Fitting Constrained Additive Ordination (CAO)
A constrained additive ordination (CAO) model is fitted using the reduced-rank vector generalized additive model (RR-VGAM) framework.
cao(formula, family = stop("argument 'family' needs to be assigned"),
data = list(),
weights = NULL, subset = NULL, na.action = na.fail,
etastart = NULL, mustart = NULL, coefstart = NULL,
control = cao.control(...), offset = NULL,
method = "cao.fit", model = FALSE, x.arg = TRUE, y.arg = TRUE,
contrasts = NULL, constraints = NULL,
extra = NULL, qr.arg = FALSE, smart = TRUE, ...)formula |
a symbolic description of the model to be fit. The RHS of the
formula is used to construct the latent variables, upon which the
smooths are applied. All the variables in the formula are used
for the construction of latent variables except for those specified
by the argument |
family |
a function of class |
data |
an optional data frame containing the variables in the model.
By default the variables are taken from |
weights |
an optional vector or matrix of (prior) weights to be used in the
fitting process. For |
subset |
an optional logical vector specifying a subset of observations to be used in the fitting process. |
na.action |
a function which indicates what should happen when the data contain
|
etastart |
starting values for the linear predictors. It is a M-column
matrix. If M=1 then it may be a vector. For |
mustart |
starting values for the fitted values. It can be a vector or a
matrix. Some family functions do not make use of this argument.
For |
coefstart |
starting values for the coefficient vector. For |
control |
a list of parameters for controlling the fitting process.
See |
offset |
a vector or M-column matrix of offset values. These are
a priori known and are added to the linear predictors during
fitting. For |
method |
the method to be used in fitting the model. The default
(and presently only) method |
model |
a logical value indicating whether the model frame should be
assigned in the |
x.arg, y.arg |
logical values indicating whether the model matrix and response
vector/matrix used in the fitting process should be assigned in the
|
contrasts |
an optional list. See the |
constraints |
an optional list of constraint matrices. For |
extra |
an optional list with any extra information that might be needed by
the family function. For |
qr.arg |
For |
smart |
logical value indicating whether smart prediction
( |
... |
further arguments passed into |
The arguments of cao are a mixture of those from
vgam and cqo, but with some extras
in cao.control. Currently, not all of the
arguments work properly.
CAO can be loosely be thought of as the result of fitting generalized
additive models (GAMs) to several responses (e.g., species) against
a very small number of latent variables. Each latent variable is a
linear combination of the explanatory variables; the coefficients
C (called C below) are called constrained
coefficients or canonical coefficients, and are interpreted as
weights or loadings. The C are estimated by maximum likelihood
estimation. It is often a good idea to apply scale
to each explanatory variable first.
For each response (e.g., species), each latent variable is smoothed
by a cubic smoothing spline, thus CAO is data-driven. If each smooth
were a quadratic then CAO would simplify to constrained quadratic
ordination (CQO; formerly called canonical Gaussian ordination
or CGO).
If each smooth were linear then CAO would simplify to constrained
linear ordination (CLO). CLO can theoretically be fitted with
cao by specifying df1.nl=0, however it is more efficient
to use rrvglm.
Currently, only Rank=1 is implemented, and only
noRRR = ~1 models are handled.
With binomial data, the default formula is
logit(P[Y_s=1]) = eta_s = f_s(ν), \ \ \ s=1,2,…,S
where x_2 is a vector of environmental variables, and
nu=C^T x_2 is a R-vector of latent variables.
The eta_s is an additive predictor for species s,
and it models the probabilities of presence as an additive model on
the logit scale. The matrix C is estimated from the data, as
well as the smooth functions f_s. The argument noRRR = ~
1 specifies that the vector x_1, defined for RR-VGLMs
and QRR-VGLMs, is simply a 1 for an intercept.
Here, the intercept in the model is absorbed into the functions.
A clogloglink link may be preferable over a
logitlink link.
With Poisson count data, the formula is
log(E[Y_s]) = eta_s = f_s(ν)
which models the mean response as an additive models on the log scale.
The fitted latent variables (site scores) are scaled to have
unit variance. The concept of a tolerance is undefined for
CAO models, but the optimums and maximums are defined. The generic
functions Max and Opt should work for
CAO objects, but note that if the maximum occurs at the boundary then
Max will return a NA. Inference for CAO models
is currently undeveloped.
CAO is very costly to compute. With version 0.7-8 it took 28 minutes on a fast machine. I hope to look at ways of speeding things up in the future.
Use set.seed just prior to calling
cao() to make your results reproducible.
The reason for this is finding the optimal
CAO model presents a difficult optimization problem, partly because the
log-likelihood function contains many local solutions. To obtain the
(global) solution the user is advised to try many initial values.
This can be done by setting Bestof some appropriate value
(see cao.control). Trying many initial values becomes
progressively more important as the nonlinear degrees of freedom of
the smooths increase.
CAO models are computationally expensive, therefore
setting trace = TRUE is a good idea, as well as running it
on a simple random sample of the data set instead.
Sometimes the IRLS algorithm does not converge within the FORTRAN code. This results in warnings being issued. In particular, if an error code of 3 is issued, then this indicates the IRLS algorithm has not converged. One possible remedy is to increase or decrease the nonlinear degrees of freedom so that the curves become more or less flexible, respectively.
T. W. Yee
Yee, T. W. (2006). Constrained additive ordination. Ecology, 87, 203–213.
cao.control,
Coef.cao,
cqo,
latvar,
Opt,
Max,
calibrate.qrrvglm,
persp.cao,
poissonff,
binomialff,
negbinomial,
gamma2,
set.seed,
gam() in gam,
trapO.
## Not run:
hspider[, 1:6] <- scale(hspider[, 1:6]) # Standardized environmental vars
set.seed(149) # For reproducible results
ap1 <- cao(cbind(Pardlugu, Pardmont, Pardnigr, Pardpull) ~
WaterCon + BareSand + FallTwig + CoveMoss + CoveHerb + ReflLux,
family = poissonff, data = hspider, Rank = 1,
df1.nl = c(Pardpull= 2.7, 2.5),
Bestof = 7, Crow1positive = FALSE)
sort(deviance(ap1, history = TRUE)) # A history of all the iterations
Coef(ap1)
concoef(ap1)
par(mfrow = c(2, 2))
plot(ap1) # All the curves are unimodal; some quite symmetric
par(mfrow = c(1, 1), las = 1)
index <- 1:ncol(depvar(ap1))
lvplot(ap1, lcol = index, pcol = index, y = TRUE)
trplot(ap1, label = TRUE, col = index)
abline(a = 0, b = 1, lty = 2)
trplot(ap1, label = TRUE, col = "blue", log = "xy", which.sp = c(1, 3))
abline(a = 0, b = 1, lty = 2)
persp(ap1, col = index, lwd = 2, label = TRUE)
abline(v = Opt(ap1), lty = 2, col = index)
abline(h = Max(ap1), lty = 2, col = index)
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.