(Vignette under construction!)

Each inlabru latent model component generates an *effect*,
given the latent *state* vector. The purpose of the mapper system
is to define the link from state vectors to effect vectors. For an
ordinary model component, named \(c\),
this link can be represented as a matrix-vector product \[
\eta_c(u_c) = A_c(\text{input}) u_c,
\] where \(u_c\) is the latent
state vector, \(\eta_c(u_c)\) is the
resulting component effect vector, and \(A_c(\text{input})\) is a *component
model matrix*. This matrix depends on the component inputs
(covariates, index information, etc) from `main`

,
`group`

, `replicate`

, and `weights`

in
the component definition.

The wider scope of component definitions is discussed in the component vignette. Here, we will focus on the mapper system itself, how the basic building blocks are used to construct the built-in mappers, and finally how to construct new mappers.

The in-package documentation for the mapper methods is contained in four parts:

```
?bru_mapper # Mapper constructors
?bru_mapper_generics # Generic and default methods
?bru_mapper_methods # Specialised mapper methods
?bru_get_mapper # Mapper extraction methods
```

Regular users normally at most need some of the methods from
`?bru_mapper`

and sometimes
`?bru_mapper_generics`

. The methods in
`?bru_mapper_methods`

provides more details and allows the
user to query mappers and to use them outside the context of inlabru
model definitions. The `bru_mapper_define()`

and
`bru_get_mapper()`

methods are needed for those implementing
their own mapper class.

The main purpose of each mapper class is to allow evaluating a
component effect, from given `input`

and latent
`state`

, by calling

for the `mapper`

associated with the component
definition.

Basic mappers take covariate vectors or matrices as input and numeric vectors as state.

Constructors:

`bru_mapper_const()`

`bru_mapper_linear()`

`bru_mapper_index(n)`

`bru_mapper_factor(values, factor_mapping)`

`bru_mapper_matrix()`

`bru_mapper_harmonics(order, scaling, intercept, interval)`

The `const`

mapper defines a mapping from an empty state
vector. This is used to define offset components, \(\eta(u)_j = z_j\), where \(z\) is a fixed input vector, indexed by
\(j\).

The `linear`

mapper defines a mapper from a length 1 state
vector. This is used to define ordinary “fixed” covariate effects,
linear in both the covariate and in the state variable; \(\eta(u)_j = z_j u\), where \(u\) is the state, and \(z\) is the input covariate vector.

The `index(n)`

mapper defines a direct mapping from a
length `n`

state variable. This is used for structured and
unstructured “random effect” component effects; \(\eta(u)_j = u_{z_j}\), where \(u\) is the state vector, and \(z\) is the input index vector.

The `factor(values, factor_mapping)`

mapper defines a
direct mapping between a state vector of length equal to the number of
factor levels of `values`

(for
`factor_mapping = "full"`

), or one less (for
`factor_mapping = "contrast"`

). Each factor level is
represented as a dummy 0/1 variable, or equivalently, the input is used
as a label index into the state vector; \(\eta(u)_j = u_{z_j}\). In the
`"contrast"`

case, the first factor level is removed from the
model, and the corresponding effect defined to be zero.

The `matrix`

mapper defines a direct matrix multiplication
between a pre-computed model matrix \(A_\text{input}\) with the state vector;
\(\eta(u) = A_\text{input} u\). This is
used e.g. for linear model formula input, that is converted to a
component model matrix before handing it over to the mapper system.

The `harmonics(order, scaling, intercept, interval)`

mapper defines a sum of weighted harmonics on a real interval \((L, U)\), each frequency additionally
scaled by factors determined by `scale`

; \(\eta(u_j) = \sum_{k=1}^{1+2p} A_{j,k}
u_k\), where \(A_{j,k}=s_k\) for
\(k=1\), \(A_{j,2k}=s_{k+1}\cos[2\pi k (z_j -
L)/(U-L)]\) for \(k=1,2,\dots,p\), and \(A_{j,2k+1}=s_{k+1}\sin[2\pi k (z_j -
L)/(U-L)]\) for \(k=1,2,\dots,p\), where \(p=\)`order`

. When
`intercept`

is `TRUE`

, the state vector has length
\(1+2p\). When `intercept`

is `FALSE`

, the first, constant, column is removed, and the
state vector has length \(2p\).

