R/flm.test.R
In moviedo5/fda.usc: Functional Data Analysis and Utilities for Statistical Computing
Defines functions
flm.test
Documented in
flm.test
#' @aliases flm.test
#'
#' @title Goodness-of-fit test for the Functional Linear Model with scalar response
#'
#' @description The function \code{flm.test} tests the composite null hypothesis of
#' a Functional Linear Model with scalar response (FLM),
#' \deqn{H_0:\,Y=\big<X,\beta\big>+\epsilon,}{H_0: Y=<X,\beta>+\epsilon,} versus
#' a general alternative. If \eqn{\beta=\beta_0}{\beta=\beta_0} is provided, then the
#' simple hypothesis \eqn{H_0:\,Y=\big<X,\beta_0\big>+\epsilon}{H_0: Y=<X,\beta_0>+\epsilon} is tested.
#' The testing of the null hypothesis is done by a Projected Cramer-von Mises statistic (see Details).
#'
#' @param X.fdata Functional covariate for the FLM. The object must be in the class
#' \code{\link{fdata}}.
#' @param Y Scalar response for the FLM. Must be a vector with the same number of elements
#' as functions are in \code{X.fdata}.
#' @param beta0.fdata Functional parameter for the simple null hypothesis, in the \code{\link{fdata}} class.
#' Recall that the \code{argvals} and \code{rangeval} arguments of \code{beta0.fdata} must be the same
#' of \code{X.fdata}. A possibility to do this is to consider, for example for \eqn{\beta_0=0}{\beta_0=0}
#' (the simple null hypothesis of no interaction),\if{latex}{\cr}
#' \code{beta0.fdata=fdata(mdata=rep(0,length(X.fdata$argvals)),}\if{latex}{\cr}\code{argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)}.\if{latex}{\cr}
#' If \code{beta0.fdata=NULL} (default), the function will test for the composite null hypothesis.
#'
#' @param B Number of bootstrap replicates to calibrate the distribution of the test statistic.
#' \code{B=5000} replicates are the recommended for carry out the test, although for exploratory analysis
#' (\bold{not inferential}), an acceptable less time-consuming option is \code{B=500}.
#' @param est.method Estimation method for the unknown parameter \eqn{\beta}{\beta},
#' only used in the composite case. Mainly, there are two options: specify the number of basis
#' elements for the estimated \eqn{\beta}{\beta} by \code{p} or optimally select \code{p} by a
#' data-driven criteria (see Details section for discussion). Then, it must be one of the following
#' methods:
#' \itemize{
#' \item \code{"pc"}: If \code{p}, the number of basis elements, is given, then \eqn{\beta}{\beta} is estimated by \code{\link{fregre.pc}}. Otherwise, an optimum \code{p} is chosen using \code{\link{fregre.pc.cv}} and the \code{"SICc"} criteria.
#' \item \code{"pls"}: If \code{p} is given, \eqn{\beta}{\beta} is estimated by \code{\link{fregre.pls}}. Otherwise, an optimum \code{p} is chosen using \code{\link{fregre.pls.cv}} and the \code{"SICc"} criteria.
#' This is the default argument as it has been checked empirically that provides a good balance between the performance of the test and the estimation of \eqn{\beta}{\beta}.
#' \item \code{"basis"}: If \code{p} is given, \eqn{\beta}{\beta} is estimated by \code{\link{fregre.basis}}. Otherwise, an optimum \code{p} is chosen using \code{\link{fregre.basis.cv}} and the \code{"GCV.S"} criteria. In these functions, the same basis for the arguments \code{basis.x} and \code{basis.b} is considered.
#' The type of basis used will be the given by the argument \code{type.basis} and must be one of the class of \code{create.basis}. Further arguments passed to \code{\link{create.basis}} (not \code{rangeval} that is taken as the \code{rangeval} of \code{X.fdata}), can be passed throughout \code{\dots}.
#' }
#' @param p Number of elements of the basis considered. If it is not given, an optimal \code{p} will be chosen using a specific criteria (see \code{est.method} and \code{type.basis} arguments).
#' @param type.basis Type of basis used to represent the functional process. Depending on the hypothesis, it will have a different interpretation:
#' \itemize{
#' \item Simple hypothesis. One of these options:
#' \itemize{
#' \item \code{"bspline"}: If \code{p} is given, the functional process is expressed in a basis of \code{p} B-splines. If not, an optimal \code{p} will be chosen by \code{\link{optim.basis}}, using the \code{"GCV.S"} criteria.
#' \item \code{"fourier"}: If \code{p} is given, the functional process is expressed in a basis of \code{p} Fourier functions. If not, an optimal \code{p} will be chosen by \code{\link{optim.basis}}, using the \code{"GCV.S"} criteria.
#' \item \code{"pc"}: \code{p} must be given. Expresses the functional process in a basis of \code{p} principal components.
#' \item \code{"pls"}: \code{p} must be given. Expresses the functional process in a basis of \code{p} partial least squares.
#' }
#' Although other basis types supported by \code{\link{create.basis}} are possible, \code{"bspline"} and \code{"fourier"} are recommended. Other basis types may cause incompatibilities.
#' \item Composite hypothesis. This argument is only used when \if{latex}{\cr}\code{est.method="basis"} and, in this case, it specifies the type of basis used in the basis estimation method of the functional parameter. Again, basis
#' \code{"bspline"} and \code{"fourier"} are recommended, as other basis types may cause incompatibilities.
#' }
#'
#' @param verbose Either to show or not information about computing progress.
#' @param plot.it Either to show or not a graph of the observed trajectory,
#' and the bootstrap trajectories under the null composite hypothesis, of the
#' process \eqn{R_n(\cdot)}{R_n(.)} (see Details). Note that if \code{plot.it=TRUE},
#' the function takes more time to run.
#' @param B.plot Number of bootstrap trajectories to show in the resulting plot of the test.
#' As the trajectories shown are the first \code{B.plot} of \code{B}, \code{B.plot} must be
#' lower or equal to \code{B}.
#' @param G Number of projections used to compute the trajectories of the process
#' \eqn{R_n(\cdot)}{R_n(.)} by Monte Carlo.
#' @param \dots Further arguments passed to \code{\link{create.basis}}.
#'
#' @details The Functional Linear Model with scalar response (FLM), is defined as
#' \eqn{Y=\big<X,\beta\big>+\epsilon}{Y=<X,\beta>+\epsilon}, for a functional process
#' \eqn{X}{X} such that \eqn{E[X(t)]=0}{E[X(t)]=0}, \eqn{E[X(t)\epsilon]=0}{E[X(t)\epsilon]=0}
#' for all \eqn{t}{t} and for a scalar variable \eqn{Y}{Y} such that \eqn{E[Y]=0}{E[Y]=0}.
#' Then, the test assumes that \code{Y} and \code{X.fdata} are \bold{centred} and will automatically
#' center them. So, bear in mind that when you apply the test for \code{Y} and \code{X.fdata},
#' actually, you are applying it to \code{Y-mean(Y)} and \code{fdata.cen(X.fdata)$Xcen}.
