VGAM (version 1.0-4)

# qrrvglm.control: Control Function for QRR-VGLMs (CQO)

## Description

Algorithmic constants and parameters for a constrained quadratic ordination (CQO), by fitting a quadratic reduced-rank vector generalized linear model (QRR-VGLM), are set using this function. It is the control function for cqo.

## Usage

qrrvglm.control(Rank = 1,
Bestof = if (length(Cinit)) 1 else 10,
checkwz = TRUE,
Cinit = NULL,
Crow1positive = TRUE,
epsilon = 1.0e-06,
EqualTolerances = NULL,
eq.tolerances = TRUE,
Etamat.colmax = 10,
FastAlgorithm = TRUE,
Hstep = 0.001,
isd.latvar = rep_len(c(2, 1, rep_len(0.5, Rank)), Rank),
iKvector = 0.1,
iShape = 0.1,
ITolerances = NULL,
I.tolerances = FALSE,
maxitl = 40,
imethod = 1,
Maxit.optim = 250,
MUXfactor = rep_len(7, Rank),
noRRR = ~ 1, Norrr = NA,
optim.maxit = 20,
Parscale = if (I.tolerances) 0.001 else 1.0,
sd.Cinit = 0.02,
SmallNo = 5.0e-13,
trace = TRUE,
Use.Init.Poisson.QO = TRUE,
wzepsilon = .Machine$double.eps^0.75, ...) ## Arguments Rank The numerical rank $$R$$ of the model, i.e., the number of ordination axes. Must be an element from the set {1,2,…,min($$M$$,$$p_2$$)} where the vector of explanatory variables $$x$$ is partitioned into ($$x_1$$,$$x_2$$), which is of dimension $$p_1+p_2$$. The variables making up $$x_1$$ are given by the terms in the noRRR argument, and the rest of the terms comprise $$x_2$$. Bestof Integer. The best of Bestof models fitted is returned. This argument helps guard against local solutions by (hopefully) finding the global solution from many fits. The argument has value 1 if an initial value for $$C$$ is inputted using Cinit. checkwz logical indicating whether the diagonal elements of the working weight matrices should be checked whether they are sufficiently positive, i.e., greater than wzepsilon. If not, any values less than wzepsilon are replaced with this value. Cinit Optional initial $$C$$ matrix, which must be a $$p_2$$ by $$R$$ matrix. The default is to apply .Init.Poisson.QO() to obtain initial values. Crow1positive Logical vector of length Rank (recycled if necessary): are the elements of the first row of $$C$$ positive? For example, if Rank is 4, then specifying Crow1positive = c(FALSE, TRUE) will force $$C[1,1]$$ and $$C[1,3]$$ to be negative, and $$C[1,2]$$ and $$C[1,4]$$ to be positive. This argument allows for a reflection in the ordination axes because the coefficients of the latent variables are unique up to a sign. epsilon Positive numeric. Used to test for convergence for GLMs fitted in C. Larger values mean a loosening of the convergence criterion. If an error code of 3 is reported, try increasing this value. eq.tolerances Logical indicating whether each (quadratic) predictor will have equal tolerances. Having eq.tolerances = TRUE can help avoid numerical problems, especially with binary data. Note that the estimated (common) tolerance matrix may or may not be positive-definite. If it is then it can be scaled to the $$R$$ by $$R$$ identity matrix, i.e., made equivalent to I.tolerances = TRUE. Setting I.tolerances = TRUE will force a common $$R$$ by $$R$$ identity matrix as the tolerance matrix to the data even if it is not appropriate. In general, setting I.tolerances = TRUE is preferred over eq.tolerances = TRUE because, if it works, it is much faster and uses less memory. However, I.tolerances = TRUE requires the environmental variables to be scaled appropriately. See Details for more details. EqualTolerances Defunct argument. Use eq.tolerances instead. Etamat.colmax Positive integer, no smaller than Rank. Controls the amount of memory used by .Init.Poisson.QO(). It is the maximum number of columns allowed for the pseudo-response and its weights. In general, the larger the value, the better the initial value. Used only if Use.Init.Poisson.QO = TRUE. FastAlgorithm Logical. Whether a new fast algorithm is to be used. The fast algorithm results in a large speed increases compared to Yee (2004). Some details of the fast algorithm are found in Appendix A of Yee (2006). Setting FastAlgorithm = FALSE will give an error. GradientFunction Logical. Whether optim's argument gr is used or not, i.e., to compute gradient values. Used only if FastAlgorithm is TRUE. The default value is usually faster on most problems. Hstep Positive value. Used as the step size in the finite difference approximation to the derivatives by optim. isd.latvar Initial standard deviations for the latent variables (site scores). Numeric, positive and of length $$R$$ (recycled if necessary). This argument is used only if I.tolerances = TRUE. Used by .Init.Poisson.QO() to obtain initial values for the constrained coefficients $$C$$ adjusted to a reasonable value. It adjusts the spread of the site scores relative to a common species tolerance of 1 for each ordination axis. A value between 0.5 and 10 is recommended; a value such as 10 means that the range of the environmental space is very large relative to the niche width of the species. The successive values should decrease because the first ordination axis should have the most spread of site scores, followed by the second ordination axis, etc. iKvector, iShape Numeric, recycled to length $$S$$ if necessary. Initial values used for estimating the positive $$k$$ and $$\lambda$$ parameters of the negative binomial and 2-parameter gamma distributions respectively. For further information see negbinomial and gamma2. These arguments override the ik and ishape arguments in negbinomial and gamma2. I.tolerances Logical. If TRUE then the (common) tolerance matrix is the $$R$$ by $$R$$ identity matrix by definition. Note that having I.tolerances = TRUE implies eq.tolerances = TRUE, but not vice versa. Internally, the quadratic terms will be treated as offsets (in GLM jargon) and so the models can potentially be fitted very efficiently. However, it is a very good idea to center and scale all numerical variables in the $$x_2$$ vector. See Details for more details. The success of I.tolerances = TRUE often depends on suitable values for isd.latvar and/or MUXfactor. ITolerances Defunct argument. Use I.tolerances instead. maxitl Maximum number of times the optimizer is called or restarted. Most users should ignore this argument. imethod Method of initialization. A positive integer 1 or 2 or 3 etc. depending on the VGAM family function. Currently it is used for negbinomial and gamma2 only, and used within the C. Maxit.optim Positive integer. Number of iterations given to the function optim at each of the optim.maxit iterations. MUXfactor Multiplication factor for detecting large offset values. Numeric, positive and of length $$R$$ (recycled if necessary). This argument is used only if I.tolerances = TRUE. Offsets are $$-0.5$$ multiplied by the sum of the squares of all $$R$$ latent variable values. If the latent variable values are too large then this will result in numerical problems. By too large, it is meant that the standard deviation of the latent variable values are greater than MUXfactor[r] * isd.latvar[r] for r=1:Rank (this is why centering and scaling all the numerical predictor variables in $$x_2$$ is recommended). A value about 3 or 4 is recommended. If failure to converge occurs, try a slightly lower value. optim.maxit Positive integer. Number of times optim is invoked. At iteration i, the ith value of Maxit.optim is fed into optim. noRRR Formula giving terms that are not to be included in the reduced-rank regression (or formation of the latent variables), i.e., those belong to $$x_1$$. Those variables which do not make up the latent variable (reduced-rank regression) correspond to the $$B_1$$ matrix. The default is to omit the intercept term from the latent variables. Norrr Defunct. Please use noRRR. Use of Norrr will become an error soon. Parscale Numerical and positive-valued vector of length $$C$$ (recycled if necessary). Passed into optim(..., control = list(parscale = Parscale)); the elements of $$C$$ become $$C$$ / Parscale. Setting I.tolerances = TRUE results in line searches that are very large, therefore $$C$$ has to be scaled accordingly to avoid large step sizes. See Details for more information. It's probably best to leave this argument alone. sd.Cinit Standard deviation of the initial values for the elements of $$C$$. These are normally distributed with mean zero. This argument is used only if Use.Init.Poisson.QO = FALSE and $$C$$ is not inputted using Cinit. trace Logical indicating if output should be produced for each iteration. The default is TRUE because the calculations are numerically intensive, meaning it may take a long time, so that the user might think the computer has locked up if trace = FALSE. SmallNo Positive numeric between .Machine$double.eps and 0.0001. Used to avoid under- or over-flow in the IRLS algorithm. Used only if FastAlgorithm is TRUE.

Use.Init.Poisson.QO

Logical. If TRUE then the function .Init.Poisson.QO() is used to obtain initial values for the canonical coefficients $$C$$. If FALSE then random numbers are used instead.

wzepsilon

Small positive number used to test whether the diagonals of the working weight matrices are sufficiently positive.

