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R/baselearnersX.R
In FDboost: Boosting Functional Regression Models
#################################
#### Base-learner for historic effect of functional covariate
### with integral over specific limits, e.g. s<=t
### hyper parameters for signal baselearner with P-splines
hyper_histx <- function(mf, vary, knots = 10, boundary.knots = NULL, degree = 3,
differences = 1, df = 4, lambda = NULL, center = FALSE,
cyclic = FALSE, constraint = "none", deriv = 0L,
Z=NULL, limits=NULL,
standard="no", intFun=integrationWeightsLeft,
inS="smooth", inTime="smooth",
penalty = "ps", check.ident = FALSE,
format="long") {
knotf <- function(x, knots, boundary.knots) {
if (is.null(boundary.knots))
boundary.knots <- range(x, na.rm = TRUE)
## <fixme> At the moment only NULL or 2 boundary knots can be specified.
## Knot expansion is done automatically on an equidistand grid.</fixme>
if ((length(boundary.knots) != 2) || !boundary.knots[1] < boundary.knots[2])
stop("boundary.knots must be a vector (or a list of vectors) ",
"of length 2 in increasing order")
if (length(knots) == 1) {
knots <- seq(from = boundary.knots[1],
to = boundary.knots[2], length = knots + 2)
knots <- knots[2:(length(knots) - 1)]
}
list(knots = knots, boundary.knots = boundary.knots)
}
nm <- colnames(mf)[colnames(mf) != vary]
if (is.list(knots)) if(!all(names(knots) %in% nm))
stop("variable names and knot names must be the same")
if (is.list(boundary.knots)) if(!all(names(boundary.knots) %in% nm))
stop("variable names and boundary.knot names must be the same")
if (!identical(center, FALSE) && cyclic)
stop("centering of cyclic covariates not yet implemented")
ret <- vector(mode = "list", length = length(nm))
names(ret) <- nm
for (n in nm)
ret[[n]] <- knotf(getTime(mf[[n]]),
knots=if(is.list(knots)) knots[[n]] else knots,
boundary.knots = if(is.list(boundary.knots)) boundary.knots[[n]] else boundary.knots)
if (cyclic & constraint != "none")
stop("constraints not implemented for cyclic B-splines")
stopifnot(is.numeric(deriv) & length(deriv) == 1)
## prediction is usually set in/by newX()
list(knots = ret, degree = degree, differences = differences,
df = df, lambda = lambda, center = center, cyclic = cyclic,
Ts_constraint = constraint, deriv = deriv,
Z = Z, limits = limits,
standard = standard, intFun = intFun,
inS = inS, inTime = inTime,
penalty = penalty, check.ident = check.ident, format = format,
prediction = FALSE)
}
### model.matrix for P-splines base-learner of signal matrix mf
### for response observed over a common grid, args$format="wide"
### or irregularly observed reponse, args$format="long"
X_histx <- function(mf, vary, args) {
stopifnot(is.data.frame(mf))
varnames <- names(mf)
xname <- getXLab(mf[[1]])
X1 <- getX(mf[[1]])
xind <- getArgvals(mf[[1]])
yind <- getTime(mf[[1]])
# force user to supply indices
#if(is.null(xind)) xind <- args$s # if the attribute is NULL use the s of the model fit
#if(is.null(yind)) yind <- args$time # if the attribute is NULL use the time of the model fit
nobs <- nrow(X1)
## get id-variable
id <- getId(mf[[1]]) ### id is always there, in long and wide format!!!
## id has values 1, 2, 3, ... for response in long format
# compute design-matrix in s-direction
Bs <- switch(args$inS,
# B-spline basis of specified degree
#"smooth" = bsplines(xind, knots=args$knots[[varnames]]$knots,
# boundary.knots=args$knots[[varnames]]$boundary.knots,
# degree=args$degree),
"smooth" = mboost_intern(xind, knots = args$knots[[varnames]]$knots,
boundary.knots = args$knots[[varnames]]$boundary.knots,
degree = args$degree,
fun = "bsplines"),
"linear" = matrix(c(rep(1, length(xind)), xind), ncol = 2),
"constant"= matrix(c(rep(1, length(xind))), ncol = 1))
colnames(Bs) <- paste(xname, 1:ncol(Bs), sep="")
# integration weights
L <- args$intFun(X1=X1, xind=xind)
# print(L[1,])
## Weighting with matrix of functional covariate
#X1 <- L*X1 ## -> do the integration weights more sophisticated!!
# # set up design matrix for historical model and s<=t with s and t equal to xind
# # expand matrix of original observations to lower triangular matrix
# X1des0 <- matrix(0, ncol=ncol(X1), nrow=ncol(X1)*nrow(X1))
# for(i in 1:ncol(X1des0)){
# #print(nrow(X1)*(i-1)+1)
# X1des0[(nrow(X1)*(i-1)+1):nrow(X1des0) ,i] <- X1[,i] # use fun. variable * integration weights
# }
## set up design matrix for historical model according to args$limits()
# use the argument limits (Code taken of function ff(), package refund)
limits <- args$limits
if (!is.null(limits)) {
if (!is.function(limits)) {
if (!(limits %in% c("s<t", "s<=t"))) {
stop("supplied <limits> argument unknown")
}
if (limits == "s<t") {
limits <- function(s, t) {
s < t
}
}
else {
if (limits == "s<=t") {
limits <- function(s, t) {
(s < t) | (s == t)
}
}
}
}
}else{
stop("<limits> argument cannot be NULL.")
