Function tsallis find Tsallis diversities with any scale or the corresponding evenness measures. Function tsallisaccum finds these statistics with accumulating sites.
tsallis(x, scales = seq(0, 2, 0.2), norm = FALSE, hill = FALSE)
tsallisaccum(x, scales = seq(0, 2, 0.2), permutations = 100,
raw = FALSE, subset, ...)
# S3 method for tsallisaccum
persp(x, theta = 220, phi = 15, col = heat.colors(100), zlim, ...)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.
Community data matrix or plotting object.
Scales of Tsallis diversity.
Logical, if TRUE diversity values are normalized
by their maximum (diversity value at equiprobability conditions).
Calculate Hill numbers.
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.
If FALSE then return summary statistics of
permutations, and if TRUE then returns the individual
permutations.
logical expression indicating sites (rows) to keep:
missing values are taken as FALSE.
angles defining the viewing
direction. theta gives the azimuthal direction and
phi the colatitude.
Colours used for surface.
Limits of vertical axis.
Other arguments which are passed to tsallis and
to graphical functions.
Péter Sólymos,
solymos@ualberta.ca, based on the code of Roeland Kindt and
Jari Oksanen written for renyi
The Tsallis diversity (also equivalent to Patil and Taillie diversity) is a one-parametric generalised entropy function, defined as:
$$H_q = \frac{1}{q-1} (1-\sum_{i=1}^S p_i^q)$$
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:
$$H_q(max) = \frac{S^{1-q}-1}{1-q}$$
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.
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 displayed with dynamic 3D function
rgl.renyiaccum in the vegan3d package. See also settings for
persp.
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)
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