grapes-Xc-grapes function

Constrained row tensor product

Combining single base-learners to form new, more complex base-learners, with an identifiability constraint to center the interaction around the intercept and around the two main effects. Suitable for functional response.

bl1 %Xc% bl2

Arguments

• bl1: base-learner 1, e.g. bols(x1)
• bl2: base-learner 2, e.g. bols(x2)

Returns

An object of class blg (base-learner generator) with a dpp function as for other baselearners.

Details

Similar to %X% in package mboost, see %X%, a row tensor product of linear base-learners is returned by %Xc%. %Xc% applies a sum-to-zero constraint to the design matrix suitable for functional response if an interaction of two scalar covariates is specified in the case that the model contains a global intercept and both main effects, as the interaction is centered around the intercept and centered around the two main effects. See Web Appendix A of Brockhaus et al. (2015) for details on how to enforce the constraint for the functional intercept. Use, e.g., in a model call to FDboost, following the scheme, y ~ 1 + bolsc(x1) + bolsc(x2) + bols(x1) %Xc% bols(x2), where 1 induces a global intercept and x1, x2 are factor variables, see Ruegamer et al. (2018).

Examples

######## Example for function-on-scalar-regression with interaction effect of two scalar covariates 
data("viscosity", package = "FDboost") 
## set time-interval that should be modeled
interval <- "101"

## model time until "interval" and take log() of viscosity
end <- which(viscosity$timeAll == as.numeric(interval))
viscosity$vis <- log(viscosity$visAll[,1:end])
viscosity$time <- viscosity$timeAll[1:end]
# with(viscosity, funplot(time, vis, pch = 16, cex = 0.2))

## fit model with interaction that is centered around the intercept 
## and the two main effects 
mod1 <- FDboost(vis ~ 1 + bolsc(T_C, df=1) + bolsc(T_A, df=1) + 
                bols(T_C, df=1) %Xc% bols(T_A, df=1),
                timeformula = ~bbs(time, df=6),
                numInt = "equal", family = QuantReg(),
                offset = NULL, offset_control = o_control(k_min = 9),
                data = viscosity, control=boost_control(mstop = 100, nu = 0.4))
                
## check centering around intercept
colMeans(predict(mod1, which = 4))

## check centering around main effects
colMeans(predict(mod1, which = 4)[viscosity$T_A == "low", ])
colMeans(predict(mod1, which = 4)[viscosity$T_A == "high", ])
colMeans(predict(mod1, which = 4)[viscosity$T_C == "low", ])
colMeans(predict(mod1, which = 4)[viscosity$T_C == "low", ])

## find optimal mstop using cvrsik() or validateFDboost()
## ... 

## look at interaction effect in one plot
# funplot(mod1$yind, predict(mod1, which=4))

References

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

Ruegamer D., Brockhaus, S., Gentsch K., Scherer, K., Greven, S. (2018). Boosting factor-specific functional historical models for the detection of synchronization in bioelectrical signals. Journal of the Royal Statistical Society: Series C (Applied Statistics), 67, 621-642.

Author(s)

Sarah Brockhaus, David Ruegamer