Similar to glm(), gam() and inla() bru uses formula objects to describe response data and latent (unknonw) components of the model to be fitted. However, in addition to the syntax compatible with inla, bru components offer addtitional functionality which facilitates modeling.
bru.components()In inla, a simple random effect model would be expressed as
formula = y ~ f(x, model = "linear"),
where f is the inla specific function to set up random effects of all kinds. The underlying
predictor would again be \(\eta = \beta * x + c\) but the result of fitting the model would state
x as the random effect's name. bru allows to rewrite this formula in order to explicitly state
the name of the random effect and the name of the associated. This is achived by replacing f
with an arbitrary name that we wish to assign to the effect, e.g.
components = y ~ psi(x, model = "linear").
Being able to disciminate between \(x\) and \(\psi\) is relevant because of two functionalities
bru offers. The formula parameters of both, bru and the prediction method predict.bru
are interpreted in the mathematical sense. For instance, predict may be used to analyze the
an analytical combination of the covariate \(x\) and the intercept using
predict(fit, data.frame(x=1)), ~ exp(x + Intercept).
On the other hand, predict may be used to only look at a transformation of the random effect \(\psi\)
predict(fit, NULL, ~ exp(psi).
It is not unusual for a random effect act on a transformation of a covariate. In other frameworks this
would mean that the transformed covariate would have to be calculated in advance and added to the
data frame that is usually provided via the data parameter. inlabru provides the option to do
this transformation automatically. For instance, one might be interested in the effect of a covariate
\(x^2\). In inla and other frameworks this would require to add a column xsquared to the
input data frame and use the formula
formula = y ~ f(xsquared, model = "linear"),
In inlabru this can be achived using two ways of using the map parameter.
components = y ~ psi(map = x^2, model = "linear")
components = y ~ psi(map = mySquareFun(x), model = "linear"),
components = y ~ psi(map = myOtherSquareFun, model = "linear"),
In the first example inlabru will interpret the map parameter as an expression to be evaluated within
the data provided. Since \(x\) is a knonwn covariate it will know how to calculate it. The second
example is an expression as well but it uses a function alled mySquareFun. This function is
defined by user but has wo be accessible within the work space when setting up the compoonents.
The third example provides the function myOtherSquareFun directly and not within an expression.
In this case, inlabru will call the function using the data provided via the data parameter.
inlabru expects that the output of this function is a data.frame with "psi" being the name of the
single existing column. For instance,
myOtherSquareFun = function(data) {
data = data[,"x", drop = FALSE] ;
colnames(data) = "psi" ;
return(data)}
When fitting spatial models it is common to work with covariates that depend on space, e.g. sea
surface temperature or elevation. Although it is straight forward to add this data to the input
data frame or write a covariate function like in the previous section there is an even more
convenient way in inlabru. Spatial covariates are often stored as SpatialPixelDataFrame,
SpatialPixelDataFrame or RasterLayer objects. These can be provided directly via
the map parameter if the input data is a SpatialPointsDataFrame. inlabru will automatically
evaluate and/or interpolate the coariate at your data locations when using code like
components = y ~ psi(mySpatialPixels, model = "linear").
A common spatial modelling component when using inla are SPDE models. An important feature of
inlabru is that it will automatically calculate the so called A-matrix which maps SPDE
values at the mesh vertices to values at the data locations. For this purpose, the map parameter
can be se to coordinates, which is the sp package function that extracts point
coordinates from the SpatialPointsDataFrame that was provided as input to bru. The code for
this would look as follows:
components = y ~ mySPDE(map = coordinates, model = inla.spde2.matern(...)).
bru will understand formulae describing fixed effect models just like the other methods. For instance, the
formula y ~ x will fit the linear combination of an effect named x and an intercept to
the response y with respect to the likelihood family stated when calling bru. Mathematically,
the linear predictor \(\eta\) would be written down as
$$\eta = \beta * x + c,$$
where:
\(c\) is the intercept
\(x \)is a covariate
\(\beta\) is a random variable associated with \(x\) and
\(\psi = \beta * x \) is called the random effect of \(x\)
A problem that arises when using this kind of R formula is that it does not clearly relect the mathematical
formula. For instance, when providing the formula to inla, the resulting object will refer to the random
effect \(\psi = \beta * x \) as x. Hence, it is not clear if x refers to the covariate
or the effect of the covariate.