}
## save the limits function in the arguments
args$limits <- limits
# ### use function limits to set up design matrix according to function limits
# ### by setting 0 at the time-points that should not be used
# if(args$format == "wide"){
# ## expand yind by replication to the yind of all observations together
# ind0 <- !t(outer( xind, rep(yind, each=nobs), limits) )
# yindHelp <- rep(yind, each=nobs)
# }else{
#### yind is over all observations in long format
ind0 <- !t(outer( xind, yind, limits) )
yindHelp <- yind
# }
### Compute the design matrix as sparse or normal matrix
### depending on dimensions of the final design matrix
MATRIX <- any(c(nrow(ind0), ncol(Bs)) > c(500, 50)) #MATRIX <- any(dim(X) > c(500, 50))
MATRIX <- MATRIX && options("mboost_useMatrix")$mboost_useMatrix
if(MATRIX){
#message("use sparse matrix in X_hist")
diag <- Diagonal
cbind <- cBind
###### <FIXME> construction of X1des directly as sparse matrix does not work
# ### compute the design matrix as sparse matrix
# if(args$format == "wide"){
# tempIndexDesign <- which(!ind0, arr.ind=TRUE)
# tempIndexX1 <- cbind(rep(1:nobs, length.out=nrow(tempIndexDesign)), tempIndexDesign[,2] )
# X1des <- sparseMatrix(i=tempIndexDesign[,1], j=tempIndexDesign[,2],
# x=X1[tempIndexX1], dims=dim(ind0))
# # object.size(X1des)
# rm(tempIndexX1, tempIndexDesign)
# }else{ # long format
# tempj <- unlist(apply(!ind0, 1, which)) # in which columns are the values?
# ## i: row numbers: one row number per observation of response,
# # repeat the row number for each entry
# X1des <- sparseMatrix(i=rep(1:length(id), times=rowSums(!ind0)), j=tempj,
# x=X1[cbind(rep(id, t=rowSums(!ind0)), tempj)], dims=dim(ind0))
# # object.size(X1des)
# rm(tempj)
# }
# ###### <FIXME> instead: build the matrix as dense matrix and convert it into a sparse matrix
# if(args$format == "wide"){
# ### expand the design matrix for all observations (yind is equal for all observations!)
# ### the response is a vector (y1(t1), y2(t1), ... , yn(t1), yn(tG))
# X1des <- X1[rep(1:nobs, times=length(yind)), ]
# } else{ # yind is over all observations in long format
# }
X1des <- X1[id, ]
X1des[ind0] <- 0
X1des <- Matrix(X1des, sparse=TRUE) # convert into sparse matrix
## if(!is(X1, "sparseMatrix"))
# X1 <- Matrix(X1, sparse=TRUE) # convert into sparse matrix
# X1des <- X1[as.numeric(id), ]
# X1des[ind0] <- 0
}else{ # small matrices: do not use Matrix
# if(args$format == "wide"){
# ### expand the design matrix for all observations (yind is equal for all observations!)
# ### the response is a vector (y1(t1), y2(t1), ... , yn(t1), yn(tG))
# X1des <- X1[rep(1:nobs, times=length(yind)), ]
# } else{ # yind is over all observations in long format
X1des <- X1[id, ]
# }
X1des[ind0] <- 0
}
## set up matrix with adequate integration and standardization weights
## start with a matrix of integration weights
## case of "no" standardization
Lnew <- args$intFun(X1des, xind)
Lnew[ind0] <- 0
## Standardize with exact length of integration interval
## (1/t-t0) \int_{t0}^t f(s) ds
if(args$standard == "length"){
## use fundamental theorem of calculus
## \lim t->t0- (1/t-t0) \int_{t0}^t f(s) ds = f(t0)
## -> integration weight in s-direction should be 1
## integration weights in s-direction always sum exactly to 1,
## good for small number of observations!
args$vecStand <- rowSums(Lnew)
args$vecStand[args$vecStand==0] <- 1 ## cannnot divide 0/0, instead divide 0/1
Lnew <- Lnew * 1/args$vecStand
}
## use time of current observation for standardization
## (1/t) \int_{t0}^t f(s) ds
if(args$standard=="time"){
if(any(yindHelp <= 0)) stop("For standardization with time, time must be positive.")
## Lnew <- matrix(1, ncol=ncol(X1des), nrow=nrow(X1des))
## Lnew[ind0] <- 0
## use fundamental theorem of calculus
## \lim t->0+ (1/t) \int_0^t f(s) ds = f(0), if necessary
## (as previously X*L, use now X*(1/L) for cases with one single point)
yindHelp[yindHelp==0] <- L[1,1] # impossible!
# standFact <- 1/yindHelp
args$vecStand <- yindHelp
Lnew <- Lnew * 1/yindHelp
}
## print(round(Lnew, 2))
## print(rowSums(Lnew))
## print(args$vecStand)
# multiply design matrix with integration weights and standardization weights
X1des <- X1des * Lnew
# Design matrix is product of expanded X1 and basis expansion over xind
X1des <- X1des %*% Bs
## see Scheipl and Greven (2016) Identifiability in penalized function-on-function regression models
if(args$check.ident && args$inS == "smooth"){
K1 <- diff(diag(ncol(Bs)), differences = args$differences)
K1 <- crossprod(K1)
# use the limits function to compute check measures on corresponding subsets of x(s) and B_j
res_check <- check_ident(X1 = X1, L = L, Bs = Bs, K = K1, xname = xname,
penalty = args$penalty,
limits = args$limits,
yind = yind, id = id, # yind is always in long format
X1des = X1des, ind0 = ind0, xind = xind)
args$penalty <- res_check$penalty
args$logCondDs <- res_check$logCondDs
args$logCondDs_hist <- res_check$logCondDs_hist
args$overlapKe <- res_check$overlapKe
args$cumOverlapKe <- res_check$cumOverlapKe
args$maxK <- res_check$maxK
}
# wide: design matrix over index of response for one response
# long: design matrix over index of response (yind has long format!)
Bt <- switch(args$inTime,
# B-spline basis of specified degree
# "smooth" = bsplines(yind, knots=args$knots[[varnames]]$knots,
# boundary.knots=args$knots[[varnames]]$boundary.knots,
# degree=args$degree),
"smooth" = mboost_intern(yind, knots = args$knots[[varnames]]$knots,
boundary.knots = args$knots[[varnames]]$boundary.knots,
degree = args$degree,
fun = "bsplines"),
"linear" = matrix(c(rep(1, length(yind)), yind), ncol=2),
"constant"= matrix(c(rep(1, length(yind))), ncol=1))
# # stack design-matrix of response nobs times in wide format
# if(args$format == "wide"){
# Bt <- Bt[rep(1:length(yind), each=nobs), ]
# }
if(! mboost_intern(Bt, fun = "isMATRIX") ) Bt <- matrix(Bt, ncol=1)
# calculate row-tensor
# X <- (X1 %x% t(rep(1, ncol(X2))) ) * ( t(rep(1, ncol(X1))) %x% X2 )
dimnames(Bt) <- NULL # otherwise warning "dimnames [2] mismatch..."
