# 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

1 2 3 4 5 |

### Arguments

`x` |
Community data matrix or plotting object. |

`scales` |
Scales of Tsallis diversity. |

`norm` |
Logical, if |

`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 |

`raw` |
If |

`subset` |
logical expression indicating sites (rows) to keep:
missing values are taken as |

`theta, phi` |
angles defining the viewing
direction. |

`col` |
Colours used for surface. |

`zlim` |
Limits of vertical axis. |

`...` |
Other arguments which are passed to |

### 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)

Péter Sólymos,
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.

### See Also

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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 | ```
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)
``` |