This vignette demonstrates some of the covariance structures available in the `glmmTMB`

package. Currently the available covariance structures are:

Covariance | Notation | Parameter count | Requirement |
---|---|---|---|

Heterogeneous unstructured | `us` |
\(n(n+1)/2\) | |

Heterogeneous Toeplitz | `toep` |
\(2n-1\) | |

Heterogeneous compound symmetry | `cs` |
\(n+1\) | |

Heterogeneous diagonal | `diag` |
\(n\) | |

AR(1) | `ar1` |
\(2\) | |

Ornstein–Uhlenbeck | `ou` |
\(2\) | Coordinates |

Spatial exponential | `exp` |
\(2\) | Coordinates |

Spatial Gaussian | `gau` |
\(2\) | Coordinates |

Spatial Matern | `mat` |
\(3\) | Coordinates |

The word ‘heterogeneous’ refers to the marginal variances of the model. Beyond correlation parameters, a heterogeneous structure uses \(n\) additional variance parameters where \(n\) is the dimension.

Some of the structures require temporal or spatial coordinates. We will show examples of this in a later section.

First, let’s consider a simple time series model. Assume that our measurements \(Y(t)\) are given at discrete times \(t \in \{1,...,n\}\) by

\[Y(t) = \mu + X(t) + \varepsilon(t)\]

where

- \(\mu\) is the mean value parameter.
- \(X(t)\) is a stationary AR(1) process, i.e. has covariance \(cov(X(s), X(t)) = \sigma^2\exp(-\theta |t-s|)\).
- \(\varepsilon(t)\) is iid. \(N(0,\sigma_0^2)\) measurement error.

A simulation experiment is set up using the parameters

Description | Parameter | Value |
---|---|---|

Mean | \(\mu\) | 0 |

Process variance | \(\sigma^2\) | 1 |

Measurement variance | \(\sigma_0^2\) | 1 |

One-step correlation | \(e^{-\theta}\) | 0.7 |

The following R-code draws a simulation based on these parameter values. For illustration purposes we consider a very short time series.

```
n <- 6 ## Number of time points
x <- mvrnorm(mu = rep(0,n),
Sigma = .7 ^ as.matrix(dist(1:n)) ) ## Simulate the process using the MASS package
y <- x + rnorm(n) ## Add measurement noise
```

In order to fit the model with `glmmTMB`

we must first specify a time variable as a *factor*. The factor *levels* correspond to unit spaced time points.

```
times <- factor(1:n)
levels(times)
```

We also need a grouping variable. In the current case there is only one time-series so the grouping is:

`group <- factor(rep(1,n))`

We combine the data into a single data frame (not absolutely required, but good practice):

`dat0 <- data.frame(y,times,group)`

Now fit the model using

`glmmTMB(y ~ ar1(times + 0 | group), data=dat0)`

This formula notation follows that of the `lme4`

package.

- The left hand side of the bar
`times + 0`

corresponds to a design matrix \(Z\) linking observation vector \(y\) (rows) with a random effects vector \(u\) (columns). - The distribution of \(u\) is
`ar1`

(this is the only`glmmTMB`

specific part of the formula). - The right hand side of the bar splits the above specification independently among groups. Each group has its own separate \(u\) vector but shares the same parameters for the covariance structure.

After running the model, we find the parameter estimates \(\mu\) (intercept), \(\sigma_0^2\) (dispersion), \(\sigma\) (Std. Dev.) and \(e^{-\theta}\) (First off-diagonal of “Corr”) in the output:

FIXME: Try a longer time series when the print.VarCorr is fixed.

A single time series of 6 time points is not sufficient to identify the parameters. We could either increase the length of the time series or increase the number of groups. We’ll try the latter:

```
simGroup <- function(g, n=6, rho=0.7) {
x <- mvrnorm(mu = rep(0,n),
Sigma = rho ^ as.matrix(dist(1:n)) ) ## Simulate the process
y <- x + rnorm(n) ## Add measurement noise
times <- factor(1:n)
group <- factor(rep(g,n))
data.frame(y, times, group)
}
simGroup(1)
```

Generate a dataset with 1000 groups:

`dat1 <- do.call("rbind", lapply(1:1000, simGroup) )`

And fitting the model on this larger dataset gives estimates close to the true values (AR standard deviation=1, residual (measurement) standard deviation=1, autocorrelation=0.7):

`(fit.ar1 <- glmmTMB(y ~ ar1(times + 0 | group), data=dat1))`

We can try to fit an unstructured covariance to the previous dataset `dat`

. For this case an unstructured covariance has 15 correlation parameters and 6 variance parameters. Adding \(\sigma_0^2 I\) on top would cause a strict overparameterization, as these would be redundant with the diagonal elements in the covariance matrix. Hence, when fitting the model with `glmmTMB`

, we have to disable the \(\varepsilon\) term (the dispersion) by setting `dispformula=~0`

:

```
fit.us <- glmmTMB(y ~ us(times + 0 | group), data=dat1, dispformula=~0)
fit.us$sdr$pdHess ## Converged ?
```

The estimated variance and correlation parameters are:

`VarCorr(fit.us)`

{#1_{{#2}}} The estimated correlation is approximately constant along diagonals (apparent Toeplitz structure) and we note that the first off-diagonal is now ca. half the true value (0.7) because the dispersion is effectively included in the estimated covariance matrix (i.e. \(\rho' = \rho {2}{\sigma^2}{AR}/({2}{\sigma^2}{AR} + {2}{sigma^2}{meas})\)).

