bootstrapCI {FDboost}

Function to compute bootstrap confidence intervals

Description

The model is fitted on bootstrapped samples of the data to compute bootstrapped coefficient estimates. To determine the optimal stopping iteration an inner bootstrap is run within each bootstrap fold. As estimation by boosting shrinks the coefficient estimates towards zero, to bootstrap confidence intervals are biased towards zero.

Usage

bootstrapCI(
  object,
  which = NULL,
  resampling_fun_outer = NULL,
  resampling_fun_inner = NULL,
  B_outer = 100,
  B_inner = 25,
  type_inner = c("bootstrap", "kfold", "subsampling"),
  levels = c(0.05, 0.95),
  verbose = TRUE,
  ...
)

Arguments

object

a fitted model object of class FDboost, for which the confidence intervals should be computed.

which

a subset of base-learners to take into account for computing confidence intervals.

resampling_fun_outer

function for the outer resampling procedure. resampling_fun_outer must be a function with arguments object and fun, where object corresponds to the fitted FDboost object and fun is passed to the fun argument of the resampling function (see examples). If NULL, applyFolds is used with 100-fold boostrap. Further arguments to applyFolds can be passed via .... Although the function can be defined very flexible, it is recommended to use applyFolds and, in particular, not cvrisk, as in this case, weights of the inner and outer fold will interact, probably causing the inner resampling to crash. For bootstrapped confidence intervals the outer function should usually be a bootstrap type of resampling.

resampling_fun_inner

function for the inner resampling procudure, which determines the optimal stopping iteration in each fold of the outer resampling procedure. Should be a function with one argument object for the fitted FDboost object. If NULL, cvrisk is used with 25-fold bootstrap.

B_outer

Number of resampling folds in the outer loop. Argument is overwritten, when a custom resampling_fun_outer is supplied.

B_inner

Number of resampling folds in the inner loop. Argument is overwritten, when a custom resampling_fun_inner is supplied.

type_inner

character argument for specifying the cross-validation method for the inner resampling level. Default is "bootstrap". Currently bootstrap, k-fold cross-validation and subsampling are implemented.

levels

the confidence levels required. If NULL, the raw results are returned.

verbose

if TRUE, information will be printed in the console

...

further arguments passed to applyFolds if the default for resampling_fun_outer is used

Value

A list containing the elements raw_results, the quantiles and mstops. In raw_results and quantiles, each baselearner selected with which in turn corresponds to a list element. The quantiles are given as vector, matrix or list of matrices depending on the nature of the effect. In case of functional effects the list element inquantiles is a length(levels) times length(effect) matrix, i.e. the rows correspond to the quantiles. In case of coefficient surfaces, quantiles comprises a list of matrices, where each list element corresponds to a quantile.

Note

Note that parallelization can be achieved by defining the resampling_fun_outer or _inner accordingly. See, e.g., cvrisk on how to parallelize resampling functions or the examples below. Also note that by defining a custum inner or outer resampling function the respective argument B_inner or B_outer is ignored. For models with complex baselearners, e.g., created by combining several baselearners with the Kronecker or row-wise tensor product, it is also recommended to use levels = NULL in order to let the function return the raw results and then manually compute confidence intervals. If a baselearner is not selected in any fold, the function treats its effect as constantly zero.

Author(s)

David Ruegamer, Sarah Brockhaus

Examples

if(require(refund)){
#########
# model with linear functional effect, use bsignal()
# Y(t) = f(t) + \int X1(s)\beta(s,t)ds + eps
set.seed(2121)
data1 <- pffrSim(scenario = "ff", n = 40)
data1$X1 <- scale(data1$X1, scale = FALSE)
dat_list <- as.list(data1)
dat_list$t <- attr(data1, "yindex")
dat_list$s <- attr(data1, "xindex")

## model fit by FDboost 
m1 <- FDboost(Y ~ 1 + bsignal(x = X1, s = s, knots = 8, df = 3), 
              timeformula = ~ bbs(t, knots = 8), data = dat_list)

}
              
             
# a short toy example with to few folds  
# and up to 200 boosting iterations 
bootCIs <- bootstrapCI(m1[200], B_inner = 2, B_outer = 5) 

# look at stopping iterations
bootCIs$mstops

# plot bootstrapped coefficient estimates
plot(bootCIs, ask = FALSE)


my_inner_fun <- function(object){ 
cvrisk(object, folds = cvLong(id = object$id, weights = 
model.weights(object), B = 2) # 10-fold for inner resampling
) 
}


bootCIs <- bootstrapCI(m1, resampling_fun_inner = my_inner_fun, 
                       B_outer = 5) # small B_outer to speed up


## We can also use the ... argument to parallelize the applyFolds
## function in the outer resampling 


bootCIs <- bootstrapCI(m1, B_inner = 5, B_outer = 3)


## Now let's parallelize the outer resampling and use 
## crossvalidation instead of bootstrap for the inner resampling

my_inner_fun <- function(object){ 
cvrisk(object, folds = cvLong(id = object$id, weights = 
model.weights(object), type = "kfold", # use CV
B = 5, # 5-fold for inner resampling
)) # use five cores
}

# use applyFolds for outer function to avoid messing up weights
my_outer_fun <- function(object, fun){
applyFolds(object = object,
folds = cv(rep(1, length(unique(object$id))), 
type = "bootstrap", B = 10), fun = fun) # parallelize on 10 cores
}


bootCIs <- bootstrapCI(m1, resampling_fun_inner = my_inner_fun,
                       resampling_fun_outer = my_outer_fun,
                       B_inner = 5, B_outer = 10) 


######## Example for scalar-on-function-regression with bsignal() 
data("fuelSubset", package = "FDboost")

## center the functional covariates per observed wavelength
fuelSubset$UVVIS <- scale(fuelSubset$UVVIS, scale = FALSE)
fuelSubset$NIR <- scale(fuelSubset$NIR, scale = FALSE)

## to make mboost:::df2lambda() happy (all design matrix entries < 10)
## reduce range of argvals to [0,1] to get smaller integration weights
fuelSubset$uvvis.lambda <- with(fuelSubset, (uvvis.lambda - min(uvvis.lambda)) /
(max(uvvis.lambda) - min(uvvis.lambda) ))
fuelSubset$nir.lambda <- with(fuelSubset, (nir.lambda - min(nir.lambda)) /
(max(nir.lambda) - min(nir.lambda) ))

## model fit with scalar response and two functional linear effects 
## include no intercept as all base-learners are centered around 0    

mod2 <- FDboost(heatan ~ bsignal(UVVIS, uvvis.lambda, knots = 40, df = 4, check.ident = FALSE) 
               + bsignal(NIR, nir.lambda, knots = 40, df=4, check.ident = FALSE), 
               timeformula = NULL, data = fuelSubset) 



# takes some time, because of defaults: B_outer = 100, B_inner = 25
bootCIs <- bootstrapCI(mod2, B_outer = 10, B_inner = 5)
           # in practice, rather set B_outer = 1000