#' The test statistic corresponds to the Cramer-von Mises norm of the \emph{Residual Marked
#' empirical Process based on Projections} \eqn{R_n(u,\gamma)}{R_n(u,\gamma)} defined in
#' Garcia-Portugues \emph{et al.} (2014).
#' The expression of this process in a \eqn{p}{p}-truncated basis of the space \eqn{L^2[0,T]}{L^2[0,T]}
#' leads to the \eqn{p}{p}-multivariate process \eqn{R_{n,p}\big(u,\gamma^{(p)}\big)}{R_{n,p}(u,\gamma^{(p)})},
#' whose Cramer-von Mises norm is computed.
#' The choice of an appropriate \eqn{p}{p} to represent the functional process \eqn{X}{X},
#' in case that is not provided, is done via the estimation of \eqn{\beta}{\beta} for the composite
#' hypothesis. For the simple hypothesis, as no estimation of \eqn{\beta}{\beta} is done, the choice
#' of \eqn{p}{p} depends only on the functional process \eqn{X}{X}. As the result of the test may
#' change for different \eqn{p}{p}'s, we recommend to use an automatic criterion to select \eqn{p}{p}
#' instead of provide a fixed one.
#' The distribution of the test statistic is approximated by a wild bootstrap resampling on the
#' residuals, using the \emph{golden section bootstrap}.
#' Finally, the graph shown if \code{plot.it=TRUE} represents the observed trajectory, and the
#' bootstrap trajectories under the null, of the process RMPP \emph{integrated on the projections}:
#' \deqn{R_n(u)\approx\frac{1}{G}\sum_{g=1}^G R_n(u,\gamma_g),}{R_n(u) \approx \frac{1}{G} \sum_{g=1}^G R_n(u,\gamma_g),}
#' where \eqn{\gamma_g}{\gamma_g} are simulated as Gaussians processes. This gives a graphical idea of
#' how \emph{distant} is the observed trajectory from the null hypothesis.
#'
#' @return An object with class \code{"htest"} whose underlying structure is a list containing
#' the following components:
#' \itemize{
#' \item \code{statistic}: The value of the test statistic.
#' \item \code{boot.statistics}: A vector of length \code{B} with the values of the bootstrap test statistics.
#' \item \code{p.value}: The p-value of the test.
#' \item \code{method}: The method used.
#' \item \code{B}: The number of bootstrap replicates used.
#' \item \code{type.basis}: The type of basis used.
#' \item \code{beta.est}: The estimated functional parameter \eqn{\beta}{\beta} in the composite
#' hypothesis. For the simple hypothesis, the given \code{beta0.fdata}.
#' \item \code{p}: The number of basis elements passed or automatically chosen.
#' \item \code{ord}: The optimal order for PC and PLS given by \code{\link{fregre.pc.cv}} and \if{latex}{\cr} \code{\link{fregre.pls.cv}}. For other methods, it is set to \code{1:p}.
#' \item \code{data.name}: The character string "Y=<X,b>+e".
#' }
#'
#' @references
#' Escanciano, J. C. (2006). A consistent diagnostic test for regression models using projections. Econometric Theory, 22, 1030-1051. \doi{10.1017/S0266466606060506}
#'
#' Garcia-Portugues, E., Gonzalez-Manteiga, W. and Febrero-Bande, M. (2014). A goodness--of--fit test for the functional linear model with scalar response. Journal of Computational and Graphical Statistics, 23(3), 761-778. \doi{10.1080/10618600.2013.812519}
#'
#' @note No NA's are allowed neither in the functional covariate nor in the scalar response.
#'
#' @author Eduardo Garcia-Portugues. Please, report bugs and suggestions to
#' \if{latex}{\cr}\email{edgarcia@@est-econ.uc3m.es}
#'
#' @seealso \code{\link{Adot}}, \code{\link{PCvM.statistic}}, \code{\link{rwild}},
#' \code{\link{flm.Ftest}}, \code{\link{dfv.test}},
#' \code{\link{fregre.pc}}, \code{\link{fregre.pls}},\if{latex}{\cr} \code{\link{fregre.basis}},
#' \code{\link{fregre.pc.cv}}, \code{\link{fregre.pls.cv}},
#' \code{\link{fregre.basis.cv}}, \code{\link{optim.basis}}, \if{latex}{\cr}
#' \link[fda]{create.basis}
#' @examples
#' # Simulated example #
#' X=rproc2fdata(n=100,t=seq(0,1,l=101),sigma="OU")
#' beta0=fdata(mdata=cos(2*pi*seq(0,1,l=101))-(seq(0,1,l=101)-0.5)^2+
#' rnorm(101,sd=0.05),argvals=seq(0,1,l=101),rangeval=c(0,1))
#' Y=inprod.fdata(X,beta0)+rnorm(100,sd=0.1)
#'
#' dev.new(width=21,height=7)
#' par(mfrow=c(1,3))
#' plot(X,main="X")
#' plot(beta0,main="beta0")
#' plot(density(Y),main="Density of Y",xlab="Y",ylab="Density")
#' rug(Y)
#'
#' \dontrun{
#' # Composite hypothesis: do not reject FLM
#' pcvm.sim=flm.test(X,Y,B=50,B.plot=50,G=100,plot.it=TRUE)
#' pcvm.sim
#' flm.test(X,Y,B=5000)
#'
#' # Estimated beta
#' dev.new()
#' plot(pcvm.sim$beta.est)
#'
#' # Simple hypothesis: do not reject beta=beta0
#' flm.test(X,Y,beta0.fdata=beta0,B=50,B.plot=50,G=100)
#' flm.test(X,Y,beta0.fdata=beta0,B=5000)
#'
#' # AEMET dataset #
#' data(aemet)
#' # Remove the 5\% of the curves with less depth (i.e. 4 curves)
#' dev.new()
#' res.FM=depth.FM(aemet$temp,draw=TRUE)
#' qu=quantile(res.FM$dep,prob=0.05)
#' l=which(res.FM$dep<=qu)
#' lines(aemet$temp[l],col=3)
#' aemet$df$name[l]
#'
#' # Data without outliers
#' wind.speed=apply(aemet$wind.speed$data,1,mean)[-l]
#' temp=aemet$temp[-l]
#' # Exploratory analysis: accept the FLM
#' pcvm.aemet=flm.test(temp,wind.speed,est.method="pls",B=100,B.plot=50,G=100)
#' pcvm.aemet
#'
#' # Estimated beta
#' dev.new()
#' plot(pcvm.aemet$beta.est,lwd=2,col=2)
#' # B=5000 for more precision on calibration of the test: also accept the FLM
#' flm.test(temp,wind.speed,est.method="pls",B=5000)
#'
#' # Simple hypothesis: rejection of beta0=0? Limiting p-value...
