Constrained Base-learners for Scalar Covariates
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 |
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 |
df |
trace of the hat matrix for the base-learner defining the
base-learner complexity. Low values of |
lambda |
smoothing parameter of the penalty, computed from |
center |
See |
cyclic |
if |
intercept |
if |
K |
in |
weights |
experiemtnal! weights that are used for the computation of the transformation matrix Z. |
contrasts.arg |
Note that a special |
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.
#### 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()
cvm <- cvrisk(m1, grid = 1:500)
m1[mstop(cvm)]
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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.