Ignored at present.

## Value

A list with components matching the input names.

## Warning

The default value of Bestof is a bare minimum for many datasets, therefore it will be necessary to increase its value to increase the chances of obtaining the global solution.

## Details

Recall that the central formula for CQO is $$\eta = B_1^T x_1 + A \nu + \sum_{m=1}^M (\nu^T D_m \nu) e_m$$ where $$x_1$$ is a vector (usually just a 1 for an intercept), $$x_2$$ is a vector of environmental variables, $$\nu=C^T x_2$$ is a $$R$$-vector of latent variables, $$e_m$$ is a vector of 0s but with a 1 in the $$m$$th position. QRR-VGLMs are an extension of RR-VGLMs and allow for maximum likelihood solutions to constrained quadratic ordination (CQO) models.

Having I.tolerances = TRUE means all the tolerance matrices are the order-$$R$$ identity matrix, i.e., it forces bell-shaped curves/surfaces on all species. This results in a more difficult optimization problem (especially for 2-parameter models such as the negative binomial and gamma) because of overflow errors and it appears there are more local solutions. To help avoid the overflow errors, scaling $$C$$ by the factor Parscale can help enormously. Even better, scaling $$C$$ by specifying isd.latvar is more understandable to humans. If failure to converge occurs, try adjusting Parscale, or better, setting eq.tolerances = TRUE (and hope that the estimated tolerance matrix is positive-definite). To fit an equal-tolerances model, it is firstly best to try setting I.tolerances = TRUE and varying isd.latvar and/or MUXfactor if it fails to converge. If it still fails to converge after many attempts, try setting eq.tolerances = TRUE, however this will usually be a lot slower because it requires a lot more memory.

With a $$R > 1$$ model, the latent variables are always uncorrelated, i.e., the variance-covariance matrix of the site scores is a diagonal matrix.

If setting eq.tolerances = TRUE is used and the common estimated tolerance matrix is positive-definite then that model is effectively the same as the I.tolerances = TRUE model (the two are transformations of each other). In general, I.tolerances = TRUE is numerically more unstable and presents a more difficult problem to optimize; the arguments isd.latvar and/or MUXfactor often must be assigned some good value(s) (possibly found by trial and error) in order for convergence to occur. Setting I.tolerances = TRUE forces a bell-shaped curve or surface onto all the species data, therefore this option should be used with deliberation. If unsuitable, the resulting fit may be very misleading. Usually it is a good idea for the user to set eq.tolerances = FALSE to see which species appear to have a bell-shaped curve or surface. Improvements to the fit can often be achieved using transformations, e.g., nitrogen concentration to log nitrogen concentration.

Fitting a CAO model (see cao) first is a good idea for pre-examining the data and checking whether it is appropriate to fit a CQO model.

## References

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.

cqo, rcqo, Coef.qrrvglm, Coef.qrrvglm-class, optim, binomialff, poissonff, negbinomial, gamma2, gaussianff.

## Examples

Run this code
# NOT RUN {
# Poisson CQO with equal tolerances
set.seed(111)  # This leads to the global solution
hspider[,1:6] <- scale(hspider[,1:6])  # Good idea when I.tolerances = TRUE
p1 <- cqo(cbind(Alopacce, Alopcune, Alopfabr, Arctlute, Arctperi, Auloalbi,
Pardlugu, Pardmont, Pardnigr, Pardpull, Trocterr, Zoraspin) ~
WaterCon + BareSand + FallTwig + CoveMoss + CoveHerb + ReflLux,
quasipoissonff, data = hspider, eq.tolerances = TRUE)
sort(deviance(p1, history = TRUE))  # A history of all the iterations

(isd.latvar <- apply(latvar(p1), 2, sd))  # Should be approx isd.latvar

# Refit the model with better initial values
set.seed(111)  # This leads to the global solution
p1 <- cqo(cbind(Alopacce, Alopcune, Alopfabr, Arctlute, Arctperi, Auloalbi,
Pardlugu, Pardmont, Pardnigr, Pardpull, Trocterr, Zoraspin) ~
WaterCon + BareSand + FallTwig + CoveMoss + CoveHerb + ReflLux,
I.tolerances = TRUE, quasipoissonff, data = hspider,
isd.latvar = isd.latvar)  # Note the use of isd.latvar here
sort(deviance(p1, history = TRUE))  # A history of all the iterations
# }


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