X <- X1des[,rep(1:ncol(Bs), each=ncol(Bt))] * Bt[,rep(1:ncol(Bt), times=ncol(Bs))]
if(! mboost_intern(X, fun = "isMATRIX") ) X <- matrix(X, ncol=1)
colnames(X) <- paste0(xname, 1:ncol(X))
### Penalty matrix: product differences matrix for smooth effect
if(args$inS == "smooth"){
K1 <- diff(diag(ncol(Bs)), differences = args$differences)
K1 <- crossprod(K1)
if(args$penalty == "pss"){
# <FIXME> allow for variable shrinkage parameter in penalty_pss()?
K1 <- penalty_pss(K = K1, difference = args$difference, shrink = 0.1)
}
}else{ # Ridge-penalty
K1 <- diag(ncol(Bs))
}
#K1 <- matrix(0, ncol=ncol(Bs), nrow=ncol(Bs))
#print(args$penalty)
if(args$inTime == "smooth"){
K2 <- diff(diag(ncol(Bt)), differences = args$differences)
K2 <- crossprod(K2)
}else{
K2 <- diag(ncol(Bt))
}
# compute penalty matrix for the whole effect
suppressMessages(K <- kronecker(K1, diag(ncol(Bt))) +
kronecker(diag(ncol(Bs)), K2))
## compare specified degrees of freedom to dimension of null space
if (!is.null(args$df)){
rns <- ncol(K) - qr(as.matrix(K))$rank # compute rank of null space
if (rns == args$df)
warning( sQuote("df"), " equal to rank of null space ",
"(unpenalized part of P-spline);\n ",
"Consider larger value for ", sQuote("df"),
" or set ", sQuote("center = TRUE"), ".", immediate.=TRUE)
if (rns > args$df)
stop("not possible to specify ", sQuote("df"),
" smaller than the rank of the null space\n ",
"(unpenalized part of P-spline). Use larger value for ",
sQuote("df"), " or set ", sQuote("center = TRUE"), ".")
}
# save matrices to compute numbers for identifiability checks
args$Bs <- Bs
args$X1des <- X1des
args$K1 <- K1
args$L <- L
# tidy up workspace
rm(Bs, Bt, ind0, X1des, X1, L)
return(list(X = X, K = K, args = args))
}
###############################################################################
#' Base-learners for Functional Covariates
#'
#' Base-learners that fit historical functional effects that can be used with the
#' tensor product, as e.g., hbistx(...) \%X\% bolsc(...).
#' For expert use only! May show unexpected behavior
#' compared to other base-learners for functional data!
#'
#' @param x object of type \code{hmatrix} containing time, index and functional covariate;
#' note that \code{timeLab} in the \code{hmatrix}-object must be equal to
#' the name of the time-variable in \code{timeformula} in the \code{FDboost}-call
#' @param knots either the number of knots or a vector of the positions
#' of the interior knots (for more details see \code{\link[mboost]{bbs})}.
#' @param 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.
#' @param degree degree of the regression spline.
#' @param differences a non-negative integer, typically 1, 2 or 3. Defaults to 1.
#' If \code{differences} = \emph{k}, \emph{k}-th-order differences are used as
#' a penalty (\emph{0}-th order differences specify a ridge penalty).
#' @param df trace of the hat matrix for the base-learner defining the
#' base-learner complexity. Low values of \code{df} correspond to a
#' large amount of smoothing and thus to "weaker" base-learners.
#' @param lambda smoothing parameter of the penalty, computed from \code{df} when \code{df} is specified.
#' @param penalty by default, \code{penalty="ps"}, the difference penalty for P-splines is used,
#' for \code{penalty="pss"} the penalty matrix is transformed to have full rank,
#' so called shrinkage approach by Marra and Wood (2011)
#' @param check.ident use checks for identifiability of the effect, based on Scheipl and Greven (2016);
#' see Brockhaus et al. (2016) for identifiability checks that take into account the integration limits
#' @param standard the historical effect can be standardized with a factor.
#' "no" means no standardization, "time" standardizes with the current value of time and
#' "lenght" standardizes with the lenght of the integral
#' @param intFun specify the function that is used to compute integration weights in \code{s}
#' over the functional covariate \eqn{x(s)}
#' @param inS historical effect can be smooth, linear or constant in s,
#' which is the index of the functional covariates x(s).
#' @param inTime historical effect can be smooth, linear or constant in time,
#' which is the index of the functional response y(time).
#' @param limits defaults to \code{"s<=t"} for an historical effect with s<=t;
#' either one of \code{"s<t"} or \code{"s<=t"} for [l(t), u(t)] = [T1, t];
#' otherwise specify limits as a function for integration limits [l(t), u(t)]:
#' function that takes \eqn{s} as the first and \code{t} as the second argument and returns
#' \code{TRUE} for combinations of values (s,t) if \eqn{s} falls into the integration range for
#' the given \eqn{t}.
#'
#' @details \code{bhistx} implements a base-learner for functional covariates with
#' flexible integration limits \code{l(t)}, \code{r(t)} and the possibility to
#' standardize the effect by \code{1/t} or the length of the integration interval.
#' The effect is \code{stand * int_{l(t)}^{r_{t}} x(s)beta(t,s) ds}.
#' The base-learner defaults to a historical effect of the form
#' \eqn{\int_{T1}^{t} x_i(s)beta(t,s) ds},
#' where \eqn{T1} is the minimal index of \eqn{t} of the response \eqn{Y(t)}.
#' \code{bhistx} can only be used if \eqn{Y(t)} and \eqn{x(s)} are observd over
#' the same domain \eqn{s,t \in [T1, T2]}.
#'
#' Note that the data has to be supplied as a \code{hmatrix} object for
#' model fit and predictions.
#'
#' @return Equally to the base-learners of package mboost:
#'
#' An object of class \code{blg} (base-learner generator) with a
#' \code{dpp} function (dpp, data pre-processing).
#'
#' The call of \code{dpp} returns an object of class
#' \code{bl} (base-learner) with a \code{fit} function. The call to
#' \code{fit} finally returns an object of class \code{bm} (base-model).
#'
#' @seealso \code{\link{FDboost}} for the model fit.
#' @keywords models
#'
#' @references
#' Brockhaus, S., Melcher, M., Leisch, F. and Greven, S. (2016):
#' Boosting flexible functional regression models with a high number of functional historical effects,
#' Statistics and Computing, accepted.