#' dat=rep(0,length(temp$argvals))
#' flm.test(temp,wind.speed, beta0.fdata=fdata(mdata=dat,argvals=temp$argvals,
#' rangeval=temp$rangeval),B=100)
#' flm.test(temp,wind.speed, beta0.fdata=fdata(mdata=dat,argvals=temp$argvals,
#' rangeval=temp$rangeval),B=5000)
#'
#' # Tecator dataset #
#' data(tecator)
#' names(tecator)
#' absorp=tecator$absorp.fdata
#' ind=1:129 # or ind=1:215
#' x=absorp[ind,]
#' y=tecator$y$Fat[ind]
#' tt=absorp[["argvals"]]
#'
#' # Exploratory analysis for composite hypothesis with automatic choose of p
#' pcvm.tecat=flm.test(x,y,B=100,B.plot=50,G=100)
#' pcvm.tecat
#'
#' # B=5000 for more precision on calibration of the test: also reject the FLM
#' flm.test(x,y,B=5000)
#'
#' # Distribution of the PCvM statistic
#' plot(density(pcvm.tecat$boot.statistics),lwd=2,xlim=c(0,10),
#' main="PCvM distribution", xlab="PCvM*",ylab="Density")
#' rug(pcvm.tecat$boot.statistics)
#' abline(v=pcvm.tecat$statistic,col=2,lwd=2)
#' legend("top",legend=c("PCvM observed"),lwd=2,col=2)
#'
#' # Simple hypothesis: fixed p
#' dat=rep(0,length(x$argvals))
#' flm.test(x,y,beta0.fdata=fdata(mdata=dat,argvals=x$argvals,
#' rangeval=x$rangeval),B=100,p=11)
#'
#' # Simple hypothesis, automatic choose of p
#' flm.test(x,y,beta0.fdata=fdata(mdata=dat,argvals=x$argvals,
#' rangeval=x$rangeval),B=100)
#' flm.test(x,y,beta0.fdata=fdata(mdata=dat,argvals=x$argvals,
#' rangeval=x$rangeval),B=5000)
#' }
#' @keywords htest models regression
# PCvM test for the composite hypothesis with bootstrap calibration
#' @rdname flm.test
#' @export
flm.test=function(X.fdata,Y,beta0.fdata=NULL,B=5000,est.method="pls",
p=NULL,type.basis="bspline",verbose=TRUE,plot.it=TRUE,
B.plot=100,G=200,...){
# Check B.plot
if(plot.it & B.plot>B) stop("B.plot must be less or equal than B")
# Number of functions
n=dim(X.fdata)[1]
if(verbose) cat("Computing estimation of beta... ")
## COMPOSITE HYPOTHESIS ##
if(is.null(beta0.fdata)){
# Center the data first
X.fdata=fdata.cen(X.fdata)$Xcen
Y=Y-mean(Y)
## 1. Optimal estimation of beta and the basis order ##
if(est.method=="pc"){
if(is.null(p)){
# Method
meth="PCvM test for the functional linear model using optimal PC basis representation"
# Choose the number of basis elements: SICc is probably the best criteria
mod.pc=fregre.pc.cv(fdataobj=X.fdata,y=Y,kmax=1:10,criteria="SICc")
p.opt=length(mod.pc$pc.opt)
ord.opt=mod.pc$pc.opt
# PC components to be passed to the bootstrap
pc.comp=mod.pc$fregre.pc$fdata.comp # pc.comp=mod.pc$fregre.pc$pc
pc.comp$l=mod.pc$pc.opt
# Express X.fdata and beta.est in the PC basis
basis.pc=mod.pc$fregre.pc$fdata.comp$basis
if(length(pc.comp$l)!=1){
X.est=fdata(mdata=mod.pc$fregre.pc$fdata.comp$coefs[,mod.pc$fregre.pc$l]%*%mod.pc$fregre.pc$fdata.comp$basis$data[mod.pc$fregre.pc$l,],argvals=X.fdata$argvals,rangeval=X.fdata$rangeval) # X.est=fdata(mdata=mod.pc$fregre.pc$pc$coefs[,mod.pc$fregre.pc$l]%*%mod.pc$fregre.pc$pc$basis$data[mod.pc$fregre.pc$l,],argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)
}else{
X.est=fdata(mdata=mod.pc$fregre.pc$fdata.comp$coefs[,mod.pc$fregre.pc$l]%*%t(mod.pc$fregre.pc$fdata.comp$basis$data[mod.pc$fregre.pc$l,]),argvals=X.fdata$argvals,rangeval=X.fdata$rangeval) # X.est=fdata(mdata=mod.pc$fregre.pc$pc$coefs[,mod.pc$fregre.pc$l]%*%t(mod.pc$fregre.pc$pc$basis$data[mod.pc$fregre.pc$l,]),argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)
}
beta.est=mod.pc$fregre.pc$beta.est
norm.beta.est=norm.fdata(beta.est)
# Compute the residuals
e=mod.pc$fregre.pc$residuals
}else{
# Method
meth=paste("PCvM test for the functional linear model using a representation in a PC basis of ",p,"elements")
# Estimation of beta on the given fixed basis
mod.pc=fregre.pc(fdataobj=X.fdata,y=Y,l=1:p)
p.opt=p
ord.opt=mod.pc$l
# PC components to be passed to the bootstrap
pc.comp=mod.pc$pc
pc.comp$l=mod.pc$l
# Express X.fdata and beta.est in the basis
if(p!=1){
X.est=fdata(mdata=mod.pc$fdata.comp$coefs[,mod.pc$l]%*%mod.pc$fdata.comp$basis$data[mod.pc$l,],argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)
}else{
X.est=fdata(mdata=mod.pc$fdata.comp$coefs[,mod.pc$l]%*%t(mod.pc$fdata.comp$basis$data[mod.pc$l,]),argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)
}
beta.est=mod.pc$beta.est
norm.beta.est=norm.fdata(beta.est)
# Compute the residuals
e=mod.pc$residuals
}
}else if(est.method=="pls"){
if(is.null(p)){
# print("PLS1")
# Method
meth="PCvM test for the functional linear model using optimal PLS basis representation"
# Choose the number of the basis: SICc is probably the best criteria
mod.pls=fregre.pls.cv(fdataobj=X.fdata,y=Y,kmax=10,criteria="SICc")
# print("PLS2")
p.opt=length(mod.pls$pls.opt)
ord.opt=mod.pls$pls.opt
# PLS components to be passed to the bootstrap
pls.comp=mod.pls$fregre.pls$fdata.comp
pls.comp$l=mod.pls$pls.opt
# Express X.fdata and beta.est in the PLS basis
basis.pls=mod.pls$fregre.pls$fdata.comp$basis
if(length(pls.comp$l)!=1){
X.est=fdata(mdata=mod.pls$fregre.pls$fdata.comp$coefs[,mod.pls$fregre.pls$l]%*%mod.pls$fregre.pls$fdata.comp$basis$data[mod.pls$fregre.pls$l,],argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)
}else{
X.est=fdata(mdata=mod.pls$fregre.pls$fdata.comp$coefs[,mod.pls$fregre.pls$l]%*%t(mod.pls$fregre.pls$fdata.comp$basis$data[mod.pls$fregre.pls$l,]),argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)
}
beta.est=mod.pls$fregre.pls$beta.est
norm.beta.est=norm.fdata(beta.est)
# Compute the residuals
e=mod.pls$fregre.pls$residuals
}else{
# Method
meth=paste("PCvM test for the functional linear model using a representation in a PLS basis of ",p,"elements")
# Estimation of beta on the given fixed basis
mod.pls=fregre.pc(fdataobj=X.fdata,y=Y,l=1:p)
p.opt=p
ord.opt=mod.pls$l
# PLS components to be passed to the bootstrap
pls.comp=mod.pls$fdata.comp
pls.comp$l=mod.pls$l
# Express X.fdata and beta.est in the basis
if(p!=1){
X.est=fdata(mdata=mod.pls$fdata.comp$coefs[,mod.pls$l]%*%mod.pls$fdata.comp$basis$data[mod.pls$l,],argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)
}else{
X.est=fdata(mdata=mod.pls$fdata.comp$coefs[,mod.pls$l]%*%t(mod.pls$fdata.comp$basis$data[mod.pls$l,]),argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)
}
beta.est=mod.pls$beta.est
norm.beta.est=norm.fdata(beta.est)
# Compute the residuals
e=mod.pls$residuals
}
}else if(est.method=="basis"){
if(is.null(p)){
# Method
meth=paste("PCvM test for the functional linear model using optimal",type.basis,"basis representation")
# Choose the number of the bspline basis with GCV.S
mod.basis=fregre.basis.cv(fdataobj=X.fdata,y=Y,basis.x=seq(5,30,by=1),basis.b=NULL,type.basis=type.basis,type.CV=GCV.S,verbose=FALSE,...)
