tsallis {vegan} R Documentation

## Tsallis Diversity and Corresponding Accumulation Curves

### Description

Function `tsallis` find Tsallis diversities with any scale or the corresponding evenness measures. Function `tsallisaccum` finds these statistics with accumulating sites.

### Usage

```tsallis(x, scales = seq(0, 2, 0.2), norm = FALSE, hill = FALSE)
tsallisaccum(x, scales = seq(0, 2, 0.2), permutations = 100, raw = FALSE, ...)
## S3 method for class 'tsallisaccum':
persp(x, theta = 220, phi = 15, col = heat.colors(100), zlim, ...)
```

### Arguments

 `x` Community data matrix or plotting object. `scales` Scales of Tsallis diversity. `norm` Logical, if `TRUE` diversity values are normalized by their maximum (diversity value at equiprobability conditions). `hill` Calculate Hill numbers. `permutations` Number of random permutations in accumulating sites. `raw` If `FALSE` then return summary statistics of permutations, and if TRUE then returns the individual permutations. `theta, phi` angles defining the viewing direction. `theta` gives the azimuthal direction and `phi` the colatitude. `col` Colours used for surface. `zlim` Limits of vertical axis. `...` Other arguments which are passed to `tsallis` and to graphical functions.

### Details

The Tsallis diversity (also equivalent to Patil and Taillie diversity) is a one-parametric generalised entropy function, defined as:

H.q = 1/(q-1)(1-sum(p^q))

where q is a scale parameter, S the number of species in the sample (Tsallis 1988, Tothmeresz 1995). This diversity is concave for all q>0, but non-additive (Keylock 2005). For q=0 it gives the number of species minus one, as q tends to 1 this gives Shannon diversity, for q=2 this gives the Simpson index (see function `diversity`).

If `norm = TRUE`, `tsallis` gives values normalized by the maximum:

H.q(max) = (S^(1-q)-1)/(1-q)

where S is the number of species. As q tends to 1, maximum is defined as ln(S).

If `hill = TRUE`, `tsallis` gives Hill numbers (numbers equivalents, see Jost 2007):

D.q = (1-(q-1)*H)^(1/(1-q))

Details on plotting methods and accumulating values can be found on the help pages of the functions `renyi` and `renyiaccum`.

### Value

Function `tsallis` returns a data frame of selected indices. Function `tsallisaccum` 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)

P'eter S'olymos, solymos@ualberta.ca, based on the code of Roeland Kindt and Jari Oksanen written for `renyi`

### References

Tsallis, C. (1988) Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phis. 52, 479–487.

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

Patil, G. P. and Taillie, C. (1982) Diversity as a concept and its measurement. J. Am. Stat. Ass. 77, 548–567.

Keylock, C. J. (2005) Simpson diversity and the Shannon-Wiener index as special cases of a generalized entropy. Oikos 109, 203–207.

Jost, L (2007) Partitioning diversity into independent alpha and beta components. Ecology 88, 2427–2439.

Plotting methods and accumulation routines are based on functions `renyi` and `renyiaccum`. An object of class 'tsallisaccum' can be used with function `rgl.renyiaccum` as well. See also settings for `persp`.

### Examples

```data(BCI)
i <- sample(nrow(BCI), 12)
x1 <- tsallis(BCI[i,])
x1
diversity(BCI[i,],"simpson") == x1[["2"]]
plot(x1)
x2 <- tsallis(BCI[i,],norm=TRUE)
x2
plot(x2)
mod1 <- tsallisaccum(BCI[i,])
plot(mod1, as.table=TRUE, col = c(1, 2, 2))
persp(mod1)
mod2 <- tsallisaccum(BCI[i,], norm=TRUE)
persp(mod2,theta=100,phi=30)
```

[Package vegan version 1.16-32 Index]