#'
#' Marra, G. and Wood, S.N. (2011): Practical variable selection for generalized additive models.
#' Computational Statistics & Data Analysis, 55, 2372-2387.
#'
#' Scheipl, F., Staicu, A.-M. and Greven, S. (2015):
#' Functional Additive Mixed Models, Journal of Computational and Graphical Statistics, 24(2), 477-501.
#' \url{http://arxiv.org/abs/1207.5947}
#'
#' Scheipl, F. and Greven, S. (2016): Identifiability in penalized function-on-function regression models.
#' Electronic Journal of Statistics, 10(1), 495-526.
#'
#' @examples
#' if(require(refund)){
#' ## simulate some data from a historical model
#' ## the interaction effect is in this case not necessary
#' n <- 100
#' nygrid <- 35
#' data1 <- pffrSim(scenario = c("int", "ff"), limits = function(s,t){ s <= t },
#' n = n, nygrid = nygrid)
#' data1$X1 <- scale(data1$X1, scale = FALSE) ## center functional covariate
#' dataList <- as.list(data1)
#' dataList$tvals <- attr(data1, "yindex")
#'
#' ## create the hmatrix-object
#' X1h <- with(dataList, hmatrix(time = rep(tvals, each = n), id = rep(1:n, nygrid),
#' x = X1, argvals = attr(data1, "xindex"),
#' timeLab = "tvals", idLab = "wideIndex",
#' xLab = "myX", argvalsLab = "svals"))
#' dataList$X1h <- I(X1h)
#' dataList$svals <- attr(data1, "xindex")
#' ## add a factor variable
#' dataList$zlong <- factor(gl(n = 2, k = n/2, length = n*nygrid), levels = 1:3)
#' dataList$z <- factor(gl(n = 2, k = n/2, length = n), levels = 1:3)
#'
#' ## do the model fit with main effect of bhistx() and interaction of bhistx() and bolsc()
#' mod <- FDboost(Y ~ 1 + bhistx(x = X1h, df = 5, knots = 5) +
#' bhistx(x = X1h, df = 5, knots = 5) %X% bolsc(zlong),
#' timeformula = ~ bbs(tvals, knots = 10), data = dataList)
#'
#' ## alternative parameterization: interaction of bhistx() and bols()
#' mod <- FDboost(Y ~ 1 + bhistx(x = X1h, df = 5, knots = 5) %X% bols(zlong),
#' timeformula = ~ bbs(tvals, knots = 10), data = dataList)
#'
#' \dontrun{
#' # find the optimal mstop over 5-fold bootstrap (small example to reduce run time)
#' cv <- cvrisk(mod, folds = cv(model.weights(mod), B = 5))
#' mstop(cv)
#' mod[mstop(cv)]
#'
#' appl1 <- applyFolds(mod, folds = cv(rep(1, length(unique(mod$id))), type = "bootstrap", B = 5))
#'
#' # plot(mod)
#' }
#'}
#'
#' @export
### P-spline base-learner for effect with time-specific integration limits using a hmatrix-object
bhistx <- function(x,
limits = "s<=t", standard = c("no", "time", "length"),
intFun = integrationWeightsLeft,
inS = c("smooth","linear","constant"), inTime = c("smooth", "linear", "constant"),
knots = 10, boundary.knots = NULL, degree = 3, differences = 1, df = 4,
lambda = NULL, #center = FALSE, cyclic = FALSE
penalty = c("ps", "pss"), check.ident = FALSE
){
index <- NULL # index is treated within hmatrix
if (!is.null(lambda)) df <- NULL
cll <- match.call()
cll[[1]] <- as.name("bhistx")
#print(cll)
penalty <- match.arg(penalty)
#print(penalty)
standard <- match.arg(standard)
inS <- match.arg(inS)
inTime <- match.arg(inTime)
#print(inS)
if(!is.hmatrix(x)) stop("x has to be a hmatrix-object.")
varnames <- all.vars(cll)[1] # only the first name, the name of x is a variable name
# Reshape mfL so that it is the dataframe of the signal with
# the index of the signal and the index of the response as attributes
xname <- getXLab(x)
indname <- getArgvalsLab(x)
indnameY <- getTimeLab(x)
## save the hmatrix-object into a data.frame
mf <- data.frame(z=I(x))
names(mf) <- varnames
if(!all( abs(colMeans(getX(x), na.rm = TRUE)) < .Machine$double.eps*10^10)){
message(xname, " is not centered per column, inducing a non-centered effect.")
}
vary <- ""
# CC <- all(Complete.cases(mf))
CC <- all(mboost_intern(mf, fun = "Complete.cases"))
if (!CC)
warning("base-learner contains missing values;\n",
"missing values are excluded per base-learner, ",
"i.e., base-learners may depend on different",
" numbers of observations.")
## call X_histx in order to compute parameter settings, e.g.
## the transformation matrix Z, shrinkage penalty, identifiability problems...
# X_histx for data in long format, as hmatrix is always in long-format
temp <- X_histx(mf, vary,
args = hyper_histx(mf, vary, knots = knots, boundary.knots = boundary.knots,
degree = degree, differences = differences,
df = df, lambda = lambda, center = FALSE, cyclic = FALSE,
limits = limits,
standard = standard, intFun = intFun,
inS = inS, inTime = inTime,
penalty = penalty, check.ident = check.ident,
format="long"))
temp$args$check.ident <- FALSE
ret <- list(model.frame = function() mf,
get_call = function(){
cll <- deparse(cll, width.cutoff=500L)
if (length(cll) > 1)
cll <- paste(cll, collapse="")
cll
},
get_data = function() mf,
## set index to NULL, as the index is treated within X_hist()
## get_index = function() index,
get_index = function() NULL,
get_vary = function() vary,
get_names = function(){
varnames
}, #colnames(mf),
set_names = function(value) {
#if(length(value) != length(colnames(mf)))
if(length(value) != names(mf[1]))
stop(sQuote("value"), " must have same length as ",
sQuote("names(mf[1])"))
for (i in 1:length(value)){
cll[[i+1]] <<- as.name(value[i])
}
attr(mf, "names") <<- value
})
class(ret) <- "blg"
### call fitter with X_histx
# ret$dpp <- bl_lin(ret, Xfun = X_histx, args = temp$args)
ret$dpp <- mboost_intern(ret, Xfun = X_histx, args = temp$args, fun = "bl_lin")
return(ret)
}
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#################################
#### Base-learner for historic effect of functional covariate
### with integral over specific limits, e.g. s<=t
### hyper parameters for signal baselearner with P-splines
hyper_histx <- function(mf, vary, knots = 10, boundary.knots = NULL, degree = 3,
differences = 1, df = 4, lambda = NULL, center = FALSE,
cyclic = FALSE, constraint = "none", deriv = 0L,
Z=NULL, limits=NULL,
standard="no", intFun=integrationWeightsLeft,
inS="smooth", inTime="smooth",
penalty = "ps", check.ident = FALSE,
format="long") {
knotf <- function(x, knots, boundary.knots) {
if (is.null(boundary.knots))
boundary.knots <- range(x, na.rm = TRUE)