p.opt=mod.basis$basis.x.opt$nbasis
ord.opt=1:p.opt
# Express X.fdata and beta.est in the optimal basis
basis.opt=mod.basis$basis.x.opt
X.est=mod.basis$x.fd
beta.est=mod.basis$beta.est
norm.beta.est=norm.fd(beta.est)
# Compute the residuals
e=mod.basis$residuals
}else{
# Method
meth=paste("PCvM test for the functional linear model using a representation in a",type.basis,"basis of ",p,"elements")
# Estimation of beta on the given fixed basis
basis.opt=do.call(what=paste("create.",type.basis,".basis",sep=""),args=list(rangeval=X.fdata$rangeval,nbasis=p,...))
mod.basis=fregre.basis(fdataobj=X.fdata,y=Y,basis.x=basis.opt,basis.b=basis.opt)
p.opt=p
ord.opt=1:p.opt
# Express X.fdata and beta.est in the basis
X.est=mod.basis$x.fd
beta.est=mod.basis$beta.est
norm.beta.est=norm.fd(beta.est)
# Compute the residuals
e=mod.basis$residuals
}
}else{
stop(paste("Estimation method",est.method,"not implemented."))
}
## SIMPLE HYPOTHESIS ##
}else{
## 1. Optimal representation of X and beta0 ##
# Choose the number of basis elements
if(type.basis!="pc" & type.basis!="pls"){
# Basis method: select the number of elements of the basis is it is not given
if(is.null(p)){
# Method
meth=paste("PCvM test for the simple hypothesis in a functional linear model, using a representation in an optimal ",type.basis,"basis")
p.opt=optim.basis(X.fdata,type.basis=type.basis,numbasis=seq(31,71,by=2),verbose=FALSE)$numbasis.opt
if(p.opt==31){
cat("Readapting interval...\n")
p.opt=optim.basis(X.fdata,type.basis=type.basis,numbasis=seq(5,31,by=2),verbose=FALSE)$numbasis.opt
}else if(p.opt==71){
cat("Readapting interval...\n")
p.opt=optim.basis(X.fdata,type.basis=type.basis,numbasis=seq(71,101,by=2),verbose=FALSE)$numbasis.opt
}
ord.opt=1:p.opt
}else{
# Method
meth=paste("PCvM test for the simple hypothesis in a functional linear model, using a representation in a ",type.basis," basis of ",p,"elements")
p.opt=p
ord.opt=1:p.opt
}
# Express X.fdata in the basis
X.est=fdata2fd(X.fdata,type.basis=type.basis,nbasis=p)
beta.est=fdata2fd(beta0.fdata,type.basis=type.basis,nbasis=p)
# Compute the residuals
e=Y-inprod(X.est,beta.est)
}else if (type.basis=="pc"){
if(is.null(p)) stop("Simple hypothesis with type.basis=\"pc\" need the number of components p")
# Method
meth=paste("PCvM test for the simple hypothesis in a functional linear model, using a representation in a PC basis of ",p,"elements")
# Express X.fdata in a PC basis
fd2pc=fdata2pc(X.fdata,ncomp=p)
if(length(fd2pc$l)!=1){
X.est=fdata(mdata=fd2pc$coefs[,fd2pc$l]%*%fd2pc$basis$data[fd2pc$l,],argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)
}else{
X.est=fdata(mdata=fd2pc$coefs[,fd2pc$l]%*%t(fd2pc$basis$data[fd2pc$l,]),argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)
}
beta.est=beta0.fdata
p.opt=p
ord.opt=1:p.opt
# Compute the residuals
e=Y-inprod.fdata(X.est,beta.est)
}else if(type.basis=="pls"){
if(is.null(p)) stop("Simple hypothesis with type.basis=\"pls\" need the number of components p")
# Method
meth=paste("PCvM test for the simple hypothesis in a functional linear model, using a representation in a PLS basis of ",p,"elements")
# Express X.fdata in a PLS basis
fd2pls=fdata2pls(X.fdata,Y,ncomp=p)
if(length(fd2pls$l)!=1){
X.est=fdata(mdata=fd2pls$coefs[,fd2pls$l]%*%fd2pls$basis$data[fd2pls$l,],argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)
}else{
X.est=fdata(mdata=fd2pls$coefs[,fd2pls$l]%*%t(fd2pls$basis$data[fd2pls$l,]),argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)
}
beta.est=beta0.fdata
p.opt=p
ord.opt=1:p.opt
# Compute the residuals
e=Y-inprod.fdata(X.est,beta.est)
}else{
stop(paste("Type of basis method",type.basis,"not implemented."))
