# bbsc

##### Constrained Base-learners for Scalar Covariates

Constrained base-learners for fitting effects of scalar covariates in models with functional response

- Keywords
- models

##### Usage

```
bbsc(..., by = NULL, index = NULL, knots = 10, boundary.knots = NULL,
degree = 3, differences = 2, df = 4, lambda = NULL, center = FALSE,
cyclic = FALSE)
```bolsc(..., by = NULL, index = NULL, intercept = TRUE, df = NULL,
lambda = 0, K = NULL, weights = NULL,
contrasts.arg = "contr.treatment")

brandomc(..., contrasts.arg = "contr.dummy", df = 4)

##### Arguments

- ...
one or more predictor variables or one matrix or data frame of predictor variables.

- by
an optional variable defining varying coefficients, either a factor or numeric variable.

- index
a vector of integers for expanding the variables in

`...`

.- knots
either the number of knots or a vector of the positions of the interior knots (for more details see

`bbs`

).- boundary.knots
boundary points at which to anchor the B-spline basis (default the range of the data). A vector (of length 2) for the lower and the upper boundary knot can be specified.

- degree
degree of the regression spline.

- differences
a non-negative integer, typically 1, 2 or 3. If

`differences`

=*k*,*k*-th-order differences are used as a penalty (*0*-th order differences specify a ridge penalty).- df
trace of the hat matrix for the base-learner defining the base-learner complexity. Low values of

`df`

correspond to a large amount of smoothing and thus to "weaker" base-learners.- lambda
smoothing parameter of the penalty, computed from

`df`

when`df`

is specified.- center
experimental! See

`bbs`

.- cyclic
if

`cyclic = TRUE`

the fitted values coincide at the boundaries (useful for cyclic covariates such as day time etc.).- intercept
if

`intercept = TRUE`

an intercept is added to the design matrix of a linear base-learner.- K
in

`bolsc`

it is possible to specify the penalty matrix K- weights
experiemtnal! weights that are used for the computation of the transformation matrix Z.

- contrasts.arg
Note that a special

`contrasts.arg`

exists in package`mboost`

, namely "contr.dummy". This contrast is used per default in`brandomc`

. It leads to a dummy coding as returned by`model.matrix(~ x - 1)`

were the intercept is implicitly included but each factor level gets a separate effect estimate (for more details see`brandom`

).

##### Details

The base-learners `bbsc`

, `bolsc`

and `brandomc`

are
the base-learners `bbs`

, `bols`

and
`brandom`

with additional identifiability constraints.
The constraints enforce that
\(\sum_{i} \hat h(x_i, t) = 0\) for all \(t\), so that
effects varying over \(t\) can be interpreted as deviations
from the global functional intercept, see Web Appendix A of
Scheipl et al. (2015).
The constraint is enforced by a basis transformation of the design and penalty matrix.
In particular, it is sufficient to apply the constraint on the covariate-part of the design
and penalty matrix and thus, it is not necessary to change the basis in $t$-direction.
See Appendix A of Brockhaus et al. (2015) for technical details on how to enforce this sum-to-zero constraint.

Cannot deal with any missing values in the covariates.

##### Value

Equally to the base-learners of package `mboost`

:

An object of class `blg`

(base-learner generator) with a
`dpp`

function (data pre-processing) and other functions.

The call to `dpp`

returns an object of class
`bl`

(base-learner) with a `fit`

function. The call to
`fit`

finally returns an object of class `bm`

(base-model).

##### References

Brockhaus, S., Scheipl, F., Hothorn, T. and Greven, S. (2015): The functional linear array model. Statistical Modelling, 15(3), 279-300.

Scheipl, F., Staicu, A.-M. and Greven, S. (2015): Functional Additive Mixed Models, Journal of Computational and Graphical Statistics, 24(2), 477-501.

##### See Also

`FDboost`

for the model fit.
`bbs`

, `bols`

and `brandom`

for the
corresponding base-learners in `mboost`

.

##### Examples

```
n <- 60 ## number of cases
Gy <- 27 ## number of observation poionts per response trajectory
dat <- list()
dat$t <- (1:Gy-1)^2/(Gy-1)^2
set.seed(123)
dat$z1 <- rep(c(-1, 1), length = n)
dat$z1_fac <- factor(dat$z1, levels = c(-1, 1), labels = c("1", "2"))
# dat$z1 <- runif(n)
# dat$z1 <- dat$z1 - mean(dat$z1)
mut <- matrix(2*sin(pi*dat$t), ncol=Gy, nrow=n, byrow=TRUE) +
outer(dat$z1, dat$t, function(z1, t) z1*cos(pi*t) ) ## true linear predictor
## function(z1, t) z1*cos(4*pi*t)
sigma <- 0.1
## draw respone y_i(t) ~ N(mu_i(t), sigma)
dat$y <- apply(mut, 2, function(x) rnorm(mean = x, sd = sigma, n = n))
## fit model
m1 <- FDboost(y ~ 1 + bolsc(z1_fac, df=1), timeformula = ~ bbs(t, df = 6), data=dat)
## look for optimal mSTOP using cvrisk() or validateFDboost()
## plot estimated coefficients
plot(dat$t, 2*sin(pi*dat$t), col = 2, type = "l")
plot(m1, which = 1, lty = 2, add = TRUE)
plot(dat$t, 1*cos(pi*dat$t), col = 2, type = "l")
lines(dat$t, -1*cos(pi*dat$t), col = 2, type = "l")
plot(m1, which = 2, lty = 2, col = 1, add = TRUE)
```

*Documentation reproduced from package FDboost, version 0.2-0, License: GPL-2*