cqo: Fitting Constrained Quadratic Ordination (CQO)

Description

A constrained quadratic ordination (CQO; formerly called canonical Gaussian ordination or CGO) model is fitted using the quadratic reduced-rank vector generalized linear model (QRR-VGLM) framework.

Usage

cqo(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 = qrrvglm.control(...), offset = NULL,
    method = "cqo.fit", model = FALSE, x.arg = TRUE, y.arg = TRUE,
    contrasts = NULL, constraints = NULL, extra = NULL,
    smart = TRUE, ...)

Value

An object of class "qrrvglm".

Arguments

formula

a symbolic description of the model to be fit. The RHS of the formula is applied to each linear predictor. Different variables in each linear predictor can be chosen by specifying constraint matrices.

family

a function of class "vglmff" (see vglmff-class) describing what statistical model is to be fitted. This is called a ``VGAM family function''. See CommonVGAMffArguments for general information about many types of arguments found in this type of function. Currently the following families are supported: poissonff, binomialff (logitlink and clogloglink links available), negbinomial, gamma2. Sometimes special arguments are required for cqo(), e.g., binomialff(multiple.responses = TRUE).

data

an optional data frame containing the variables in the model. By default the variables are taken from environment(formula), typically the environment from which cqo is called.

weights

an optional vector or matrix of (prior) weights to be used in the fitting process. Currently, this argument should not be used.

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 NAs. The default is set by the na.action setting of options, and is na.fail if that is unset. The ``factory-fresh'' default is na.omit.

etastart

starting values for the linear predictors. It is a $M$-column matrix. If $M = 1$ then it may be a vector. Currently, this argument probably should not be used.

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. Currently, this argument probably should not be used.

coefstart

starting values for the coefficient vector. Currently, this argument probably should not be used.

control

a list of parameters for controlling the fitting process. See qrrvglm.control for details.

offset

This argument must not be used.

method

the method to be used in fitting the model. The default (and presently only) method cqo.fit uses iteratively reweighted least squares (IRLS).

model

a logical value indicating whether the model frame should be assigned in the model slot.

x.arg, y.arg

logical values indicating whether the model matrix and response matrix used in the fitting process should be assigned in the x and y slots. Note the model matrix is the LM model matrix.

contrasts

an optional list. See the contrasts.arg of model.matrix.default.

constraints

an optional list of constraint matrices. The components of the list must be named with the term it corresponds to (and it must match in character format). Each constraint matrix must have $M$ rows, and be of full-column rank. By default, constraint matrices are the $M$ by $M$ identity matrix unless arguments in the family function itself override these values. If constraints is used it must contain all the terms; an incomplete list is not accepted. Constraint matrices for $x_2$ variables are taken as the identity matrix.

extra

an optional list with any extra information that might be needed by the family function.

smart: logical value indicating whether smart prediction (smartpred) will be used.
...: further arguments passed into qrrvglm.control.

Author

Thomas W. Yee. Thanks to Alvin Sou for converting a lot of the original FORTRAN code into C.

Warning

Local solutions are not uncommon when fitting CQO models. To increase the chances of obtaining the global solution, increase the value of the argument Bestof in qrrvglm.control. For reproducibility of the results, it pays to set a different random number seed before calling cqo (the function set.seed does this). The function cqo chooses initial values for C using .Init.Poisson.QO() if Use.Init.Poisson.QO = TRUE, else random numbers.

Unless I.tolerances = TRUE or eq.tolerances = FALSE, CQO is computationally expensive with memory and time. It pays to keep the rank down to 1 or 2. If eq.tolerances = TRUE and I.tolerances = FALSE then the cost grows quickly with the number of species and sites (in terms of memory requirements and time). The data needs to conform quite closely to the statistical model, and the environmental range of the data should be wide in order for the quadratics to fit the data well (bell-shaped response surfaces). If not, RR-VGLMs will be more appropriate because the response is linear on the transformed scale (e.g., log or logit) and the ordination is called constrained linear ordination or CLO.

Like many regression models, CQO is sensitive to outliers (in the environmental and species data), sparse data, high leverage points, multicollinearity etc. For these reasons, it is necessary to examine the data carefully for these features and take corrective action (e.g., omitting certain species, sites, environmental variables from the analysis, transforming certain environmental variables, etc.). Any optimum lying outside the convex hull of the site scores should not be trusted. Fitting a CAO is recommended first, then upon transformations etc., possibly a CQO can be fitted.

For binary data, it is necessary to have `enough' data. In general, the number of sites $n$ ought to be much larger than the number of species S, e.g., at least 100 sites for two species. Compared to count (Poisson) data, numerical problems occur more frequently with presence/absence (binary) data. For example, if Rank = 1 and if the response data for each species is a string of all absences, then all presences, then all absences (when enumerated along the latent variable) then infinite parameter estimates will occur. In general, setting I.tolerances = TRUE may help.