}
}
## 2. Bootstrap calibration ##
# Start up
pcvm.star=numeric(B)
e.hat.star=matrix(ncol=n,nrow=B)
# Calculus of the Adot.vec
Adot.vec=Adot(X.est)
# REAL WORLD
pcvm=PCvM.statistic(X=X.est,residuals=e,p=p.opt,Adot.vec=Adot.vec)
# BOOTSTRAP WORLD
if(verbose) cat("Done.\nBootstrap calibration...\n ")
if(verbose) pb=txtProgressBar(style=3)
## COMPOSITE HYPOTHESIS ##
if(is.null(beta0.fdata)){
# Calculate the design matrix of the linear model
# This allows to resample efficiently the residuals without estimating again the beta
if(est.method=="pc"){
if(is.null(p)){
# Design matrix for the PC estimation
X.matrix=mod.pc$fregre.pc$lm$x
}else{
# Design matrix for the PC estimation
X.matrix=mod.pc$lm$x
}
}else if(est.method=="pls"){
if(is.null(p)){
# Design matrix for the PLS estimation
X.matrix=mod.pls$fregre.pls$lm$x
}else{
# Design matrix for the PLS estimation
X.matrix=mod.pls$lm$x
}
}else if(est.method=="basis"){
if(is.null(p)){
# Design matrix for the basis estimation
X.matrix=mod.basis$lm$x
}else{
# Design matrix for the basis estimation
X.matrix=mod.basis$lm$x
}
}
# Projection matrix
P=(diag(rep(1,n))-X.matrix%*%solve(t(X.matrix)%*%X.matrix)%*%t(X.matrix))
# Bootstrap resampling
for(i in 1:B){
# Generate bootstrap errors
e.hat=rwild(e,"golden")
# Calculate Y.star
Y.star=Y-e+e.hat
# Residuals from the bootstrap estimated model
e.hat.star[i,]=P%*%Y.star
# Calculate PCVM.star
pcvm.star[i]=PCvM.statistic(X=X.est,residuals=e.hat.star[i,],p=p.opt,Adot.vec=Adot.vec)
if(verbose) setTxtProgressBar(pb,i/B)
}
## SIMPLE HYPOTHEIS ##
}else{
# Bootstrap resampling
for(i in 1:B){
# Generate bootstrap errors
e.hat.star[i,]=rwild(e,"golden")
# Calculate PCVM.star
pcvm.star[i]=PCvM.statistic(X=X.est,residuals=e.hat.star[i,],p=p.opt,Adot.vec=Adot.vec)
if(verbose) setTxtProgressBar(pb,i/B)
}
}
## 3. MC estimation of the p-value and order the result ##
# Compute the p-value
pvalue=sum(pcvm.star>pcvm)/B
## 4. Graphical representation of the integrated process ##
if(verbose) cat("\nDone.\nComputing graphical representation... ")
if(is.null(beta0.fdata) & plot.it){
gamma=rproc2fdata(n=G,t=X.fdata$argvals,sigma="OU",par.list=list(theta=2/diff(range(X.fdata$argvals))))
gamma=gamma/drop(norm.fdata(gamma))
ind=drop(inprod.fdata(X.fdata,gamma))
r=0.9*max(max(ind),-min(ind))
u=seq(-r,r,l=200)
mean.proc=numeric(length(u))
mean.boot.proc=matrix(ncol=length(u),nrow=B.plot)
res=apply(ind,2,function(iind){
iind.sort=sort(iind,index.return=TRUE)
stepfun(x=iind.sort$x,y=c(0,cumsum(e[iind.sort$ix])))(u)/sqrt(n)
}
)
mean.proc=apply(res,1,mean)
for(i in 1:B.plot){
res=apply(ind,2,function(iind){
iind.sort=sort(iind,index.return=TRUE)
stepfun(x=iind.sort$x,y=c(0,cumsum(e.hat.star[i,iind.sort$ix])))(u)/sqrt(n)
}
)
mean.boot.proc[i,]=apply(res,1,mean)
}
# Plot
dev.new()
plot(u,mean.proc,ylim=c(min(mean.proc,mean.boot.proc),max(mean.proc,mean.boot.proc))*1.05,type="l",xlab=expression(paste(symbol("\341"),list(X, gamma),symbol("\361"))),ylab=expression(R[n](u)),main="")
for(i in 1:B.plot) lines(u,mean.boot.proc[i,],lty=2,col=gray(0.8))
lines(u,mean.proc)
text(x=0.75*u[1],y=0.75*min(mean.proc,mean.boot.proc),labels=sprintf("p-value=%.3f",pvalue))
}
if(verbose) cat("Done.\n")
# Result: class htest
names(pcvm)="PCvM statistic"
result=structure(list(statistic=pcvm,boot.statistics=pcvm.star,p.value=pvalue,method=meth,B=B,type.basis=type.basis,beta.est=beta.est,p=p.opt,ord=ord.opt,data.name="Y=<X,b>+e"))
class(result) <- "htest"
return(result)
}
moviedo5/fda.usc documentation built on Nov. 6, 2024, 1:21 a.m.
R Package Documentation
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Defines functions flm.test
Documented in flm.test
#' @aliases flm.test
#'
#' @title Goodness-of-fit test for the Functional Linear Model with scalar response
#'
#' @description The function \code{flm.test} tests the composite null hypothesis of
#' a Functional Linear Model with scalar response (FLM),
#' \deqn{H_0:\,Y=\big<X,\beta\big>+\epsilon,}{H_0: Y=<X,\beta>+\epsilon,} versus
#' a general alternative. If \eqn{\beta=\beta_0}{\beta=\beta_0} is provided, then the
#' simple hypothesis \eqn{H_0:\,Y=\big<X,\beta_0\big>+\epsilon}{H_0: Y=<X,\beta_0>+\epsilon} is tested.
#' The testing of the null hypothesis is done by a Projected Cramer-von Mises statistic (see Details).
#'
#' @param X.fdata Functional covariate for the FLM. The object must be in the class
#' \code{\link{fdata}}.
#' @param Y Scalar response for the FLM. Must be a vector with the same number of elements
#' as functions are in \code{X.fdata}.
#' @param beta0.fdata Functional parameter for the simple null hypothesis, in the \code{\link{fdata}} class.
#' Recall that the \code{argvals} and \code{rangeval} arguments of \code{beta0.fdata} must be the same
#' of \code{X.fdata}. A possibility to do this is to consider, for example for \eqn{\beta_0=0}{\beta_0=0}
#' (the simple null hypothesis of no interaction),\if{latex}{\cr}
#' \code{beta0.fdata=fdata(mdata=rep(0,length(X.fdata$argvals)),}\if{latex}{\cr}\code{argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)}.\if{latex}{\cr}
#' If \code{beta0.fdata=NULL} (default), the function will test for the composite null hypothesis.
#'
#' @param B Number of bootstrap replicates to calibrate the distribution of the test statistic.
#' \code{B=5000} replicates are the recommended for carry out the test, although for exploratory analysis
#' (\bold{not inferential}), an acceptable less time-consuming option is \code{B=500}.
