiaccum` is similar to `specaccum` but finds Rényi or Hill diversities at given `scales` for random permutations of accumulated sites. Its `plot` function uses lattice function `xyplot` to display the accumulation curves for each value of `scales` in a separate panel. In addition, it has a `persp` method to plot the diversity surface against scale and number and sites. Similar dynamic graphics can be made with `rgl.renyiaccum` in vegan3d package.

## Value

Function `renyi` returns a data frame of selected indices. Function `renyiaccum` with argument `raw = FALSE` returns a three-dimensional array, where the first dimension are the accumulated sites, second dimension are the diversity scales, and third dimension are the summary statistics `mean`, `stdev`, `min`, `max`, `Qnt 0.025` and `Qnt 0.975`. With argument `raw = TRUE` the statistics on the third dimension are replaced with individual permutation results.

## Author(s)

Roeland Kindt and Jari Oksanen

## References

Hill, M.O. (1973). Diversity and evenness: a unifying notation and its consequences. Ecology 54, 427–473.

Kindt, R., Van Damme, P., Simons, A.J. (2006). Tree diversity in western Kenya: using profiles to characterise richness and evenness. Biodiversity and Conservation 15, 1253–1270.

Tóthmérész, B. (1995). Comparison of different methods for diversity ordering. Journal of Vegetation Science 6, 283–290.

`diversity` for diversity indices, and `specaccum` for ordinary species accumulation curves, and `xyplot`, `persp` and `rgl.renyiaccum`.

## Examples

 ```1 2 3 4 5 6 7``` ```data(BCI) i <- sample(nrow(BCI), 12) mod <- renyi(BCI[i,]) plot(mod) mod <- renyiaccum(BCI[i,]) plot(mod, as.table=TRUE, col = c(1, 2, 2)) persp(mod) ```

### Example output

```Loading required package: permute