Generic function for the tolerances of a model.
An object for which the computation or extraction of a tolerance or tolerances is meaningful.
Other arguments fed into the specific
methods function of the model. Sometimes they are fed
into the methods function for
Different models can define an optimum in different ways. Many models have no such notion or definition.
Tolerances occur in quadratic ordination, i.e., CQO and UQO. They have ecological meaning because a high tolerance for a species means the species can survive over a large environmental range (stenoecous species), whereas a small tolerance means the species' niche is small (eurycous species). Mathematically, the tolerance is like the variance of a normal distribution.
The value returned depends specifically on the methods
cqo binomial or Poisson fit, this
function returns a
R \times R \times S array, where R is the rank
and S is the number of species.
Each tolerance matrix ought to be positive-definite, and
for a rank-1 fit, taking the square root of each tolerance
matrix results in each species' tolerance (like a standard
There is a direct inverse relationship between the scaling of
the latent variables (site scores) and the tolerances.
One normalization is for the latent variables to have unit
Another normalization is for all the tolerances to be unit.
These two normalization cannot simultaneously hold in general.
For rank-R>1 models it becomes more complicated because
the latent variables are also uncorrelated. An important
argument when fitting quadratic ordination models is whether
See Yee (2004) for details.
Tolerances are undefined for ‘linear’ and additive ordination models. They are well-defined for quadratic ordination models.
Thomas W. Yee
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.
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## Not run: set.seed(111) # This leads to the global solution hspider[,1:6] <- scale(hspider[, 1:6]) # Standardized environmental vars p1 <- cqo(cbind(Alopacce, Alopcune, Alopfabr, Arctlute, Arctperi, Auloalbi, Pardlugu, Pardmont, Pardnigr, Pardpull, Trocterr, Zoraspin) ~ WaterCon + BareSand + FallTwig + CoveMoss + CoveHerb + ReflLux, poissonff, data = hspider, Crow1positive = FALSE) Tol(p1) ## End(Not run)
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