vegan (version 2.4-0)

eventstar: Scale Parameter at the Minimum of the Tsallis Evenness Profile

Description

The function eventstar finds the minimum ($q*$) of the evenness profile based on the Tsallis entropy. This scale factor of the entropy represents a specific weighting of species relative frequencies that leads to minimum evenness of the community (Mendes et al. 2008).

Usage

eventstar(x, qmax = 5)

Arguments

x

A community matrix or a numeric vector.

qmax

Maximum scale parameter of the Tsallis entropy to be used in finding the minimum of Tsallis based evenness in the range c(0, qmax).

Value

A data frame with columns:

• qstar scale parameter value $q*$ corresponding to minimum value of Tsallis based evenness profile.
• Estar Value of evenness based on normalized Tsallis entropy at $q*$.
• Hstar Value of Tsallis entropy at $q*$.
• Dstar Value of Tsallis entropy at $q*$ converted to numbers equivalents (also called as Hill numbers, effective number of species, ‘true’ diversity; cf. Jost 2007).

See tsallis for calculation details.

Details

The function eventstar finds a characteristic value of the scale parameter $q$ of the Tsallis entropy corresponding to minimum of the evenness (equitability) profile based on Tsallis entropy. This value was proposed by Mendes et al. (2008) as $q*$.

The $q*$ index represents the scale parameter of the one parameter Tsallis diversity family that leads to the greatest deviation from the maximum equitability given the relative abundance vector of a community.

The value of $q*$ is found by identifying the minimum of the evenness profile over scaling factor $q$ by one-dimensional minimization. Because evenness profile is known to be a convex function, it is guaranteed that underlying optimize function will find a unique solution if it is in the range c(0, qmax).

The scale parameter value $q*$ is used to find corresponding values of diversity ($H.q*$), evenness ($H.q*(max)$), and numbers equivalent ($D.q*$). For calculation details, see tsallis and Examples below.

Mendes et al. (2008) advocated the use of $q*$ and corresponding diversity, evenness, and Hill numbers, because it is a unique value representing the diversity profile, and is is positively associated with rare species in the community, thus it is a potentially useful indicator of certain relative abundance distributions of the communities.

References

Mendes, R.S., Evangelista, L.R., Thomaz, S.M., Agostinho, A.A. and Gomes, L.C. (2008) A unified index to measure ecological diversity and species rarity. Ecography 31, 450--456.

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

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

See Also

Tsallis entropy: tsallis

Examples

data(BCI)
(x <- eventstar(BCI[1:5,]))
## profiling
y <- as.numeric(BCI[10,])
(z <- eventstar(y))
q <- seq(0, 2, 0.05)
Eprof <- tsallis(y, scales=q, norm=TRUE)
Hprof <- tsallis(y, scales=q)
Dprof <- tsallis(y, scales=q, hill=TRUE)
opar <- par(mfrow=c(3,1))
plot(q, Eprof, type="l", main="Evenness")
abline(v=z$qstar, h=tsallis(y, scales=z$qstar, norm=TRUE), col=2)
plot(q, Hprof, type="l", main="Diversity")
abline(v=z$qstar, h=tsallis(y, scales=z$qstar), col=2)
plot(q, Dprof, type="l", main="Effective number of species")
abline(v=z$qstar, h=tsallis(y, scales=z$qstar, hill=TRUE), col=2)
par(opar)