## <fixme> At the moment only NULL or 2 boundary knots can be specified.
## Knot expansion is done automatically on an equidistand grid.</fixme>
if ((length(boundary.knots) != 2) || !boundary.knots[1] < boundary.knots[2])
stop("boundary.knots must be a vector (or a list of vectors) ",
"of length 2 in increasing order")
if (length(knots) == 1) {
knots <- seq(from = boundary.knots[1],
to = boundary.knots[2], length = knots + 2)
knots <- knots[2:(length(knots) - 1)]
}
list(knots = knots, boundary.knots = boundary.knots)
}
nm <- colnames(mf)[colnames(mf) != vary]
if (is.list(knots)) if(!all(names(knots) %in% nm))
stop("variable names and knot names must be the same")
if (is.list(boundary.knots)) if(!all(names(boundary.knots) %in% nm))
stop("variable names and boundary.knot names must be the same")
if (!identical(center, FALSE) && cyclic)
stop("centering of cyclic covariates not yet implemented")
ret <- vector(mode = "list", length = length(nm))
names(ret) <- nm
for (n in nm)
ret[[n]] <- knotf(getTime(mf[[n]]),
knots=if(is.list(knots)) knots[[n]] else knots,
boundary.knots = if(is.list(boundary.knots)) boundary.knots[[n]] else boundary.knots)
if (cyclic & constraint != "none")
stop("constraints not implemented for cyclic B-splines")
stopifnot(is.numeric(deriv) & length(deriv) == 1)
## prediction is usually set in/by newX()
list(knots = ret, degree = degree, differences = differences,
df = df, lambda = lambda, center = center, cyclic = cyclic,
Ts_constraint = constraint, deriv = deriv,
Z = Z, limits = limits,
standard = standard, intFun = intFun,
inS = inS, inTime = inTime,
penalty = penalty, check.ident = check.ident, format = format,
prediction = FALSE)
}
### model.matrix for P-splines base-learner of signal matrix mf
### for response observed over a common grid, args$format="wide"
### or irregularly observed reponse, args$format="long"
X_histx <- function(mf, vary, args) {
stopifnot(is.data.frame(mf))
varnames <- names(mf)
xname <- getXLab(mf[[1]])
X1 <- getX(mf[[1]])
xind <- getArgvals(mf[[1]])
yind <- getTime(mf[[1]])
# force user to supply indices
#if(is.null(xind)) xind <- args$s # if the attribute is NULL use the s of the model fit
#if(is.null(yind)) yind <- args$time # if the attribute is NULL use the time of the model fit
nobs <- nrow(X1)
## get id-variable
id <- getId(mf[[1]]) ### id is always there, in long and wide format!!!
## id has values 1, 2, 3, ... for response in long format
# compute design-matrix in s-direction
Bs <- switch(args$inS,
# B-spline basis of specified degree
#"smooth" = bsplines(xind, knots=args$knots[[varnames]]$knots,
# boundary.knots=args$knots[[varnames]]$boundary.knots,
# degree=args$degree),
"smooth" = mboost_intern(xind, knots = args$knots[[varnames]]$knots,
boundary.knots = args$knots[[varnames]]$boundary.knots,
degree = args$degree,
fun = "bsplines"),
"linear" = matrix(c(rep(1, length(xind)), xind), ncol = 2),
"constant"= matrix(c(rep(1, length(xind))), ncol = 1))
colnames(Bs) <- paste(xname, 1:ncol(Bs), sep="")
# integration weights
L <- args$intFun(X1=X1, xind=xind)
# print(L[1,])
## Weighting with matrix of functional covariate
#X1 <- L*X1 ## -> do the integration weights more sophisticated!!
# # set up design matrix for historical model and s<=t with s and t equal to xind
# # expand matrix of original observations to lower triangular matrix
# X1des0 <- matrix(0, ncol=ncol(X1), nrow=ncol(X1)*nrow(X1))
# for(i in 1:ncol(X1des0)){
# #print(nrow(X1)*(i-1)+1)
# X1des0[(nrow(X1)*(i-1)+1):nrow(X1des0) ,i] <- X1[,i] # use fun. variable * integration weights
# }
## set up design matrix for historical model according to args$limits()
# use the argument limits (Code taken of function ff(), package refund)
limits <- args$limits
if (!is.null(limits)) {
if (!is.function(limits)) {
if (!(limits %in% c("s<t", "s<=t"))) {
stop("supplied <limits> argument unknown")
}
if (limits == "s<t") {
limits <- function(s, t) {
s < t
}
}
else {
if (limits == "s<=t") {
limits <- function(s, t) {
(s < t) | (s == t)
}
}
}
}
}else{
stop("<limits> argument cannot be NULL.")