Transformation mappers are mappers that would normally be combined with other mappers, as steps in a sequence of transformations, or as individual transformation mappers.

`bru_mapper_scale()`

`bru_mapper_marginal(qfun, pfun, ...)`

`bru_mapper_aggregate(rescale)`

`bru_mapper_logsumexp(rescale)`

The `scale`

mapper multiplies the `input`

vector with the `state`

vector. This is used for INLA models
to implement the `weights`

component scaling, as the final
stage of a `pipe`

mapper, see below.

The `marginal`

mapper transforms the `state`

vector from standardised Gaussian marginal distribution to the
distribution defined by a quantile function. This can be used to define
components with latent marginal `N(0,1)`

distribution but
non-Gaussian effect.

```
mapper <- bru_mapper_marginal(qfun = ..., pfun = ..., dfun = ..., ..., inverse = ...)
ibm_eval(mapper,
input = NULL, # If a list is given, it overrides the parameter specification
state = ...
)
# Examples:
mapper <- bru_mapper_marginal(qfun = qexp, rate = 1 / 8)
mapper <- bru_mapper_marginal(qfun = qexp, pfun = pexp, dfun = dexp, rate = 1 / 8)
```

The `aggregate`

mapper aggregates the `state`

vector elements by blockwise summation after scaling the elements by
weights. This can be used for summation, integration, and averaging (for
`rescale=TRUE`

).

The `logsumexp`

mapper aggregates the
`exp(state)`

vector elements by blockwise summation after
scaling the elements by weights, and takes the logarithm. The
implementation takes care to avoid numerical overflow. This can be used
for summation, integration, and averaging (for
`rescale=TRUE`

).

Compound mappers define collections or chains of mappings, and can take various forms of input. The state vector is normally a numeric vector, but can in some cases be a list of vectors.

`bru_mapper_collect(mappers, hidden)`

`bru_mapper_multi(mappers)`

`bru_mapper_pipe(mappers)`

The `collect`

mapper defines a compound mapper where the
Jacobian is block-diagonal, and each block is defined by a separate
sub-mapper. When `hidden = TRUE`

, it can be used for INLA
models where the user-visible part of the latent state is only a part of
the internal state, such as for `"bym"`

models.

```
mapper <- bru_mapper_collect(list(name1 = ..., name2 = ..., ...),
hidden = FALSE
)
ibm_eval(mapper,
input = list(name1 = ..., name2 = ..., ...),
state = ...
)
# If hidden = TRUE, the inla_f=TRUE argument "hides" all but the first mapper:
ibm_eval(mapper,
input = name1_input,
state = ...,
inla_f = TRUE
)
```

The `multi`

mapper defines a compound mapper where the
Jacobian is the row-wise Kronecker product between the sub-mapper
Jacobians. This can be used for INLA models to construct the internal
mapper for `main`

, `group`

, and
`replicate`

, but also for user-defined models where a single
`rgeneric`

or `cgeneric`

model is defined on e.g.
a space-time product domain.

Pipe mappers chain multiple mappers into a sequence, where the evaluation result of each mapper is given as the state vector for the next mapper.

All model component mappers are currently defined as a
`pipe`

mapper containing a `multi`

mapper followed
by a `scale`

mapper:

For object with predefined mapper conversion methods, use
`bru_mapper(object)`

to obtain a suitable mapper object.
Current predefined object mappers are defined for objects of these
classes:

`fm_mesh_2d`

and`inla.mesh`

; Use`bru_mapper(object)`

`fm_mesh_1d`

and`inla.mesh.1d`

; Use`bru_mapper(object, indexed)`

More details are given below.

For inla model objects with predefined mappers, use
`bru_get_mapper(object)`

Current predefined model object
mappers are defined for models of these classes:

`inla.spde`

; this includes`inla.spde2.matern()`

and`inla.spde2.pcmatern()`

models. The conversion method calls`bru_mapper()`

in the appropriate way for the mesh type used for the model`inla.rgeneric`

; This relies on the rgeneric definition including a`"mapper"`

callback argument. A less invasive method, that works for both rgeneric and cgeneric models, is to define a`bru_get_mapper()`

method for the class, which needs to include a unique class identifier, e.g. add`class(object) <- c("my_unique_model_class", class(object))`

and define

and register it in the namespace.