The next natural step would be to reduce the number of parameters by collecting correlation parameters within the same off-diagonal. This amounts to 5 correlation parameters and 6 variance parameters.

FIXME: Explain why dispformula=~1 causes over-parameterization

```
fit.toep <- glmmTMB(y ~ toep(times + 0 | group), data=dat1, dispformula=~0)
fit.toep$sdr$pdHess ## Converged ?
```

The estimated variance and correlation parameters are:

`(vc.toep <- VarCorr(fit.toep))`

The diagonal elements are all approximately equal to the true total variance (\({2}{\sigma^2}{AR} + {2}{sigma^2}{meas}\)=2), and the off-diagonal elements are approximately equal to the expected value of 0.7/2=0.35.

```
vc1 <- vc.toep$cond[[1]] ## first term of var-cov for RE of conditional model
summary(diag(vc1))
summary(vc1[row(vc1)!=col(vc1)])
```

We can get a *slightly* better estimate of the variance by using REML estimation (however, the estimate of the correlations seems to have gotten slightly worse):

```
fit.toep.reml <- update(fit.toep, REML=TRUE)
vc1R <- VarCorr(fit.toep.reml)$cond[[1]]
summary(diag(vc1R))
summary(vc1R[row(vc1R)!=col(vc1R)])
```

The compound symmetry structure collects all off-diagonal elements of the correlation matrix to one common value.

FIXME: Explain why dispformula=~1 causes over-parameterization

```
fit.cs <- glmmTMB(y ~ cs(times + 0 | group), data=dat1, dispformula=~0)
fit.cs$sdr$pdHess ## Converged ?
```

The estimated variance and correlation parameters are:

`VarCorr(fit.cs)`

The models `ar1`

, `toep`

, and `us`

are nested so we can use:

`anova(fit.ar1, fit.toep, fit.us)`

`ar1`

has the lowest AIC (it’s the simplest model, and fits the data adequately); we can’t reject the (true in this case!) null model that an AR1 structure is adequate to describe the data.

The model `cs`

is a sub-model of `toep`

:

`anova(fit.cs, fit.toep)`

Here we *can* reject the null hypothesis of compound symmetry (i.e., that all the pairwise correlations are the same).

Coordinate information can be added to a variable using the `glmmTMB`

function `numFactor`

. This is necessary in order to use those covariance structures that require coordinates. For example, if we have the numeric coordinates

```
x <- sample(1:2, 10, replace=TRUE)
y <- sample(1:2, 10, replace=TRUE)
```

we can generate a factor representing \((x,y)\) coordinates by

`(pos <- numFactor(x,y))`

Numeric coordinates can be recovered from the factor levels:

`parseNumLevels(levels(pos))`

In order to try the remaining structures on our test data we re-interpret the time factor using `numFactor`

:

```
dat1$times <- numFactor(dat1$times)
levels(dat1$times)
```

Having the numeric times encoded in the factor levels we can now try the Ornstein–Uhlenbeck covariance structure.

```
fit.ou <- glmmTMB(y ~ ou(times + 0 | group), data=dat1)
fit.ou$sdr$pdHess ## Converged ?
```

It should give the exact same results as `ar1`

in this case since the times are equidistant:

`VarCorr(fit.ou)`

However, note the differences between `ou`

and `ar1`

:

`ou`

can handle irregular time points.`ou`

only allows positive correlation between neighboring time points.

The structures `exp`

, `gau`

and `mat`

are meant to used for spatial data. They all require a Euclidean distance matrix which is calculated internally based on the coordinates. Here, we will try these models on the simulated time series data.

An example with spatial data is presented in a later section.

```
fit.mat <- glmmTMB(y ~ mat(times + 0 | group), data=dat1, dispformula=~0)
fit.mat$sdr$pdHess ## Converged ?
```

`VarCorr(fit.mat)`

“Gaussian” refers here to a Gaussian decay in correlation with distance, i.e. \(\rho = \exp(-d x^2)\), not to the conditional distribution (“family”).

```
fit.gau <- glmmTMB(y ~ gau(times + 0 | group), data=dat1, dispformula=~0)
fit.gau$sdr$pdHess ## Converged ?
```

`VarCorr(fit.gau)`

```
fit.exp <- glmmTMB(y ~ exp(times + 0 | group), data=dat1)
fit.exp$sdr$pdHess ## Converged ?
```