This function was formerly called cgo. It has been renamed to reinforce a new nomenclature described in Yee (2006).

Details

QRR-VGLMs or constrained quadratic ordination (CQO) models are estimated here by maximum likelihood estimation. Optimal linear combinations of the environmental variables are computed, called latent variables (these appear as latvar for $R=1$ else latvar1, latvar2, etc. in the output). Here, $R$ is the rank or the number of ordination axes. Each species' response is then a regression of these latent variables using quadratic polynomials on a transformed scale (e.g., log for Poisson counts, logit for presence/absence responses). The solution is obtained iteratively in order to maximize the log-likelihood function, or equivalently, minimize the deviance.

The central formula (for Poisson and binomial species data) is given by $$\eta = B_1^T x_1 + A \nu + \sum_{m=1}^M (\nu^T D_m \nu) e_m$$ 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. The $\eta$ are a vector of linear/additive predictors, e.g., the $m$th element is $\eta_m = \log(E[Y_m])$ for the $m$th species. The matrices $B_1$, $A$, $C$ and $D_m$ are estimated from the data, i.e., contain the regression coefficients. The tolerance matrices satisfy $T_s = -\frac12 D_s^{-1}$. Many important CQO details are directly related to arguments in qrrvglm.control, e.g., the argument noRRR specifies which variables comprise $x_1$.

Theoretically, the four most popular VGAM family functions to be used with cqo correspond to the Poisson, binomial, normal, and negative binomial distributions. The latter is a 2-parameter model. All of these are implemented, as well as the 2-parameter gamma.

For initial values, the function .Init.Poisson.QO should work reasonably well if the data is Poisson with species having equal tolerances. It can be quite good on binary data too. Otherwise the Cinit argument in qrrvglm.control can be used.

It is possible to relax the quadratic form to an additive model. The result is a data-driven approach rather than a model-driven approach, so that CQO is extended to constrained additive ordination (CAO) when $R=1$. See cao for more details.

In this documentation, $M$ is the number of linear predictors, $S$ is the number of responses (species). Then $M=S$ for Poisson and binomial species data, and $M=2S$ for negative binomial and gamma distributed species data.

Incidentally, Unconstrained quadratic ordination (UQO) may be performed by, e.g., fitting a Goodman's RC association model; see uqo and the Yee and Hadi (2014) referenced there. For UQO, the response is the usual site-by-species matrix and there are no environmental variables; the site scores are free parameters. UQO can be performed under the assumption that all species have the same tolerance matrices.

References

Yee, T. W. (2004). A new technique for maximum-likelihood canonical Gaussian ordination. Ecological Monographs, 74, 685--701.

ter Braak, C. J. F. and Prentice, I. C. (1988). A theory of gradient analysis. Advances in Ecological Research, 18, 271--317.

Yee, T. W. (2006). Constrained additive ordination. Ecology, 87, 203--213.

Examples

Run this code

if (FALSE) {
# Example 1; Fit an unequal tolerances model to the hunting spiders data
hspider[,1:6] <- scale(hspider[,1:6])  # Standardized environmental variables
set.seed(1234)  # For reproducibility of the results
p1ut <- cqo(cbind(Alopacce, Alopcune, Alopfabr, Arctlute, Arctperi,
                  Auloalbi, Pardlugu, Pardmont, Pardnigr, Pardpull,
                  Trocterr, Zoraspin) ~
            WaterCon + BareSand + FallTwig + CoveMoss + CoveHerb + ReflLux,
            fam = poissonff, data = hspider, Crow1positive = FALSE,
            eq.tol = FALSE)
sort(deviance(p1ut, history = TRUE))  # A history of all the iterations
if (deviance(p1ut) > 1177) warning("suboptimal fit obtained")

S <- ncol(depvar(p1ut))  # Number of species
clr <- (1:(S+1))[-7]  # Omits yellow
lvplot(p1ut, y = TRUE, lcol = clr, pch = 1:S, pcol = clr,
       las = 1)  # Ordination diagram
legend("topright", leg = colnames(depvar(p1ut)), col = clr,
       pch = 1:S, merge = TRUE, bty = "n", lty = 1:S, lwd = 2)
(cp <- Coef(p1ut))