}
## save the limits function in the arguments
args$limits <- limits
# ### use function limits to set up design matrix according to function limits
# ### by setting 0 at the time-points that should not be used
# if(args$format == "wide"){
# ## expand yind by replication to the yind of all observations together
# ind0 <- !t(outer( xind, rep(yind, each=nobs), limits) )
# yindHelp <- rep(yind, each=nobs)
# }else{
#### yind is over all observations in long format
ind0 <- !t(outer( xind, yind, limits) )
yindHelp <- yind
# }
### Compute the design matrix as sparse or normal matrix
### depending on dimensions of the final design matrix
MATRIX <- any(c(nrow(ind0), ncol(Bs)) > c(500, 50)) #MATRIX <- any(dim(X) > c(500, 50))
MATRIX <- MATRIX && options("mboost_useMatrix")$mboost_useMatrix
if(MATRIX){
#message("use sparse matrix in X_hist")
diag <- Diagonal
cbind <- cBind
###### <FIXME> construction of X1des directly as sparse matrix does not work
# ### compute the design matrix as sparse matrix
# if(args$format == "wide"){
# tempIndexDesign <- which(!ind0, arr.ind=TRUE)
# tempIndexX1 <- cbind(rep(1:nobs, length.out=nrow(tempIndexDesign)), tempIndexDesign[,2] )
# X1des <- sparseMatrix(i=tempIndexDesign[,1], j=tempIndexDesign[,2],
# x=X1[tempIndexX1], dims=dim(ind0))
# # object.size(X1des)
# rm(tempIndexX1, tempIndexDesign)
# }else{ # long format
# tempj <- unlist(apply(!ind0, 1, which)) # in which columns are the values?
# ## i: row numbers: one row number per observation of response,
# # repeat the row number for each entry
# X1des <- sparseMatrix(i=rep(1:length(id), times=rowSums(!ind0)), j=tempj,
# x=X1[cbind(rep(id, t=rowSums(!ind0)), tempj)], dims=dim(ind0))
# # object.size(X1des)
# rm(tempj)
# }
# ###### <FIXME> instead: build the matrix as dense matrix and convert it into a sparse matrix
# if(args$format == "wide"){
# ### expand the design matrix for all observations (yind is equal for all observations!)
# ### the response is a vector (y1(t1), y2(t1), ... , yn(t1), yn(tG))
# X1des <- X1[rep(1:nobs, times=length(yind)), ]
# } else{ # yind is over all observations in long format
# }
X1des <- X1[id, ]
X1des[ind0] <- 0
X1des <- Matrix(X1des, sparse=TRUE) # convert into sparse matrix
## if(!is(X1, "sparseMatrix"))
# X1 <- Matrix(X1, sparse=TRUE) # convert into sparse matrix
# X1des <- X1[as.numeric(id), ]
# X1des[ind0] <- 0
}else{ # small matrices: do not use Matrix
# if(args$format == "wide"){
# ### expand the design matrix for all observations (yind is equal for all observations!)
# ### the response is a vector (y1(t1), y2(t1), ... , yn(t1), yn(tG))
# X1des <- X1[rep(1:nobs, times=length(yind)), ]
# } else{ # yind is over all observations in long format
X1des <- X1[id, ]
# }
X1des[ind0] <- 0
}
## set up matrix with adequate integration and standardization weights
## start with a matrix of integration weights
## case of "no" standardization
Lnew <- args$intFun(X1des, xind)
Lnew[ind0] <- 0
## Standardize with exact length of integration interval
## (1/t-t0) \int_{t0}^t f(s) ds
if(args$standard == "length"){
## use fundamental theorem of calculus
## \lim t->t0- (1/t-t0) \int_{t0}^t f(s) ds = f(t0)
## -> integration weight in s-direction should be 1
## integration weights in s-direction always sum exactly to 1,
## good for small number of observations!
args$vecStand <- rowSums(Lnew)
args$vecStand[args$vecStand==0] <- 1 ## cannnot divide 0/0, instead divide 0/1
Lnew <- Lnew * 1/args$vecStand
}
## use time of current observation for standardization
## (1/t) \int_{t0}^t f(s) ds
if(args$standard=="time"){
if(any(yindHelp <= 0)) stop("For standardization with time, time must be positive.")
## Lnew <- matrix(1, ncol=ncol(X1des), nrow=nrow(X1des))
## Lnew[ind0] <- 0
## use fundamental theorem of calculus
## \lim t->0+ (1/t) \int_0^t f(s) ds = f(0), if necessary
## (as previously X*L, use now X*(1/L) for cases with one single point)
yindHelp[yindHelp==0] <- L[1,1] # impossible!
# standFact <- 1/yindHelp
args$vecStand <- yindHelp
Lnew <- Lnew * 1/yindHelp
}
## print(round(Lnew, 2))
## print(rowSums(Lnew))
## print(args$vecStand)
# multiply design matrix with integration weights and standardization weights
X1des <- X1des * Lnew
# Design matrix is product of expanded X1 and basis expansion over xind
X1des <- X1des %*% Bs
## see Scheipl and Greven (2016) Identifiability in penalized function-on-function regression models
if(args$check.ident && args$inS == "smooth"){
K1 <- diff(diag(ncol(Bs)), differences = args$differences)
K1 <- crossprod(K1)
# use the limits function to compute check measures on corresponding subsets of x(s) and B_j
res_check <- check_ident(X1 = X1, L = L, Bs = Bs, K = K1, xname = xname,
penalty = args$penalty,
limits = args$limits,
yind = yind, id = id, # yind is always in long format
X1des = X1des, ind0 = ind0, xind = xind)
args$penalty <- res_check$penalty
args$logCondDs <- res_check$logCondDs
args$logCondDs_hist <- res_check$logCondDs_hist
args$overlapKe <- res_check$overlapKe
args$cumOverlapKe <- res_check$cumOverlapKe
args$maxK <- res_check$maxK
}
# wide: design matrix over index of response for one response
# long: design matrix over index of response (yind has long format!)
Bt <- switch(args$inTime,
# B-spline basis of specified degree
# "smooth" = bsplines(yind, knots=args$knots[[varnames]]$knots,
# boundary.knots=args$knots[[varnames]]$boundary.knots,
# degree=args$degree),
"smooth" = mboost_intern(yind, knots = args$knots[[varnames]]$knots,
boundary.knots = args$knots[[varnames]]$boundary.knots,
degree = args$degree,
fun = "bsplines"),
"linear" = matrix(c(rep(1, length(yind)), yind), ncol=2),
"constant"= matrix(c(rep(1, length(yind))), ncol=1))
# # stack design-matrix of response nobs times in wide format
# if(args$format == "wide"){
# Bt <- Bt[rep(1:length(yind), each=nobs), ]
# }
if(! mboost_intern(Bt, fun = "isMATRIX") ) Bt <- matrix(Bt, ncol=1)
# calculate row-tensor
# X <- (X1 %x% t(rep(1, ncol(X2))) ) * ( t(rep(1, ncol(X1))) %x% X2 )
dimnames(Bt) <- NULL # otherwise warning "dimnames [2] mismatch..."