`VarCorr(fit.exp)`

Starting out with the built in `volcano`

dataset we reshape it to a `data.frame`

with pixel intensity `z`

and pixel position `x`

and `y`

:

```
d <- data.frame(z = as.vector(volcano),
x = as.vector(row(volcano)),
y = as.vector(col(volcano)))
```

Next, add random normal noise to the pixel intensities and extract a small subset of 100 pixels. This is our spatial dataset:

```
set.seed(1)
d$z <- d$z + rnorm(length(volcano), sd=15)
d <- d[sample(nrow(d), 100), ]
```

Display sampled noisy volcano data:

```
volcano.data <- array(NA, dim(volcano))
volcano.data[cbind(d$x, d$y)] <- d$z
image(volcano.data, main="Spatial data", useRaster=TRUE)
```

Based on this data, we’ll attempt to re-construct the original image.

As model, it is assumed that the original image `image(volcano)`

is a realization of a random field with correlation decaying exponentially with distance between pixels.

Denoting by \(u(x,y)\) this random field the model for the observations is

\[ z_{i} = \mu + u(x_i,y_i) + \varepsilon_i \]

To fit the model, a `numFactor`

and a dummy grouping variable must be added to the dataset:

```
d$pos <- numFactor(d$x, d$y)
d$group <- factor(rep(1, nrow(d)))
```

The model is fit by

`f <- glmmTMB(z ~ 1 + exp(pos + 0 | group), data=d)`

Recall that a standard deviation `sd=15`

was used to distort the image. A confidence interval for this parameter is

`confint(f, "sigma")`

The glmmTMB `predict`

method can predict unseen levels of the random effects. For instance to predict a 3-by-3 corner of the image one could construct the new data:

```
newdata <- data.frame( pos=numFactor(expand.grid(x=1:3,y=1:3)) )
newdata$group <- factor(rep(1, nrow(newdata)))
newdata
```

and predict using

`predict(f, newdata, type="response", allow.new.levels=TRUE)`

A specific image column can thus be predicted using the function

```
predict_col <- function(i) {
newdata <- data.frame( pos = numFactor(expand.grid(1:87,i)))
newdata$group <- factor(rep(1,nrow(newdata)))
predict(f, newdata=newdata, type="response", allow.new.levels=TRUE)
}
```

Prediction of the entire image is carried out by (this takes a while…):

`pred <- sapply(1:61, predict_col)`

Finally plot the re-constructed image by

`image(pred, main="Reconstruction")`

For various advanced purposes, such as computing likelihood profiles, it is useful to know the details of the parameterization of the models - the scale on which the parameters are defined (e.g. standard deviation, variance, or log-standard deviation for variance parameters) and their order.

For an unstructured matrix of size `n`

, parameters `1:n`

represent the log-standard deviations while the remaining `n(n-1)/2`

(i.e. `(n+1):(n:(n*(n+1)/2))`

) are the elements of the *scaled* Cholesky factor of the correlation matrix, filled in row-wise order (see TMB documentation). In particular, if \(L\) is the lower-triangular matrix with 1 on the diagonal and the correlation parameters in the lower triangle, then the correlation matrix is defined as \(\Sigma = D^{-1/2} L L^\top D^{-1/2}\), where \(D = \textrm{diag}(L L^\top)\). For a single correlation parameter \(\theta_0\), this works out to \(\rho = \theta_0/(1+\theta_0^2)\).

```
vv0 <- VarCorr(fit.us)
vv1 <- vv0$cond$group ## extract 'naked' V-C matrix
n <- nrow(vv1)
rpars <- getME(fit.us,"theta") ## extract V-C parameters
## first n parameters are log-std devs:
all.equal(unname(diag(vv1)),exp(rpars[1:n])^2)
## now try correlation parameters:
cpars <- rpars[-(1:n)]
length(cpars)==n*(n-1)/2 ## the expected number
cc <- diag(n)
cc[upper.tri(cc)] <- cpars
L <- crossprod(cc)
D <- diag(1/sqrt(diag(L)))
round(D %*% L %*% D,3)
round(unname(attr(vv1,"correlation")),3)
```

`all.equal(c(cov2cor(vv1)),c(fit.us$obj$env$report(fit.us$fit$parfull)$corr[[1]]))`

Profiling (experimental/exploratory):

```
## want $par, not $parfull: do NOT include conditional modes/'b' parameters
ppar <- fit.us$fit$par
length(ppar)
range(which(names(ppar)=="theta")) ## the last n*(n+1)/2 parameters
## only 1 fixed effect parameter
tt <- tmbprofile(fit.us$obj,2,trace=FALSE)
```

```
confint(tt)
plot(tt)
```

```
ppar <- fit.cs$fit$par
length(ppar)
range(which(names(ppar)=="theta")) ## the last n*(n+1)/2 parameters
## only 1 fixed effect parameter, 1 dispersion parameter
tt2 <- tmbprofile(fit.cs$obj,3,trace=FALSE)
```

`plot(tt2)`