#' @param est.method Estimation method for the unknown parameter \eqn{\beta}{\beta},
#' only used in the composite case. Mainly, there are two options: specify the number of basis
#' elements for the estimated \eqn{\beta}{\beta} by \code{p} or optimally select \code{p} by a
#' data-driven criteria (see Details section for discussion). Then, it must be one of the following
#' methods:
#' \itemize{
#' \item \code{"pc"}: If \code{p}, the number of basis elements, is given, then \eqn{\beta}{\beta} is estimated by \code{\link{fregre.pc}}. Otherwise, an optimum \code{p} is chosen using \code{\link{fregre.pc.cv}} and the \code{"SICc"} criteria.
#' \item \code{"pls"}: If \code{p} is given, \eqn{\beta}{\beta} is estimated by \code{\link{fregre.pls}}. Otherwise, an optimum \code{p} is chosen using \code{\link{fregre.pls.cv}} and the \code{"SICc"} criteria.
#' This is the default argument as it has been checked empirically that provides a good balance between the performance of the test and the estimation of \eqn{\beta}{\beta}.
#' \item \code{"basis"}: If \code{p} is given, \eqn{\beta}{\beta} is estimated by \code{\link{fregre.basis}}. Otherwise, an optimum \code{p} is chosen using \code{\link{fregre.basis.cv}} and the \code{"GCV.S"} criteria. In these functions, the same basis for the arguments \code{basis.x} and \code{basis.b} is considered.
#' The type of basis used will be the given by the argument \code{type.basis} and must be one of the class of \code{create.basis}. Further arguments passed to \code{\link{create.basis}} (not \code{rangeval} that is taken as the \code{rangeval} of \code{X.fdata}), can be passed throughout \code{\dots}.
#' }
#' @param p Number of elements of the basis considered. If it is not given, an optimal \code{p} will be chosen using a specific criteria (see \code{est.method} and \code{type.basis} arguments).
#' @param type.basis Type of basis used to represent the functional process. Depending on the hypothesis, it will have a different interpretation:
#' \itemize{
#' \item Simple hypothesis. One of these options:
#' \itemize{
#' \item \code{"bspline"}: If \code{p} is given, the functional process is expressed in a basis of \code{p} B-splines. If not, an optimal \code{p} will be chosen by \code{\link{optim.basis}}, using the \code{"GCV.S"} criteria.
#' \item \code{"fourier"}: If \code{p} is given, the functional process is expressed in a basis of \code{p} Fourier functions. If not, an optimal \code{p} will be chosen by \code{\link{optim.basis}}, using the \code{"GCV.S"} criteria.
#' \item \code{"pc"}: \code{p} must be given. Expresses the functional process in a basis of \code{p} principal components.
#' \item \code{"pls"}: \code{p} must be given. Expresses the functional process in a basis of \code{p} partial least squares.
#' }
#' Although other basis types supported by \code{\link{create.basis}} are possible, \code{"bspline"} and \code{"fourier"} are recommended. Other basis types may cause incompatibilities.
#' \item Composite hypothesis. This argument is only used when \if{latex}{\cr}\code{est.method="basis"} and, in this case, it specifies the type of basis used in the basis estimation method of the functional parameter. Again, basis
#' \code{"bspline"} and \code{"fourier"} are recommended, as other basis types may cause incompatibilities.
#' }
#'
#' @param verbose Either to show or not information about computing progress.
#' @param plot.it Either to show or not a graph of the observed trajectory,
#' and the bootstrap trajectories under the null composite hypothesis, of the
#' process \eqn{R_n(\cdot)}{R_n(.)} (see Details). Note that if \code{plot.it=TRUE},
#' the function takes more time to run.
#' @param B.plot Number of bootstrap trajectories to show in the resulting plot of the test.
#' As the trajectories shown are the first \code{B.plot} of \code{B}, \code{B.plot} must be
#' lower or equal to \code{B}.
#' @param G Number of projections used to compute the trajectories of the process
#' \eqn{R_n(\cdot)}{R_n(.)} by Monte Carlo.
#' @param \dots Further arguments passed to \code{\link{create.basis}}.
#'
#' @details The Functional Linear Model with scalar response (FLM), is defined as
#' \eqn{Y=\big<X,\beta\big>+\epsilon}{Y=<X,\beta>+\epsilon}, for a functional process
#' \eqn{X}{X} such that \eqn{E[X(t)]=0}{E[X(t)]=0}, \eqn{E[X(t)\epsilon]=0}{E[X(t)\epsilon]=0}
#' for all \eqn{t}{t} and for a scalar variable \eqn{Y}{Y} such that \eqn{E[Y]=0}{E[Y]=0}.
#' Then, the test assumes that \code{Y} and \code{X.fdata} are \bold{centred} and will automatically
#' center them. So, bear in mind that when you apply the test for \code{Y} and \code{X.fdata},
#' actually, you are applying it to \code{Y-mean(Y)} and \code{fdata.cen(X.fdata)$Xcen}.
#' The test statistic corresponds to the Cramer-von Mises norm of the \emph{Residual Marked
#' empirical Process based on Projections} \eqn{R_n(u,\gamma)}{R_n(u,\gamma)} defined in
#' Garcia-Portugues \emph{et al.} (2014).
#' The expression of this process in a \eqn{p}{p}-truncated basis of the space \eqn{L^2[0,T]}{L^2[0,T]}
#' leads to the \eqn{p}{p}-multivariate process \eqn{R_{n,p}\big(u,\gamma^{(p)}\big)}{R_{n,p}(u,\gamma^{(p)})},
#' whose Cramer-von Mises norm is computed.
#' The choice of an appropriate \eqn{p}{p} to represent the functional process \eqn{X}{X},
#' in case that is not provided, is done via the estimation of \eqn{\beta}{\beta} for the composite
#' hypothesis. For the simple hypothesis, as no estimation of \eqn{\beta}{\beta} is done, the choice
#' of \eqn{p}{p} depends only on the functional process \eqn{X}{X}. As the result of the test may
#' change for different \eqn{p}{p}'s, we recommend to use an automatic criterion to select \eqn{p}{p}
#' instead of provide a fixed one.
#' The distribution of the test statistic is approximated by a wild bootstrap resampling on the
#' residuals, using the \emph{golden section bootstrap}.
#' Finally, the graph shown if \code{plot.it=TRUE} represents the observed trajectory, and the
#' bootstrap trajectories under the null, of the process RMPP \emph{integrated on the projections}:
#' \deqn{R_n(u)\approx\frac{1}{G}\sum_{g=1}^G R_n(u,\gamma_g),}{R_n(u) \approx \frac{1}{G} \sum_{g=1}^G R_n(u,\gamma_g),}
#' where \eqn{\gamma_g}{\gamma_g} are simulated as Gaussians processes. This gives a graphical idea of
#' how \emph{distant} is the observed trajectory from the null hypothesis.
#'
#' @return An object with class \code{"htest"} whose underlying structure is a list containing
#' the following components:
#' \itemize{
#' \item \code{statistic}: The value of the test statistic.
#' \item \code{boot.statistics}: A vector of length \code{B} with the values of the bootstrap test statistics.
#' \item \code{p.value}: The p-value of the test.
#' \item \code{method}: The method used.
