Similar to glm(), gam() and inla(), bru() models can be constructed via a formula-like syntax, where each latent effect is specified. However, in addition to the parts of the syntax compatible with INLA::inla, bru components offer additional functionality which facilitates modelling, and the predictor expression can be specified separately, allowing more complex and non-linear predictors to be defined. The formula syntax is just a way to allow all model components to be defined in a single line of code, but the definitions can optionally be split up into separate component definitions. See Details for more information.

The component methods all rely on the component.character() method, that defines a model component with a given label/name. The user usually doesn't need to call these methods directly, but can instead supply a formula expression that can be interpreted by the component_list.formula() method, called inside bru().

## Usage

component(...)

# S3 method for character
component(
object,
main = NULL,
weights = NULL,
...,
model = NULL,
mapper = NULL,
main_layer = NULL,
main_selector = NULL,
n = NULL,
values = NULL,
season.length = NULL,
copy = NULL,
weights_layer = NULL,
weights_selector = NULL,
group = NULL,
group_mapper = NULL,
group_layer = NULL,
group_selector = NULL,
ngroup = NULL,
control.group = NULL,
replicate = NULL,
replicate_mapper = NULL,
replicate_layer = NULL,
replicate_selector = NULL,
nrep = NULL,
A.msk = NULL,
.envir = parent.frame(),
envir_extra = NULL
)

## Arguments

...

Parameters passed on to other methods

object

A character label for the component

main

main takes an R expression that evaluates to where the latent variables should be evaluated (coordinates, indices, continuous scalar (for rw2 etc)). Arguments starting with weights, group, replicate behave similarly to main, but for the corresponding features of INLA::f().

weights, weights_layer, weights_selector

Optional specification of effect scaling weights. Same syntax as for main.

model

Either one of "const" (same as "offset"), "factor_full", "factor_contrast", "linear", "fixed", or a model name or object accepted by INLA's f function. If set to NULL, then "linear" is used for vector inputs, and "fixed" for matrix input (converted internally to an iid model with fixed precision)

mapper

Information about how to do the mapping from the values evaluated in main, and to the latent variables. Auto-detects spde model objects in model and extracts the mesh object to use as the mapper, and auto-generates mappers for indexed models. (Default: NULL, for auto-determination)

main_layer, main_selector

The _layer input should evaluate to a numeric index or character name or vector of which layer/variable to extract from a covariate data object given in main. (Default: NULL if _selector is given. Otherwise the effect component name, if it exists in the covariate object, and otherwise the first column of the covariate data frame)

The _selector value should be a character name of a variable whose contents determines which layer to extract from a covariate for each data point. (Default: NULL)

n

The number of latent variables in the model. Should be auto-detected for most or all models (Default: NULL, for auto-detection). An error is given if it can't figure it out by itself.

values

Specifies for what covariate/index values INLA should build the latent model. Normally generated internally based on the mapping details. (Default: NULL, for auto-determination)

season.length

Passed on to INLA::f() for model "seasonal" (TODO: check if this parameter is still fully handled)

copy

character; label of other component that this component should be a copy of. If the fixed = FALSE, a scaling constant is estimated, via a hyperparameter. If fixed = TRUE, the component scaling is fixed, by default to 1; for fixed scaling, it's more efficient to express the scaling in the predictor expression instead of making a copy component.

group, group_mapper, group_layer, group_selector, ngroup

Optional specification of kronecker/group model indexing.

control.group

list of kronecker/group model parameters, currently passed directly on to INLA::f

replicate, replicate_mapper, replicate_layer, replicate_selector, nrep

Optional specification of indices for an independent replication model. Same syntax as for main

A.msk

TODO: check/fix/deprecate this parameter. Likely doesn't work at the moment, and I've found no examples that use it.

.envir

Evaluation environment

envir_extra

TODO: check/fix this parameter.

## Details

As shorthand, bru() will understand basic additive formulae describing fixed effect models. For instance, the components specification y ~ x will define 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 latent variable associated with $$x$$ and

• $$\psi = \beta * x$$ is called the effect of $$x$$

A problem that arises when using this kind of R formula is that it does not clearly reflect 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 when x refers to the covariate or the effect of the covariate.

The component.character method is inlabru's equivalent to INLA's f function but adds functionality that is unique to inlabru.

Deprecated parameters:

• map: Use main instead.

• mesh: Use mapper instead.

## Naming random effects

In INLA, the f() notation is used to define more complex models, but a simple linear effect model can also 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 rewriting this formula in order to explicitly state the name of the random effect and the name of the associated covariate. This is achieved 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 discriminate 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 analytical combination of the covariate $$x$$ and the intercept using

• predict(fit, data.frame(x=2)), ~ exp(psi + Intercept).

which corresponds to the mathematical expression e β + c.

On the other hand, predict may be used to only look at a transformation of the latent variable $$\beta_\psi$$

• predict(fit, NULL, ~ exp(psi_latent)).

which corresponds to the mathematical expression e β.

Other component constructors: component_list()

## Author

Fabian E. Bachl bachlfab@gmail.com and Finn Lindgren Finn.Lindgren@gmail.com

## Examples

# As an example, let us create a linear component. Here, the component is
# called "myLinearEffectOfX" while the covariate the component acts on is
# called "x". Note that a list of components is returned because the
# formula may define multiple components

cmp <- component_list(~ myLinearEffectOfX(main = x, model = "linear"))
summary(cmp)
#> Label:	myLinearEffectOfX
#>   Type:	main = linear
#>   Input:	main = x
#>   Map: Not yet initialised
#>   INLA formula:
#>     ~ . + f(myLinearEffectOfX, model =
#>       BRU_myLinearEffectOfX_main_model)
#> Label:	Intercept
#>   Type:	main = linear
#>   Input:	main = 1
#>   Map: Not yet initialised
#>   INLA formula:
#>     ~ . + f(Intercept, model = BRU_Intercept_main_model)
# Equivalent shortcuts:
cmp <- component_list(~ myLinearEffectOfX(x, model = "linear"))
cmp <- component_list(~ myLinearEffectOfX(x))
# Individual component
cmp <- component("myLinearEffectOfX", main = x, model = "linear")
summary(cmp)
#> Label:	myLinearEffectOfX
#>   Type:	main = linear
#>   Input:	main = x
#>   Map: Not yet initialised
#>   INLA formula:
#>     ~ . + f(myLinearEffectOfX, model =
#>       BRU_myLinearEffectOfX_main_model)
# \donttest{
if (bru_safe_inla(quietly = TRUE)) {
# As an example, let us create a linear component. Here, the component is
# called "myEffectOfX" while the covariate the component acts on is called "x":

cmp <- component("myEffectOfX", main = x, model = "linear")
summary(cmp)

# A more complicated component:
cmp <- component("myEffectOfX",
main = x,
model = INLA::inla.spde2.matern(INLA::inla.mesh.1d(1:10))
)

# Compound fixed effect component, where x and z are in the input data.
# The formula will be passed on to MatrixModels::model.Matrix:
cmp <- component("eff", ~ -1 + x:z, model = "fixed")
summary(cmp)
}
#> Label:	eff
#>   Type:	main = fixed
#>   Input:	main = ~-1 + x:z
#>   Map: Not yet initialised
#>   INLA formula:
#>     ~ . + f(eff, model = BRU_eff_main_model, hyper =
#>       BRU_eff_main_fixed_hyper)
# }