`bru_mapper_taylor()`

```
ibm_n(mapper, inla_f, ...)
ibm_n_output(mapper, input, ...)
ibm_values(mapper, inla_f, ...)
ibm_jacobian(mapper, input, state, ...)
ibm_eval(mapper, input, state, ...)
ibm_names(mapper, ...)
ibm_inla_subset(mapper, ...)
```

An example where the `inla_f`

argument matters is the
`bru_mapper_collect`

class, when the `hidden=TRUE`

argument is used to indicate that only the first mapper should be used
for the `INLA::f()`

inputs, e.g. for `"bym2"`

models. For `inla_f=FALSE`

(the default), the
`ibm_n`

and `ibm_values`

methods return the total
number of latent variables, but with `inla_f=TRUE`

we get the
values needed by `INLA::f()`

instead:

```
mapper <- bru_mapper_collect(
list(
a = bru_mapper_index(3),
b = bru_mapper_index(2)
),
hidden = TRUE
)
ibm_n(mapper)
#> [1] 5
```

For the `bru_mapper_multi`

class, the `multi`

argument provides access to the inner layers of the multi-mapper:

```
mapper <- bru_mapper_multi(list(
a = bru_mapper_index(3),
b = bru_mapper_index(2)
))
ibm_n(mapper)
#> [1] 6
```

```
ibm_values(mapper, multi = TRUE, inla_f = TRUE)
#> a b
#> 1 1 1
#> 2 2 1
#> 3 3 1
#> 4 1 2
#> 5 2 2
#> 6 3 2
```

The default `ibm_inla_subset`

method determines the inla
subset by comparing the full values information from
`ibm_values(mapper, inla_f = FALSE)`

to the inla specific
values information from `ibm_values(mapper, inla_f = TRUE)`

,
to determine the logical vector identifying the inla values subset.
Custom mappers should normally not need to specialise this method.

For component models referenced by a character label
(e.g. `"iid"`

, `"bym2"`

, `"rw2"`

etc),
inlabru will construct default mappers, that in most cases replicate the
default `INLA::f()`

behaviour.

For the `fm_mesh_1d`

, `fm_mesh_2d`

,
`inla.mesh`

and `inla.mesh.1d`

classes, default
mappers can be constructed by the pre-defined `bru_mapper`

S3
methods. For 2d meshes (`fm_mesh_2d`

,
`inla.mesh`

),

For 1d meshes (`fm_mesh_1d`

,
`inla.mesh.1d`

),

A mapper object should store enough information in order for the
`ibm_*`

methods to work. The simplest case of a customised
mapper is to just attached a new class label to the front of the S3
`class()`

information of an existing mapper, to obtain a
class to override some of the standard `ibm_*`

method
implementations. More commonly, one would instead start with a basic
`list()`

, that might *contain* an existing mapper
object, and then add methods that know how to use that information. In
all these cases, the `bru_mapper_define()`

method should be
used to properly set the class information:

Implementations can avoid having to define `ibm_n`

and
`ibm_values`

methods, by instead computing and storing
`n`

, `n_inla`

, `values`

, and
`values_inla`

in the mapper object during construction:

The default `ibm_n`

and `ibm_values`

methods
checks if these values are available, and return the appropriate values
depending on the `inla_f`

argument. When `*_inla`

values are requested but not available, the methods fall back to the
non-inla versions. If the needed information is not found, the default
methods give an error message.

Note: Before version 2.6.0, `ibm_amatrix`

was used instead
of `ibm_jacobian`

. Version 2.6.0 still supports this, by
having the default `ibm_jacobian`

method call
`ibm_amatrix`

, but will give a deprecation message later, and
it may be removed in a future version.