#' \item \code{B}: The number of bootstrap replicates used.
#' \item \code{type.basis}: The type of basis used.
#' \item \code{beta.est}: The estimated functional parameter \eqn{\beta}{\beta} in the composite
#' hypothesis. For the simple hypothesis, the given \code{beta0.fdata}.
#' \item \code{p}: The number of basis elements passed or automatically chosen.
#' \item \code{ord}: The optimal order for PC and PLS given by \code{\link{fregre.pc.cv}} and \if{latex}{\cr} \code{\link{fregre.pls.cv}}. For other methods, it is set to \code{1:p}.
#' \item \code{data.name}: The character string "Y=<X,b>+e".
#' }
#'
#' @references
#' Escanciano, J. C. (2006). A consistent diagnostic test for regression models using projections. Econometric Theory, 22, 1030-1051. \doi{10.1017/S0266466606060506}
#'
#' Garcia-Portugues, E., Gonzalez-Manteiga, W. and Febrero-Bande, M. (2014). A goodness--of--fit test for the functional linear model with scalar response. Journal of Computational and Graphical Statistics, 23(3), 761-778. \doi{10.1080/10618600.2013.812519}
#'
#' @note No NA's are allowed neither in the functional covariate nor in the scalar response.
#'
#' @author Eduardo Garcia-Portugues. Please, report bugs and suggestions to
#' \if{latex}{\cr}\email{edgarcia@@est-econ.uc3m.es}
#'
#' @seealso \code{\link{Adot}}, \code{\link{PCvM.statistic}}, \code{\link{rwild}},
#' \code{\link{flm.Ftest}}, \code{\link{dfv.test}},
#' \code{\link{fregre.pc}}, \code{\link{fregre.pls}},\if{latex}{\cr} \code{\link{fregre.basis}},
#' \code{\link{fregre.pc.cv}}, \code{\link{fregre.pls.cv}},
#' \code{\link{fregre.basis.cv}}, \code{\link{optim.basis}}, \if{latex}{\cr}
#' \link[fda]{create.basis}
#' @examples
#' # Simulated example #
#' X=rproc2fdata(n=100,t=seq(0,1,l=101),sigma="OU")
#' beta0=fdata(mdata=cos(2*pi*seq(0,1,l=101))-(seq(0,1,l=101)-0.5)^2+
#' rnorm(101,sd=0.05),argvals=seq(0,1,l=101),rangeval=c(0,1))
#' Y=inprod.fdata(X,beta0)+rnorm(100,sd=0.1)
#'
#' dev.new(width=21,height=7)
#' par(mfrow=c(1,3))
#' plot(X,main="X")
#' plot(beta0,main="beta0")
#' plot(density(Y),main="Density of Y",xlab="Y",ylab="Density")
#' rug(Y)
#'
#' \dontrun{
#' # Composite hypothesis: do not reject FLM
#' pcvm.sim=flm.test(X,Y,B=50,B.plot=50,G=100,plot.it=TRUE)
#' pcvm.sim
#' flm.test(X,Y,B=5000)
#'
#' # Estimated beta
#' dev.new()
#' plot(pcvm.sim$beta.est)
#'
#' # Simple hypothesis: do not reject beta=beta0
#' flm.test(X,Y,beta0.fdata=beta0,B=50,B.plot=50,G=100)
#' flm.test(X,Y,beta0.fdata=beta0,B=5000)
#'
#' # AEMET dataset #
#' data(aemet)
#' # Remove the 5\% of the curves with less depth (i.e. 4 curves)
#' dev.new()
#' res.FM=depth.FM(aemet$temp,draw=TRUE)
#' qu=quantile(res.FM$dep,prob=0.05)
#' l=which(res.FM$dep<=qu)
#' lines(aemet$temp[l],col=3)
#' aemet$df$name[l]
#'
#' # Data without outliers
#' wind.speed=apply(aemet$wind.speed$data,1,mean)[-l]
#' temp=aemet$temp[-l]
#' # Exploratory analysis: accept the FLM
#' pcvm.aemet=flm.test(temp,wind.speed,est.method="pls",B=100,B.plot=50,G=100)
#' pcvm.aemet
#'
#' # Estimated beta
#' dev.new()
#' plot(pcvm.aemet$beta.est,lwd=2,col=2)
#' # B=5000 for more precision on calibration of the test: also accept the FLM
#' flm.test(temp,wind.speed,est.method="pls",B=5000)
#'
#' # Simple hypothesis: rejection of beta0=0? Limiting p-value...
#' dat=rep(0,length(temp$argvals))
#' flm.test(temp,wind.speed, beta0.fdata=fdata(mdata=dat,argvals=temp$argvals,
#' rangeval=temp$rangeval),B=100)
#' flm.test(temp,wind.speed, beta0.fdata=fdata(mdata=dat,argvals=temp$argvals,
#' rangeval=temp$rangeval),B=5000)
#'
#' # Tecator dataset #
#' data(tecator)
#' names(tecator)
#' absorp=tecator$absorp.fdata
#' ind=1:129 # or ind=1:215
#' x=absorp[ind,]
#' y=tecator$y$Fat[ind]
#' tt=absorp[["argvals"]]
#'
#' # Exploratory analysis for composite hypothesis with automatic choose of p
#' pcvm.tecat=flm.test(x,y,B=100,B.plot=50,G=100)
#' pcvm.tecat
#'
#' # B=5000 for more precision on calibration of the test: also reject the FLM
#' flm.test(x,y,B=5000)
#'
#' # Distribution of the PCvM statistic
#' plot(density(pcvm.tecat$boot.statistics),lwd=2,xlim=c(0,10),
#' main="PCvM distribution", xlab="PCvM*",ylab="Density")
#' rug(pcvm.tecat$boot.statistics)
#' abline(v=pcvm.tecat$statistic,col=2,lwd=2)
#' legend("top",legend=c("PCvM observed"),lwd=2,col=2)
#'
#' # Simple hypothesis: fixed p
#' dat=rep(0,length(x$argvals))
#' flm.test(x,y,beta0.fdata=fdata(mdata=dat,argvals=x$argvals,
#' rangeval=x$rangeval),B=100,p=11)
#'
#' # Simple hypothesis, automatic choose of p
#' flm.test(x,y,beta0.fdata=fdata(mdata=dat,argvals=x$argvals,
#' rangeval=x$rangeval),B=100)
#' flm.test(x,y,beta0.fdata=fdata(mdata=dat,argvals=x$argvals,
#' rangeval=x$rangeval),B=5000)
#' }
#' @keywords htest models regression
# PCvM test for the composite hypothesis with bootstrap calibration
#' @rdname flm.test
#' @export
flm.test=function(X.fdata,Y,beta0.fdata=NULL,B=5000,est.method="pls",
p=NULL,type.basis="bspline",verbose=TRUE,plot.it=TRUE,
B.plot=100,G=200,...){
# Check B.plot
if(plot.it & B.plot>B) stop("B.plot must be less or equal than B")
# Number of functions
n=dim(X.fdata)[1]
if(verbose) cat("Computing estimation of beta... ")
