bbsc: Constrained Base-learners for Scalar Covariates
In FDboost: Boosting Functional Regression Models

Description Usage Arguments Details Value Author(s) References See Also Examples

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

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)

`...`	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`).

The base-learners bbsc, bolsc and brandomc are the base-learners bbs, bols and brandom with additional identifiability constraints. The constraints enforce that ∑_{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.

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).

Sarah Brockhaus, Almond Stoecker

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.

FDboost for the model fit. bbs, bols and brandom for the corresponding base-learners in mboost.

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)