X <- X1des[,rep(1:ncol(Bs), each=ncol(Bt))] * Bt[,rep(1:ncol(Bt), times=ncol(Bs))]
if(! mboost_intern(X, fun = "isMATRIX") ) X <- matrix(X, ncol=1)
colnames(X) <- paste0(xname, 1:ncol(X))
### Penalty matrix: product differences matrix for smooth effect
if(args$inS == "smooth"){
K1 <- diff(diag(ncol(Bs)), differences = args$differences)
K1 <- crossprod(K1)
if(args$penalty == "pss"){
# <FIXME> allow for variable shrinkage parameter in penalty_pss()?
K1 <- penalty_pss(K = K1, difference = args$difference, shrink = 0.1)
}
}else{ # Ridge-penalty
K1 <- diag(ncol(Bs))
}
#K1 <- matrix(0, ncol=ncol(Bs), nrow=ncol(Bs))
#print(args$penalty)
if(args$inTime == "smooth"){
K2 <- diff(diag(ncol(Bt)), differences = args$differences)
K2 <- crossprod(K2)
}else{
K2 <- diag(ncol(Bt))
}
# compute penalty matrix for the whole effect
suppressMessages(K <- kronecker(K1, diag(ncol(Bt))) +
kronecker(diag(ncol(Bs)), K2))
## compare specified degrees of freedom to dimension of null space
if (!is.null(args$df)){
rns <- ncol(K) - qr(as.matrix(K))$rank # compute rank of null space
if (rns == args$df)
warning( sQuote("df"), " equal to rank of null space ",
"(unpenalized part of P-spline);\n ",
"Consider larger value for ", sQuote("df"),
" or set ", sQuote("center = TRUE"), ".", immediate.=TRUE)
if (rns > args$df)
stop("not possible to specify ", sQuote("df"),
" smaller than the rank of the null space\n ",
"(unpenalized part of P-spline). Use larger value for ",
sQuote("df"), " or set ", sQuote("center = TRUE"), ".")
}
# save matrices to compute numbers for identifiability checks
args$Bs <- Bs
args$X1des <- X1des
args$K1 <- K1
args$L <- L
# tidy up workspace
rm(Bs, Bt, ind0, X1des, X1, L)
return(list(X = X, K = K, args = args))
}
###############################################################################
#' Base-learners for Functional Covariates
#'
#' Base-learners that fit historical functional effects that can be used with the
#' tensor product, as e.g., hbistx(...) \%X\% bolsc(...).
#' For expert use only! May show unexpected behavior
#' compared to other base-learners for functional data!
#'
#' @param x object of type \code{hmatrix} containing time, index and functional covariate;
#' note that \code{timeLab} in the \code{hmatrix}-object must be equal to
#' the name of the time-variable in \code{timeformula} in the \code{FDboost}-call
#' @param knots either the number of knots or a vector of the positions
#' of the interior knots (for more details see \code{\link[mboost]{bbs})}.
#' @param 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.
#' @param degree degree of the regression spline.
#' @param differences a non-negative integer, typically 1, 2 or 3. Defaults to 1.
#' If \code{differences} = \emph{k}, \emph{k}-th-order differences are used as
#' a penalty (\emph{0}-th order differences specify a ridge penalty).
#' @param df trace of the hat matrix for the base-learner defining the
#' base-learner complexity. Low values of \code{df} correspond to a
#' large amount of smoothing and thus to "weaker" base-learners.
#' @param lambda smoothing parameter of the penalty, computed from \code{df} when \code{df} is specified.
#' @param penalty by default, \code{penalty="ps"}, the difference penalty for P-splines is used,
#' for \code{penalty="pss"} the penalty matrix is transformed to have full rank,
#' so called shrinkage approach by Marra and Wood (2011)
#' @param check.ident use checks for identifiability of the effect, based on Scheipl and Greven (2016);
#' see Brockhaus et al. (2016) for identifiability checks that take into account the integration limits
#' @param standard the historical effect can be standardized with a factor.
#' "no" means no standardization, "time" standardizes with the current value of time and
#' "lenght" standardizes with the lenght of the integral
#' @param intFun specify the function that is used to compute integration weights in \code{s}
#' over the functional covariate \eqn{x(s)}
#' @param inS historical effect can be smooth, linear or constant in s,
#' which is the index of the functional covariates x(s).
#' @param inTime historical effect can be smooth, linear or constant in time,
#' which is the index of the functional response y(time).
#' @param limits defaults to \code{"s<=t"} for an historical effect with s<=t;
#' either one of \code{"s<t"} or \code{"s<=t"} for [l(t), u(t)] = [T1, t];
#' otherwise specify limits as a function for integration limits [l(t), u(t)]:
#' function that takes \eqn{s} as the first and \code{t} as the second argument and returns
#' \code{TRUE} for combinations of values (s,t) if \eqn{s} falls into the integration range for
#' the given \eqn{t}.
#'
#' @details \code{bhistx} implements a base-learner for functional covariates with
#' flexible integration limits \code{l(t)}, \code{r(t)} and the possibility to
#' standardize the effect by \code{1/t} or the length of the integration interval.
#' The effect is \code{stand * int_{l(t)}^{r_{t}} x(s)beta(t,s) ds}.
#' The base-learner defaults to a historical effect of the form
#' \eqn{\int_{T1}^{t} x_i(s)beta(t,s) ds},
#' where \eqn{T1} is the minimal index of \eqn{t} of the response \eqn{Y(t)}.
#' \code{bhistx} can only be used if \eqn{Y(t)} and \eqn{x(s)} are observd over
#' the same domain \eqn{s,t \in [T1, T2]}.
#'
#' Note that the data has to be supplied as a \code{hmatrix} object for
#' model fit and predictions.
#'
#' @return Equally to the base-learners of package mboost:
#'
#' An object of class \code{blg} (base-learner generator) with a
#' \code{dpp} function (dpp, data pre-processing).
#'
#' The call of \code{dpp} returns an object of class
#' \code{bl} (base-learner) with a \code{fit} function. The call to
#' \code{fit} finally returns an object of class \code{bm} (base-model).
#'
#' @seealso \code{\link{FDboost}} for the model fit.
#' @keywords models
#'
#' @references
#' Brockhaus, S., Melcher, M., Leisch, F. and Greven, S. (2016):
#' Boosting flexible functional regression models with a high number of functional historical effects,
#' Statistics and Computing, accepted.
#'
#' Marra, G. and Wood, S.N. (2011): Practical variable selection for generalized additive models.
