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.
fisherfit(x, ...)
prestonfit(x, tiesplit = TRUE, ...)
prestondistr(x, truncate = -1, ...)
"plot"(x, xlab = "Frequency", ylab = "Species", bar.col = "skyblue", line.col = "red", lwd = 2, ...)
"lines"(x, line.col = "red", lwd = 2, ...)
veiledspec(x, ...)
as.fisher(x, ...)
"plot"(x, xlab = "Frequency", ylab = "Species", bar.col = "skyblue", kind = c("bar", "hiplot", "points", "lines"), add = FALSE, ...)
as.preston(x, tiesplit = TRUE, ...)
"plot"(x, xlab = "Frequency", ylab = "Species", bar.col = "skyblue", ...)
"lines"(x, xadjust = 0.5, ...)plot functions.x and y axes."bar" is similar bar plot
as in plot.fisherfit, "hiplot" draws vertical lines
as with plot(..., type="h"), and "points" and
"lines" are obvious.prestonfit and tiesplit passed to as.preston in
prestondistr. 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
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 $f = S0 exp(-(log2(o)-mu)^2/2/sigma^2)$, where
$\mu$ is the location of the mode and $\sigma$ the width,
both in $log2$ scale, and $S0$ 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 $log2$ 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.
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.
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.
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)
Run the code above in your browser using DataLab