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, ...)
## S3 method for class 'fisherfit':
confint(object, parm, level = 0.95, ...)
## S3 method for class 'fisherfit':
profile(fitted, alpha = 0.01, maxsteps = 20, del = zmax/5,
...)
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, ...)
as.preston(x, tiesplit = TRUE, ...)plot functions.x and y axes.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 following items:as.fisher.nlm. 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. Function fisherfit estimates the standard error of
$\alpha$. However, the confidence limits cannot be directly
estimated from the standard errors, but you should use function
confint based on profile likelihood. Function confint
uses function confint.glm of the profile.fisherfit for the profile
likelihood. Function profile.fisherfit follows
profile.glm and finds the $\tau$ parameter or
signed square root of two times log-Likelihood profile. The profile can
be inspected with a plot function which shows the $\tau$
and a dotted line corresponding to the Normal assumption: if standard
errors can be directly used in Normal inference these two lines
are similar.
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 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.
Kempton, R.A. & Taylor, L.R. (1974). Log-series and log-normal parameters as diversity discriminators for Lepidoptera. Journal of Animal Ecology 43: 381--399.
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 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
plot(profile(mod))
confint(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