(a <- latvar(cp)[cp@latvar.order])  # Ordered site scores along the gradient
# Names of the ordered sites along the gradient:
rownames(latvar(cp))[cp@latvar.order]
(aa <- Opt(cp)[, cp@Optimum.order])  # Ordered optimums along the gradient
aa <- aa[!is.na(aa)]  # Delete the species that is not unimodal
names(aa)  # Names of the ordered optimums along the gradient

trplot(p1ut, which.species = 1:3, log = "xy", type = "b", lty = 1, lwd = 2,
       col = c("blue","red","green"), label = TRUE) -> ii  # Trajectory plot
legend(0.00005, 0.3, paste(ii$species[, 1], ii$species[, 2], sep = " and "),
       lwd = 2, lty = 1, col = c("blue", "red", "green"))
abline(a = 0, b = 1, lty = "dashed")

S <- ncol(depvar(p1ut))  # Number of species
clr <- (1:(S+1))[-7]  # Omits yellow
persp(p1ut, col = clr, label = TRUE, las = 1)  # Perspective plot


# Example 2; Fit an equal tolerances model. Less numerically fraught.
set.seed(1234)
p1et <- 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)
sort(deviance(p1et, history = TRUE))  # A history of all the iterations
if (deviance(p1et) > 1586) warning("suboptimal fit obtained")
S <- ncol(depvar(p1et))  # Number of species
clr <- (1:(S+1))[-7]  # Omits yellow
persp(p1et, col = clr, label = TRUE, las = 1)


# Example 3: A rank-2 equal tolerances CQO model with Poisson data
# This example is numerically fraught... need I.toler = TRUE too.
set.seed(555)
p2 <- 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,
          I.toler = TRUE, Rank = 2, Bestof = 3, isd.latvar = c(2.1, 0.9))
sort(deviance(p2, history = TRUE))  # A history of all the iterations
if (deviance(p2) > 1127) warning("suboptimal fit obtained")
lvplot(p2, ellips = FALSE, label = TRUE, xlim = c(-3,4),
       C = TRUE, Ccol = "brown", sites = TRUE, scol = "grey",
       pcol = "blue", pch = "+", chull = TRUE, ccol = "grey")


# Example 4: species packing model with presence/absence data
set.seed(2345)
n <- 200; p <- 5; S <- 5
mydata <- rcqo(n, p, S, fam = "binomial", hi.abundance = 4,
               eq.tol = TRUE, es.opt = TRUE, eq.max = TRUE)
myform <- attr(mydata, "formula")
set.seed(1234)
b1et <- cqo(myform, binomialff(multiple.responses = TRUE, link = "clogloglink"),
            data = mydata)
sort(deviance(b1et, history = TRUE))  # A history of all the iterations
lvplot(b1et, y = TRUE, lcol = 1:S, pch = 1:S, pcol = 1:S, las = 1)
Coef(b1et)

# Compare the fitted model with the 'truth'
cbind(truth = attr(mydata, "concoefficients"), fitted = concoef(b1et))


# Example 5: Plot the deviance residuals for diagnostic purposes
set.seed(1234)
p1et <- cqo(cbind(Alopacce, Alopcune, Alopfabr, Arctlute, Arctperi,
                  Auloalbi, Pardlugu, Pardmont, Pardnigr, Pardpull,
                  Trocterr, Zoraspin) ~
            WaterCon + BareSand + FallTwig + CoveMoss + CoveHerb + ReflLux,
            poissonff, data = hspider, eq.tol = TRUE, trace = FALSE)
sort(deviance(p1et, history = TRUE))  # A history of all the iterations
if (deviance(p1et) > 1586) warning("suboptimal fit obtained")
S <- ncol(depvar(p1et))
par(mfrow = c(3, 4))
for (ii in 1:S) {
  tempdata <- data.frame(latvar1 = c(latvar(p1et)),
                         sppCounts = depvar(p1et)[, ii])
  tempdata <- transform(tempdata, myOffset = -0.5 * latvar1^2)

# For species ii, refit the model to get the deviance residuals
  fit1 <- vglm(sppCounts ~ offset(myOffset) + latvar1, poissonff,
               data = tempdata, trace = FALSE)

# For checking: this should be 0
# print("max(abs(c(Coef(p1et)@B1[1,ii],Coef(p1et)@A[ii,1])-coef(fit1)))")
# print( max(abs(c(Coef(p1et)@B1[1,ii],Coef(p1et)@A[ii,1])-coef(fit1))) )

# Plot the deviance residuals
  devresid <- resid(fit1, type = "deviance")
  predvalues <- predict(fit1) + fit1@offset
  ooo <- with(tempdata, order(latvar1))
  plot(predvalues + devresid ~ latvar1, data = tempdata, col = "red",
       xlab = "latvar1", ylab = "", main = colnames(depvar(p1et))[ii])
  with(tempdata, lines(latvar1[ooo], predvalues[ooo], col = "blue"))
}
}

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