Random generation for constrained quadratic ordination (CQO).
rcqo(n, p, S, Rank = 1,
family = c("poisson", "negbinomial", "binomial-poisson",
"Binomial-negbinomial", "ordinal-poisson",
"Ordinal-negbinomial", "gamma2"),
eq.maximums = FALSE, eq.tolerances = TRUE, es.optimums = FALSE,
lo.abundance = if (eq.maximums) hi.abundance else 10,
hi.abundance = 100, sd.latvar = head(1.5/2^(0:3), Rank),
sd.optimums = ifelse(es.optimums, 1.5/Rank, 1) *
ifelse(scale.latvar, sd.latvar, 1),
sd.tolerances = 0.25, Kvector = 1, Shape = 1,
sqrt.arg = FALSE, log.arg = FALSE, rhox = 0.5, breaks = 4,
seed = NULL, optimums1.arg = NULL, Crow1positive = TRUE,
xmat = NULL, scale.latvar = TRUE)A \(n\) by \(p-1+S\) data frame with components and attributes. In the following the attributes are labelled with double quotes.
The environmental variables. This makes up the
\(n\) by \(p-1\) \(X_2\) matrix.
Note that x1 is not present; it is effectively a
vector of ones since it corresponds to an intercept term when
cqo is applied to the data.
The species data. This makes up the
\(n\) by \(S\) matrix \(Y\).
This will be of the form described by the argument
family.
The \(p-1\) by \(R\) matrix of constrained coefficients (or canonical coefficients). These are also known as weights or loadings.
The formula involving the species and environmental
variable names.
This can be used directly in the formula argument
of cqo.
The \(S\)-vector of species' maximums, on a log scale.
These are uniformly distributed between
log(lo.abundance) and log(hi.abundance).
The \(n\) by \(R\) matrix of site scores.
Each successive column (latent variable) has
sample standard deviation
equal to successive values of sd.latvar.
The linear/additive predictor value.
The \(S\) by \(R\) matrix of species' optimums.
The \(S\) by \(R\) matrix of species' tolerances. These are the square root of the diagonal elements of the tolerance matrices (recall that all tolerance matrices are restricted to being diagonal in this function).
Other attributes are "break",
"family", "Rank",
"lo.abundance", "hi.abundance",
"eq.tolerances", "eq.maximums",
"seed" as used.
Number of sites. It is denoted by \(n\) below.
Number of environmental variables, including an intercept term. It is denoted by \(p\) below. Must be no less than \(1+R\) in value.
Number of species. It is denoted by \(S\) below.
The rank or the number of latent variables or true dimension of the data on the reduced space. This must be either 1, 2, 3 or 4. It is denoted by \(R\).
What type of species data is to be returned.
The first choice is the default.
If binomial then a 0 means absence and 1 means presence.
If ordinal then the breaks argument is passed into
the breaks argument of cut.
Note that either the Poisson or
negative binomial distributions
are used to generate binomial and
ordinal data, and that
an upper-case choice is used for the
negative binomial distribution
(this makes it easier for the user).
If "gamma2" then this is the
2-parameter gamma distribution.
Logical. Does each species have the same maximum?
See arguments lo.abundance and hi.abundance.
Logical. Does each species have the
same tolerance? If TRUE then the common
value is 1 along
every latent variable, i.e., all species' tolerance matrices
are the order-\(R\) identity matrix.
Logical. Do the species have equally spaced optimums?
If TRUE then the quantity
\(S^{1/R}\) must be an
integer with value 2 or more. That is, there has to be an
appropriate number of species in total.
This is so that a grid
of optimum values is possible in \(R\)-dimensional
latent variable space
in order to place the species' optimums.
Also see the argument sd.tolerances.
Numeric. These are recycled to a vector of length \(S\).
The species have a maximum
between lo.abundance and hi.abundance. That is,
at their optimal environment, the mean abundance of each
species is between the two componentwise values.
If eq.maximums is TRUE then
lo.abundance and hi.abundance
must have the same values.
If eq.maximums is FALSE then the
logarithm of the maximums are uniformly distributed between
log(lo.abundance) and log(hi.abundance).
Numeric, of length \(R\) (recycled if necessary). Site scores along each latent variable have these standard deviation values. This must be a decreasing sequence of values because the first ordination axis contains the greatest spread of the species' site scores, followed by the second axis, followed by the third axis, etc.
Numeric, of length \(R\) (recycled if necessary).
If es.optimums = FALSE then,
for the \(r\)th latent variable axis,
the optimums of the species are generated from a
normal distribution centered about 0.
If es.optimums = TRUE then the \(S\) optimums
are equally spaced about 0 along every
latent variable axis.
Regardless of the value of es.optimums, the optimums
are then scaled to give standard deviation
sd.optimums[r].
Logical. If eq.tolerances = FALSE then, for the
\(r\)th latent variable, the
species' tolerances are
chosen from a normal distribution with mean 1 and
standard deviation
sd.tolerances[r].
However, the first species y1 has its tolerance matrix
set equal to the order-\(R\) identity matrix.
All tolerance matrices for all species are
diagonal in this function.
