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vegan (version 2.0-10)

fisherfit: Fit Fisher's Logseries and Preston's Lognormal Model to Abundance Data

Description

Function fisherfit fits Fisher's logseries to abundance data. Function prestonfit groups species frequencies into doubling octave classes and fits Preston's lognormal model, and function prestondistr fits the truncated lognormal model without pooling the data into octaves.

Usage

fisherfit(x, ...)
prestonfit(x, tiesplit = TRUE, ...)
prestondistr(x, truncate = -1, ...)
## S3 method for class 'prestonfit':
plot(x, xlab = "Frequency", ylab = "Species", bar.col = "skyblue", 
    line.col = "red", lwd = 2, ...)
## S3 method for class 'prestonfit':
lines(x, line.col = "red", lwd = 2, ...)
veiledspec(x, ...)
as.fisher(x, ...)
## S3 method for class 'fisher':
plot(x, xlab = "Frequency", ylab = "Species", bar.col = "skyblue",
             kind = c("bar", "hiplot", "points", "lines"), add = FALSE, ...)
as.preston(x, tiesplit = TRUE, ...)
## S3 method for class 'preston':
plot(x, xlab = "Frequency", ylab = "Species", bar.col = "skyblue", ...)
## S3 method for class 'preston':
lines(x, xadjust = 0.5, ...)

Arguments

x
Community data vector for fitting functions or their result object for plot functions.
tiesplit
Split frequencies $1, 2, 4, 8$ etc between adjacent octaves.
truncate
Truncation point for log-Normal model, in log2 units. Default value $-1$ corresponds to the left border of zero Octave. The choice strongly influences the fitting results.
xlab, ylab
Labels for x and y axes.
bar.col
Colour of data bars.
line.col
Colour of fitted line.
lwd
Width of fitted line.
kind
Kind of plot to drawn: "bar" is similar bar plot as in plot.fisherfit, "hiplot" draws vertical lines as with plot(..., type="h"), and "points" and "lines" are
add
Add to an existing plot.
xadjust
Adjustment of horizontal positions in octaves.
...
Other parameters passed to functions. Ignored in prestonfit and tiesplit passed to as.preston in prestondistr.

Value

  • The function prestonfit returns an object with fitted coefficients, and with observed (freq) and fitted (fitted) frequencies, and a string describing the fitting method. Function prestondistr omits the entry fitted. The function fisherfit returns the result of nlm, where item estimate is $\alpha$. The result object is amended with the nuisance parameter and item fisher for the observed data from as.fisher

Details

In Fisher's logarithmic series the expected number of species $f$ with $n$ observed individuals is $f_n = \alpha x^n / n$ (Fisher et al. 1943). The estimation is possible only for genuine counts of individuals. The parameter $\alpha$ is used as a diversity index, and $\alpha$ and its standard error can be estimated with a separate function fisher.alpha. The parameter $x$ is taken as a nuisance parameter which is not estimated separately but taken to be $n/(n+\alpha)$. Helper function as.fisher transforms abundance data into Fisher frequency table.

Preston (1948) was not satisfied with Fisher's model which seemed to imply infinite species richness, and postulated that rare species is a diminishing class and most species are in the middle of frequency scale. This was achieved by collapsing higher frequency classes into wider and wider ``octaves'' of doubling class limits: 1, 2, 3--4, 5--8, 9--16 etc. occurrences. It seems that Preston regarded frequencies 1, 2, 4, etc.. as tied between octaves (Williamson & Gaston 2005). This means that only half of the species with frequency 1 are shown in the lowest octave, and the rest are transferred to the second octave. Half of the species from the second octave are transferred to the higher one as well, but this is usually not as large a number of species. This practise makes data look more lognormal by reducing the usually high lowest octaves. This can be achieved by setting argument tiesplit = TRUE. With tiesplit = FALSE the frequencies are not split, but all ones are in the lowest octave, all twos in the second, etc. Williamson & Gaston (2005) discuss alternative definitions in detail, and they should be consulted for a critical review of log-Normal model.

Any logseries data will look like lognormal when plotted in Preston's way. The expected frequency $f$ at abundance octave $o$ is defined by $f_o = S_0 \exp(-(\log_2(o) - \mu)^2/2/\sigma^2)$, where $\mu$ is the location of the mode and $\sigma$ the width, both in $\log_2$ scale, and $S_0$ is the expected number of species at mode. The lognormal model is usually truncated on the left so that some rare species are not observed. Function prestonfit fits the truncated lognormal model as a second degree log-polynomial to the octave pooled data using Poisson (when tiesplit = FALSE) or quasi-Poisson (when tiesplit = TRUE) error. Function prestondistr fits left-truncated Normal distribution to $\log_2$ transformed non-pooled observations with direct maximization of log-likelihood. Function prestondistr is modelled after function fitdistr which can be used for alternative distribution models.

The functions have common print, plot and lines methods. The lines function adds the fitted curve to the octave range with line segments showing the location of the mode and the width (sd) of the response. Function as.preston transforms abundance data to octaves. Argument tiesplit will not influence the fit in prestondistr, but it will influence the barplot of the octaves.

The total extrapolated richness from a fitted Preston model can be found with function veiledspec. The function accepts results both from prestonfit and from prestondistr. If veiledspec is called with a species count vector, it will internally use prestonfit. Function specpool provides alternative ways of estimating the number of unseen species. In fact, Preston's lognormal model seems to be truncated at both ends, and this may be the main reason why its result differ from lognormal models fitted in Rank--Abundance diagrams with functions rad.lognormal.

References

Fisher, R.A., Corbet, A.S. & Williams, C.B. (1943). The relation between the number of species and the number of individuals in a random sample of animal population. Journal of Animal Ecology 12: 42--58.

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

Williamson, M. & Gaston, K.J. (2005). The lognormal distribution is not an appropriate null hypothesis for the species--abundance distribution. Journal of Animal Ecology 74, 409--422.

See Also

diversity, fisher.alpha, radfit, specpool. Function fitdistr of MASS package was used as the model for prestondistr. Function density can be used for smoothed ``non-parametric'' estimation of responses, and qqplot is an alternative, traditional and more effective way of studying concordance of observed abundances to any distribution model.

Examples

Run this code
data(BCI)
mod <- fisherfit(BCI[5,])
mod
# prestonfit seems to need large samples
mod.oct <- prestonfit(colSums(BCI))
mod.ll <- prestondistr(colSums(BCI))
mod.oct
mod.ll
plot(mod.oct)  
lines(mod.ll, line.col="blue3") # Different
## Smoothed density
den <- density(log2(colSums(BCI)))
lines(den$x, ncol(BCI)*den$y, lwd=2) # Fairly similar to mod.oct
## Extrapolated richness
veiledspec(mod.oct)
veiledspec(mod.ll)

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