sads (version 0.6.3)

fitsadC: ML fitting of species abundance distributions to data pooled in abundance classes

Description

Fits probability distributions for abundances of species aggregated in abundance classes in a sample or assemblage by maximum likelihood.

Usage

fitsadC(x, sad = c("exp","gamma","lnorm","pareto", "weibull"), ...)
fitexpC(x, trunc = NULL, start.value, ...)
fitgammaC(x, trunc, start.value,  ...)
fitlnormC(x, trunc, start.value,  ...)
fitparetoC(x, trunc, start.value,  upper = 20, ...)
fitweibullC(x, trunc, start.value,  ...)

Value

An object of fitsadC-class which inherits from mle2-class

and thus has methods for handling results of maximum likelihood fits from mle2

and also specific methods to handle SADs models (see fitsadC-class).

Arguments

x

object of class histogram with the number of species in abundance classes, in a sample or ecological assemblage.

sad

character; root name of community sad distribution to be fitted. "exp" for exponential distribution, "gamma" for gamma distribution, "lnorm" for lognormal, "pareto" for Pareto distribution, "weibull" for Weibull distribution.

trunc

non-negative integer, trunc > min(x); truncation point to fit a truncated distribution.

start.value

named numeric vector; starting values of free parameters to be passed to mle2. Parameters should be named as in the corresponding density function, and in the same order.

upper

real positive; upper bound for Brent's one-parameter optimization method (default), for fits that use this method by default. See details and optim.

...

in fitsad further arguments to be passed to the specific fitting function (most used are trunc, start.value) In the specific fitting functions further arguments to be passed to mle2.

Author

Paulo I Prado , Murilo Dantas Miranda and Andre Chalom, after Ben Bolker, R Core Team.

Details

fitsadC is simply a wrapper that calls the specific functions to fit the distribution chosen with the argument sad. Users can interchangeably use fitsadC or the individual functions detailed below (e.g. fitsad(x, sad="exp", ...) is the same as fitexpC(x, ...) and so on).

The distributions are fitted by the maximum likelihood method using numerical optimization, with mle2. The resulting object is of fitsadC-class which can be handled with mle2 methods for fitted models and has also some additional methods for SADs models (see fitsadC-class and examples).

For counts of species in abundances classes the likelihood function is

$$L(\theta) = \sum^C n_i \ln P_i$$ for i = 1, 2, 3, ... C, where C is the number of abundance classes, \(n_i\) is the number of species in abundance class i.

\(P_i\) is the probability attributed by the distribution model to the observation of one species in class i, which depends on the vector \(\theta\) of free parameters of the distribution model:

$$P_i = \int_{L_i}^{U_i} F(x \mid \theta) dx$$

where F(x|theta) is the value of the probability density function for a cover value x, under parameter values fixed at theta, and L_i and U_i are the lower and upper limits of the cover class i.

See fitsad for descriptions of each distribution model.

References

Bolker, B. and R Development Core Team 2012. bbmle: Tools for general maximum likelihood estimation. R package version 1.0.5.2. http://CRAN.R-project.org/package=bbmle

Chapman, D. G. 1956. Estimating the parameters of a truncated gamma distribution. The Annals of Mathematical Statistics, 27(2): 498--506.

Magurran, A.E. 1989. Ecological diversity and its measurement. Princenton University Press.

Preston, F.W. 1948. The commonness and rarity of species. Ecology 29: 254--283.

Rinne, H. 2009. The Weibull distribution: a handbook. CRC Press

See Also

dpareto,for corresponding density functions created for fitting SADs; standard distributions dexp, dgamma, dlnorm, dweibull; fitsadC-class.

Examples

## An example using data from Vieira et al, see dataset "grasslands"
## Breakpoints of the abundance classes used (cover classes)
vieira.brk <- c(0,1,3,5,seq(15,100, by=10),100)
##  creates an histogram object 
grass.h <- hist(grasslands$mids, breaks = vieira.brk, plot = FALSE)
#Fits Pareto, lognormal and gamma distributions
grass.p <- fitparetoC(grass.h)
grass.l <- fitlnormC(grass.h)
grass.g <- fitgammaC(grass.h)
## Predicted values by each model
grass.p.pred <- coverpred(grass.p)
grass.l.pred <- coverpred(grass.l)
grass.g.pred <- coverpred(grass.g)
## model selection
AICctab(grass.p, grass.l, grass.g, weights =TRUE, base = TRUE)
## A histogram with the densities predicted by each model
plot(grass.h, main = "", xlab = "Abundance (cover)")
## Adds predicted densities by each model to the plot
points(grass.p.pred, col = 1)
points(grass.l.pred, col = 2)
points(grass.g.pred, col = 3)
legend("topright", legend=c("Pareto","Log-normal", "Gamma"), col = 1:3, lty=1, pch =1)