Rank – Abundance or Dominance / Diversity Models
Functions construct rank – abundance or dominance / diversity or Whittaker plots and fit brokenstick, pre-emption, log-Normal, Zipf and Zipf-Mandelbrot models of species abundance.
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## Default S3 method: radfit(x, ...) rad.null(x, family=poisson, ...) rad.preempt(x, family = poisson, ...) rad.lognormal(x, family = poisson, ...) rad.zipf(x, family = poisson, ...) rad.zipfbrot(x, family = poisson, ...) ## S3 method for class 'radline' predict(object, newdata, total, ...) ## S3 method for class 'radfit' plot(x, BIC = FALSE, legend = TRUE, ...) ## S3 method for class 'radfit.frame' plot(x, order.by, BIC = FALSE, model, legend = TRUE, as.table = TRUE, ...) ## S3 method for class 'radline' plot(x, xlab = "Rank", ylab = "Abundance", type = "b", ...) radlattice(x, BIC = FALSE, ...) ## S3 method for class 'radfit' lines(x, ...) ## S3 method for class 'radfit' points(x, ...) as.rad(x) ## S3 method for class 'rad' plot(x, xlab = "Rank", ylab = "Abundance", log = "y", ...)
Data frame, matrix or a vector giving species abundances, or an object to be plotted.
Error distribution (passed to
A fitted result object.
Ranks used for ordinations. All models can interpolate to non-integer “ranks” (although this may be approximate), but extrapolation may fail
The new total used for predicting abundance. Observed total count is used if this is omitted.
A vector used for ordering sites in plots.
Use Bayesian Information Criterion, BIC, instead of Akaike's AIC. The penalty in BIC is k = log(S) where S is the number of species, whereas AIC uses k = 2.
Show only the specified model. If missing, AIC is used
to select the model. The model names (which can be abbreviated)
Add legend of line colours.
Arrange panels starting from upper left corner (passed
Type of the plot,
Use logarithmic scale for given axis. The default
Other parameters to functions.
Rank–Abundance Dominance (RAD) or Dominance/Diversity plots (Whittaker 1965) display logarithmic species abundances against species rank order. These plots are supposed to be effective in analysing types of abundance distributions in communities. These functions fit some of the most popular models mainly following Wilson (1991).
zipfbrot fit the individual models
(described below) for a single vector (row of data frame), and
radfit fits all models. The argument of the function
radfit can be either a vector for a single community or a data
frame where each row represents a distinct community.
rad.null fits a brokenstick model where the expected
abundance of species at rank r is a[r] = J/S sum(from x=r to S) 1/x (Pielou
1975), where J is the total number of individuals (site total)
and S is the total number of species in the community. This
gives a Null model where the individuals are randomly distributed
among observed species, and there are no fitted parameters.
rad.preempt fits the niche preemption model,
a.k.a. geometric series or Motomura model, where the expected
abundance a of species at rank r is a[r] = J*alpha*(1-alpha)^(r-1). The only
estimated parameter is the preemption coefficient α which
gives the decay rate of abundance per rank. The niche preemption
model is a straight line in a RAD plot. Function
rad.lognormal fits a log-Normal model which assumes that the
logarithmic abundances are distributed Normally, or a[r] = exp(log(mu) + log(sigma) * N),
where N is a Normal deviate. Function
the Zipf model a[r] = J*p1*r^gamma where
p1 is the fitted proportion of the most abundant species,
and γ is a decay coefficient. The Zipf–Mandelbrot model
rad.zipfbrot) adds one parameter: a[r] = J*c*(r+beta)^gamma after which p1
of the Zipf model changes into a meaningless scaling constant
Log-Normal and Zipf models are generalized linear models
glm) with logarithmic link function. Zipf–Mandelbrot
adds one nonlinear parameter to the Zipf model, and is fitted using
nlm for the nonlinear parameter and estimating other
parameters and log-Likelihood with
model is fitted as a purely nonlinear model. There are no estimated
parameters in the Null model.
poisson which is
appropriate only for genuine counts (integers), but other families
link = "log" can be used. Families
gaussian may be appropriate for
abundance data, such as cover. The “best” model is selected by
AIC. Therefore “quasi” families such as
quasipoisson cannot be used: they do not have
AIC nor log-Likelihood needed in non-linear models.
All these functions have their own
plot functions. When
radfit was applied for a data frame,
Lattice graphics, and other
functions use ordinary graphics. The ordinary graphics functions
return invisibly an
ordiplot object for observed points,
identify.ordiplot can be used to label
selected species. Alternatively,
Lattice graphics to display each
model of a single site in a separate panel together with their AIC or
as.rad is a base function to construct ordered RAD
plot is used by other RAD
which pass extra arguments (such as
zipfbrot fit each a single RAD model to a
single site. The result object has class
glm, and can be handled by some (but not
radfit fits all models either to a single site or to
all rows of a data frame or a matrix. When fitted to a single site,
the function returns an object of class
"radfit" with items
y (observed values),
which is a list of fitted
"radline" models. When applied for a
data frame or matrix,
radfit function returns an object of
"radfit.frame" which is a list of
objects, each item names by the corresponding row name.
All result objects (
"radfit.frame") can be accessed with same method functions.
The following methods are available:
addition the fit results can be accessed with
residuals (inheriting from
residuals.glm). The graphical functions were discussed
above in Details.
The RAD models are usually fitted for proportions instead of original
abundances. However, nothing in these models seems to require division
of abundances by site totals, and original observations are used in
these functions. If you wish to use proportions, you must standardize
your data by site totals, e.g. with
decostand and use
family such as
The lognormal model is fitted in a standard way, but I do think this is not quite correct – at least it is not equivalent to fitting Normal density to log abundances like originally suggested (Preston 1948).
Some models may fail. In particular, estimation of the Zipf-Mandelbrot
model is difficult. If the fitting fails,
NA is returned.
Wilson (1991) defined preemption model as a[r] = J*p1*(1 - alpha)^(r-1), where p1 is the fitted proportion of the first species. However, parameter p1 is completely defined by α since the fitted proportions must add to one, and therefore I handle preemption as a one-parameter model.
Veiled log-Normal model was included in earlier releases of this
function, but it was removed because it was flawed: an implicit veil
line also appears in the ordinary log-Normal. The latest release version
Pielou, E.C. (1975) Ecological Diversity. Wiley & Sons.
Preston, F.W. (1948) The commonness and rarity of species. Ecology 29, 254–283.
Whittaker, R. H. (1965) Dominance and diversity in plant communities. Science 147, 250–260.
Wilson, J. B. (1991) Methods for fitting dominance/diversity curves. Journal of Vegetation Science 2, 35–46.
An alternative approach is to use
qqplot with any distribution.
For controlling graphics:
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data(BCI) mod <- rad.lognormal(BCI[5,]) mod plot(mod) mod <- radfit(BCI[1,]) ## Standard plot overlaid for all models ## Pre-emption model is a line plot(mod) ## log for both axes: Zipf model is a line plot(mod, log = "xy") ## Lattice graphics separately for each model radlattice(mod) # Take a subset of BCI to save time and nerves mod <- radfit(BCI[3:5,]) mod plot(mod, pch=".")