Let’s build a mapper class `bru_mapper_p_quadratic`

for a
model component that takes input covariates with values \(a_{ij}\), \(i=1,\dots,m\) and \(j=1,\dots,p\), in a `matrix`

or
`data.frame`

and evaluates a full quadratic expression \[
\eta_i = x_0 + \sum_{j=1}^p a_{ij} x_j +
\frac{1}{2}\sum_{j=1}^p\sum_{k=1}^j \gamma_{j,k} a_{ij}a_{ik} x_{j,k},
\] where the latent component vector is \(\boldsymbol{x}=[x_0,x_1,\dots,x_p,x_{1,1},\dots,x_{p,1},x_{2,2}\dots,x_{p,p}]\),
and \[
\gamma_{j,k} = \begin{cases}
1, & j = k,\\
2, & j > k.
\end{cases}
\]

We start with the constructor method. Like the
`bru_mapper_matrix`

, we require the user to supply a vector
of covariate labels, and store them as a character vector. We also
include a `min_degree`

parameter to control if an intercept
(`min_degree <= 0`

) and linear terms
(`min_degree <= 1`

) should be included in the model.

```
bru_mapper_p_quadratic <- function(labels, min_degree = 0, ...) {
if (is.factor(labels)) {
mapper <- list(
labels = levels(labels),
min_degree = min_degree
)
} else {
mapper <- list(
labels = as.character(labels),
min_degree = min_degree
)
}
bru_mapper_define(mapper, new_class = "bru_mapper_p_quadratic")
}
```

The `ibm_n`

method can compute the value of \(n\) from \(n=1+p+\frac{p(p+1)}{2}\):

```
ibm_n.bru_mapper_p_quadratic <- function(mapper, ...) {
p <- length(mapper$labels)
(mapper$min_degree <= 0) + (mapper$min_degree <= 1) * p + p * (p + 1) / 2
}
```

For `ibm_values`

, the default method is sufficient,
returning the vector \([1,\dots,n]\)
with \(n\) obtained by
`ibm_n(mapper)`

. However, for clearer result naming, we can
use a character vector, at least for INLA `f()`

models that
allow it (we could have an option argument to the
`bru_mapper_p_quadratic`

constructor to control this):

```
ibm_values.bru_mapper_p_quadratic <- function(mapper, ...) {
p <- length(mapper$labels)
n <- ibm_n(mapper)
jk <- expand.grid(seq_len(p), seq_len(p))
jk <- jk[jk[, 2] <= jk[, 1], , drop = FALSE]
c(
if (mapper$min_degree <= 0) "Intercept" else NULL,
if (mapper$min_degree <= 1) mapper$labels else NULL,
paste0(mapper$labels[jk[, 1]], ":", mapper$labels[jk[, 2]])
)
}
```

While the approach to `ibm_n`

and `ibm_values`

above works, it is wasteful to recompute the `n`

and
`values`

information for each call. Instead we can use that
the default method will check if `n`

or `values`

are stored in the mapper object, and then return them. If we change the
`.`

in the method definitions to `_`

, we can end
the constructor method like this, making subsequent method calls
faster:

```
bru_mapper_p_quadratic <- function(labels, min_degree = 0, ...) {
...
mapper <- bru_mapper_define(mapper, new_class = "bru_mapper_p_quadratic")
mapper$n <- ibm_n_bru_mapper_p_quadratic(mapper)
mapper$values <- ibm_values_bru_mapper_p_quadratic(mapper)
mapper
}
```

Note that the order matters, since our
`ibm_values_bru_mapper_p_quadratic`

function calls
`ibm_n(mapper)`

.

We can now define the main part of the mapper interface that computes
the model matrix linking the latent variables to the component effect.
It’s required that `NULL`

input should return a \(0\)-by-\(n\) matrix.

```
ibm_jacobian.bru_mapper_p_quadratic <- function(mapper, input, ...) {
if (is.null(input)) {
return(Matrix::Matrix(0, 0, ibm_n(mapper)))
}
p <- length(mapper$labels)
n <- ibm_n(mapper)
N <- NROW(input)
A <- list()
in_ <- as(input, "Matrix")
idx <- 0
if (mapper$min_degree <= 0) {
idx <- idx + 1
A[[idx]] <- Matrix::Matrix(1, N)
}
if (mapper$min_degree <= 1) {
idx <- idx + 1
A[[idx]] <- in_
}
for (k in seq_len(p)) {
idx <- idx + 1
A[[idx]] <- in_[, seq(k, p, by = 1), drop = FALSE] * in_[, k]
A[[idx]][, k] <- A[[idx]][, k] / 2
}
A <- do.call(cbind, A)
colnames(A) <- as.character(ibm_values(mapper))
A
}
```

If the output of a mapper is of a different size than
`NROW(input)`

, the mapper should define a
`ibm_n_output(mapper, input, ...)`

method to return the
output size for a given input.