This argument is ignored if eq.tolerances is TRUE,
otherwise it is recycled to length \(R\) if necessary.
A vector of positive \(k\) values (recycled to length \(S\)
if necessary) for the negative binomial distribution
(see negbinomial for details).
Note that a natural default value does not exist,
however the default
value here is probably a realistic one, and that
for large values
of \(\mu\) one has \(Var(Y)=\mu^2/k\)
approximately.
A vector of positive \(\lambda\)
values (recycled to length
\(S\) if necessary) for the 2-parameter gamma
distribution (see
gamma2 for details).
Note that a natural default value
does not exist, however the default value
here is probably a realistic
one, and that
\(Var(Y) = \mu^2 / \lambda\).
Logical. Take the square-root of the
negative binomial counts?
Assigning sqrt.arg = TRUE
when family="negbinomial" means
that the resulting species data can be
considered very crudely to be
approximately Poisson distributed.
They will not integers in general but much easier
(less numerical
problems) to estimate using something like
cqo(..., family="poissonff").
Logical. Take the logarithm of the gamma random variates?
Assigning log.arg = TRUE
when family="gamma2" means
that the resulting species data can be
considered very crudely to be
approximately Gaussian distributed about its (quadratic) mean.
Numeric, less than 1 in absolute value.
The correlation between the environmental variables.
The correlation matrix is a matrix of 1's along the diagonal
and rhox in the off-diagonals.
Note that each environmental variable is normally distributed
with mean 0. The standard deviation of each environmental
variable is chosen so that the site scores have the determined
standard deviation, as given by argument sd.latvar.
If family is assigned an ordinal value then this argument
is used to define the cutpoints. It is fed into the
breaks argument of cut.
If given, it is passed into set.seed.
This argument can be used to obtain reproducible results.
If set, the value is saved as the "seed"
attribute of the returned value. The default will
not change the random generator state, and return
.Random.seed
as "seed" attribute.
If assigned and Rank = 1 then these are
the explicity optimums.
Recycled to length S.
See qrrvglm.control for details.
The \(n\) by \(p-1\) environmental matrix can be inputted.
Logical. If FALSE the argument
sd.latvar is ignored and no scaling of the latent variable
values is performed.
T. W. Yee
This function generates data coming from a
constrained quadratic
ordination (CQO) model. In particular,
data coming from a species packing model
can be generated
with this function.
The species packing model states that species have equal
tolerances, equal maximums, and optimums which are uniformly
distributed over the latent variable space. This can be
achieved by assigning the arguments es.optimums = TRUE,
eq.maximums = TRUE, eq.tolerances = TRUE.
At present, the Poisson and negative binomial abundances
are generated first using lo.abundance and
hi.abundance, and if family is binomial or ordinal
then it is converted into these forms.
In CQO theory the \(n\) by \(p\) matrix \(X\) is partitioned into two parts \(X_1\) and \(X_2\). The matrix \(X_2\) contains the `real' environmental variables whereas the variables in \(X_1\) are just for adjustment purposes; they contain the intercept terms and other variables that one wants to adjust for when (primarily) looking at the variables in \(X_2\). This function has \(X_1\) only being a matrix of ones, i.e., containing an intercept only.
Yee, T. W. (2004). A new technique for maximum-likelihood canonical Gaussian ordination. Ecological Monographs, 74, 685--701.
Yee, T. W. (2006). Constrained additive ordination. Ecology, 87, 203--213.
ter Braak, C. J. F. and Prentice, I. C. (1988). A theory of gradient analysis. Advances in Ecological Research, 18, 271--317.
cqo,
qrrvglm.control,
cut,
binomialff,
poissonff,
negbinomial,
gamma2.
if (FALSE) {
# Example 1: Species packing model:
n <- 100; p <- 5; S <- 5
mydata <- rcqo(n, p, S, es.opt = TRUE, eq.max = TRUE)
names(mydata)
(myform <- attr(mydata, "formula"))
fit <- cqo(myform, poissonff, mydata, Bestof = 3) # eq.tol = TRUE
matplot(attr(mydata, "latvar"), mydata[,-(1:(p-1))], col = 1:S)
persp(fit, col = 1:S, add = TRUE)
lvplot(fit, lcol = 1:S, y = TRUE, pcol = 1:S) # Same plot as above
# Compare the fitted model with the 'truth'
concoef(fit) # The fitted model
attr(mydata, "concoefficients") # The 'truth'
c(apply(attr(mydata, "latvar"), 2, sd),
apply(latvar(fit), 2, sd)) # Both values should be approx equal
# Example 2: negative binomial data fitted using a Poisson model:
n <- 200; p <- 5; S <- 5
mydata <- rcqo(n, p, S, fam = "negbin", sqrt = TRUE)
myform <- attr(mydata, "formula")
fit <- cqo(myform, fam = poissonff, dat = mydata) # I.tol = TRUE,
lvplot(fit, lcol = 1:S, y = TRUE, pcol = 1:S)
# Compare the fitted model with the 'truth'
concoef(fit) # The fitted model
attr(mydata, "concoefficients") # The 'truth'
}
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