#' Computational Statistics & Data Analysis, 55, 2372-2387.
#'
#' Scheipl, F., Staicu, A.-M. and Greven, S. (2015):
#' Functional Additive Mixed Models, Journal of Computational and Graphical Statistics, 24(2), 477-501.
#' \url{http://arxiv.org/abs/1207.5947}
#'
#' Scheipl, F. and Greven, S. (2016): Identifiability in penalized function-on-function regression models.
#' Electronic Journal of Statistics, 10(1), 495-526.
#'
#' @examples
#' if(require(refund)){
#' ## simulate some data from a historical model
#' ## the interaction effect is in this case not necessary
#' n <- 100
#' nygrid <- 35
#' data1 <- pffrSim(scenario = c("int", "ff"), limits = function(s,t){ s <= t },
#' n = n, nygrid = nygrid)
#' data1$X1 <- scale(data1$X1, scale = FALSE) ## center functional covariate
#' dataList <- as.list(data1)
#' dataList$tvals <- attr(data1, "yindex")
#'
#' ## create the hmatrix-object
#' X1h <- with(dataList, hmatrix(time = rep(tvals, each = n), id = rep(1:n, nygrid),
#' x = X1, argvals = attr(data1, "xindex"),
#' timeLab = "tvals", idLab = "wideIndex",
#' xLab = "myX", argvalsLab = "svals"))
#' dataList$X1h <- I(X1h)
#' dataList$svals <- attr(data1, "xindex")
#' ## add a factor variable
#' dataList$zlong <- factor(gl(n = 2, k = n/2, length = n*nygrid), levels = 1:3)
#' dataList$z <- factor(gl(n = 2, k = n/2, length = n), levels = 1:3)
#'
#' ## do the model fit with main effect of bhistx() and interaction of bhistx() and bolsc()
#' mod <- FDboost(Y ~ 1 + bhistx(x = X1h, df = 5, knots = 5) +
#' bhistx(x = X1h, df = 5, knots = 5) %X% bolsc(zlong),
#' timeformula = ~ bbs(tvals, knots = 10), data = dataList)
#'
#' ## alternative parameterization: interaction of bhistx() and bols()
#' mod <- FDboost(Y ~ 1 + bhistx(x = X1h, df = 5, knots = 5) %X% bols(zlong),
#' timeformula = ~ bbs(tvals, knots = 10), data = dataList)
#'
#' \dontrun{
#' # find the optimal mstop over 5-fold bootstrap (small example to reduce run time)
#' cv <- cvrisk(mod, folds = cv(model.weights(mod), B = 5))
#' mstop(cv)
#' mod[mstop(cv)]
#'
#' appl1 <- applyFolds(mod, folds = cv(rep(1, length(unique(mod$id))), type = "bootstrap", B = 5))
#'
#' # plot(mod)
#' }
#'}
#'
#' @export
### P-spline base-learner for effect with time-specific integration limits using a hmatrix-object
bhistx <- function(x,
limits = "s<=t", standard = c("no", "time", "length"),
intFun = integrationWeightsLeft,
inS = c("smooth","linear","constant"), inTime = c("smooth", "linear", "constant"),
knots = 10, boundary.knots = NULL, degree = 3, differences = 1, df = 4,
lambda = NULL, #center = FALSE, cyclic = FALSE
penalty = c("ps", "pss"), check.ident = FALSE
){
index <- NULL # index is treated within hmatrix
if (!is.null(lambda)) df <- NULL
cll <- match.call()
cll[[1]] <- as.name("bhistx")
#print(cll)
penalty <- match.arg(penalty)
#print(penalty)
standard <- match.arg(standard)
inS <- match.arg(inS)
inTime <- match.arg(inTime)
#print(inS)
if(!is.hmatrix(x)) stop("x has to be a hmatrix-object.")
varnames <- all.vars(cll)[1] # only the first name, the name of x is a variable name
# Reshape mfL so that it is the dataframe of the signal with
# the index of the signal and the index of the response as attributes
xname <- getXLab(x)
indname <- getArgvalsLab(x)
indnameY <- getTimeLab(x)
## save the hmatrix-object into a data.frame
mf <- data.frame(z=I(x))
names(mf) <- varnames
if(!all( abs(colMeans(getX(x), na.rm = TRUE)) < .Machine$double.eps*10^10)){
message(xname, " is not centered per column, inducing a non-centered effect.")
}
vary <- ""
# CC <- all(Complete.cases(mf))
CC <- all(mboost_intern(mf, fun = "Complete.cases"))
if (!CC)
warning("base-learner contains missing values;\n",
"missing values are excluded per base-learner, ",
"i.e., base-learners may depend on different",
" numbers of observations.")
## call X_histx in order to compute parameter settings, e.g.
## the transformation matrix Z, shrinkage penalty, identifiability problems...
# X_histx for data in long format, as hmatrix is always in long-format
temp <- X_histx(mf, vary,
args = hyper_histx(mf, vary, knots = knots, boundary.knots = boundary.knots,
degree = degree, differences = differences,
df = df, lambda = lambda, center = FALSE, cyclic = FALSE,
limits = limits,
standard = standard, intFun = intFun,
inS = inS, inTime = inTime,
penalty = penalty, check.ident = check.ident,
format="long"))
temp$args$check.ident <- FALSE
ret <- list(model.frame = function() mf,
get_call = function(){
cll <- deparse(cll, width.cutoff=500L)
if (length(cll) > 1)
cll <- paste(cll, collapse="")
cll
},
get_data = function() mf,
## set index to NULL, as the index is treated within X_hist()
## get_index = function() index,
get_index = function() NULL,
get_vary = function() vary,
get_names = function(){
varnames
}, #colnames(mf),
set_names = function(value) {
#if(length(value) != length(colnames(mf)))
if(length(value) != names(mf[1]))
stop(sQuote("value"), " must have same length as ",
sQuote("names(mf[1])"))
for (i in 1:length(value)){
cll[[i+1]] <<- as.name(value[i])
}
attr(mf, "names") <<- value
})
class(ret) <- "blg"
### call fitter with X_histx
# ret$dpp <- bl_lin(ret, Xfun = X_histx, args = temp$args)
ret$dpp <- mboost_intern(ret, Xfun = X_histx, args = temp$args, fun = "bl_lin")
return(ret)
}
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