FDboost (version 1.0-0)

# bbsc: Constrained Base-learners for Scalar Covariates

## Description

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

## 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

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).

## 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).

## 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.

## 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.

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

## Examples

# NOT RUN {
#### simulate data with functional response and scalar covariate (functional ANOVA)
n <- 60   ## number of cases
Gy <- 27  ## number of observation poionts per response curve
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)

# mean and standard deviation for the functional response
mut <- matrix(2*sin(pi*dat$t), ncol = Gy, nrow = n, byrow = TRUE) + outer(dat$z1, dat$t, function(z1, t) z1*cos(pi*t) ) # true linear predictor sigma <- 0.1 # draw respone y_i(t) ~ N(mu_i(t), sigma) dat$y <- apply(mut, 2, function(x) rnorm(mean = x, sd = sigma, n = n))

## fit function-on-scalar model with a linear effect of z1
m1 <- FDboost(y ~ 1 + bolsc(z1_fac, df = 1), timeformula = ~ bbs(t, df = 6), data = dat)

# look for optimal mSTOP using cvrisk() or validateFDboost()

# }
# NOT RUN {
cvm <- cvrisk(m1, grid = 1:500)
m1[mstop(cvm)]
# }
# NOT RUN {
m1[200] # use 200 boosting iterations

# plot true and estimated coefficients
plot(dat$t, 2*sin(pi*dat$t), col = 2, type = "l", main = "intercept")
plot(m1, which = 1, lty = 2, add = TRUE)

plot(dat$t, 1*cos(pi*dat$t), col = 2, type = "l", main = "effect of z1")
lines(dat$t, -1*cos(pi*dat$t), col = 2, type = "l")
plot(m1, which = 2, lty = 2, col = 1, add = TRUE)

# }