## COMPOSITE HYPOTHESIS ##
if(is.null(beta0.fdata)){
# Center the data first
X.fdata=fdata.cen(X.fdata)$Xcen
Y=Y-mean(Y)
## 1. Optimal estimation of beta and the basis order ##
if(est.method=="pc"){
if(is.null(p)){
# Method
meth="PCvM test for the functional linear model using optimal PC basis representation"
# Choose the number of basis elements: SICc is probably the best criteria
mod.pc=fregre.pc.cv(fdataobj=X.fdata,y=Y,kmax=1:10,criteria="SICc")
p.opt=length(mod.pc$pc.opt)
ord.opt=mod.pc$pc.opt
# PC components to be passed to the bootstrap
pc.comp=mod.pc$fregre.pc$fdata.comp # pc.comp=mod.pc$fregre.pc$pc
pc.comp$l=mod.pc$pc.opt
# Express X.fdata and beta.est in the PC basis
basis.pc=mod.pc$fregre.pc$fdata.comp$basis
if(length(pc.comp$l)!=1){
X.est=fdata(mdata=mod.pc$fregre.pc$fdata.comp$coefs[,mod.pc$fregre.pc$l]%*%mod.pc$fregre.pc$fdata.comp$basis$data[mod.pc$fregre.pc$l,],argvals=X.fdata$argvals,rangeval=X.fdata$rangeval) # X.est=fdata(mdata=mod.pc$fregre.pc$pc$coefs[,mod.pc$fregre.pc$l]%*%mod.pc$fregre.pc$pc$basis$data[mod.pc$fregre.pc$l,],argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)
}else{
X.est=fdata(mdata=mod.pc$fregre.pc$fdata.comp$coefs[,mod.pc$fregre.pc$l]%*%t(mod.pc$fregre.pc$fdata.comp$basis$data[mod.pc$fregre.pc$l,]),argvals=X.fdata$argvals,rangeval=X.fdata$rangeval) # X.est=fdata(mdata=mod.pc$fregre.pc$pc$coefs[,mod.pc$fregre.pc$l]%*%t(mod.pc$fregre.pc$pc$basis$data[mod.pc$fregre.pc$l,]),argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)
}
beta.est=mod.pc$fregre.pc$beta.est
norm.beta.est=norm.fdata(beta.est)
# Compute the residuals
e=mod.pc$fregre.pc$residuals
}else{
# Method
meth=paste("PCvM test for the functional linear model using a representation in a PC basis of ",p,"elements")
# Estimation of beta on the given fixed basis
mod.pc=fregre.pc(fdataobj=X.fdata,y=Y,l=1:p)
p.opt=p
ord.opt=mod.pc$l
# PC components to be passed to the bootstrap
pc.comp=mod.pc$pc
pc.comp$l=mod.pc$l
# Express X.fdata and beta.est in the basis
if(p!=1){
X.est=fdata(mdata=mod.pc$fdata.comp$coefs[,mod.pc$l]%*%mod.pc$fdata.comp$basis$data[mod.pc$l,],argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)
}else{
X.est=fdata(mdata=mod.pc$fdata.comp$coefs[,mod.pc$l]%*%t(mod.pc$fdata.comp$basis$data[mod.pc$l,]),argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)
}
beta.est=mod.pc$beta.est
norm.beta.est=norm.fdata(beta.est)
# Compute the residuals
e=mod.pc$residuals
}
}else if(est.method=="pls"){
if(is.null(p)){
# print("PLS1")
# Method
meth="PCvM test for the functional linear model using optimal PLS basis representation"
# Choose the number of the basis: SICc is probably the best criteria
mod.pls=fregre.pls.cv(fdataobj=X.fdata,y=Y,kmax=10,criteria="SICc")
# print("PLS2")
p.opt=length(mod.pls$pls.opt)
ord.opt=mod.pls$pls.opt
# PLS components to be passed to the bootstrap
pls.comp=mod.pls$fregre.pls$fdata.comp
pls.comp$l=mod.pls$pls.opt
# Express X.fdata and beta.est in the PLS basis
basis.pls=mod.pls$fregre.pls$fdata.comp$basis
if(length(pls.comp$l)!=1){
X.est=fdata(mdata=mod.pls$fregre.pls$fdata.comp$coefs[,mod.pls$fregre.pls$l]%*%mod.pls$fregre.pls$fdata.comp$basis$data[mod.pls$fregre.pls$l,],argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)
}else{
X.est=fdata(mdata=mod.pls$fregre.pls$fdata.comp$coefs[,mod.pls$fregre.pls$l]%*%t(mod.pls$fregre.pls$fdata.comp$basis$data[mod.pls$fregre.pls$l,]),argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)
}
beta.est=mod.pls$fregre.pls$beta.est
norm.beta.est=norm.fdata(beta.est)
# Compute the residuals
e=mod.pls$fregre.pls$residuals
}else{
# Method
meth=paste("PCvM test for the functional linear model using a representation in a PLS basis of ",p,"elements")
# Estimation of beta on the given fixed basis
mod.pls=fregre.pc(fdataobj=X.fdata,y=Y,l=1:p)
p.opt=p
ord.opt=mod.pls$l
# PLS components to be passed to the bootstrap
pls.comp=mod.pls$fdata.comp
pls.comp$l=mod.pls$l
# Express X.fdata and beta.est in the basis
if(p!=1){
X.est=fdata(mdata=mod.pls$fdata.comp$coefs[,mod.pls$l]%*%mod.pls$fdata.comp$basis$data[mod.pls$l,],argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)
}else{
X.est=fdata(mdata=mod.pls$fdata.comp$coefs[,mod.pls$l]%*%t(mod.pls$fdata.comp$basis$data[mod.pls$l,]),argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)
}
beta.est=mod.pls$beta.est
norm.beta.est=norm.fdata(beta.est)
# Compute the residuals
e=mod.pls$residuals
}
}else if(est.method=="basis"){
if(is.null(p)){
# Method
meth=paste("PCvM test for the functional linear model using optimal",type.basis,"basis representation")
# Choose the number of the bspline basis with GCV.S
mod.basis=fregre.basis.cv(fdataobj=X.fdata,y=Y,basis.x=seq(5,30,by=1),basis.b=NULL,type.basis=type.basis,type.CV=GCV.S,verbose=FALSE,...)
p.opt=mod.basis$basis.x.opt$nbasis
ord.opt=1:p.opt
# Express X.fdata and beta.est in the optimal basis
basis.opt=mod.basis$basis.x.opt
X.est=mod.basis$x.fd
beta.est=mod.basis$beta.est
norm.beta.est=norm.fd(beta.est)
# Compute the residuals
e=mod.basis$residuals
}else{
# Method
meth=paste("PCvM test for the functional linear model using a representation in a",type.basis,"basis of ",p,"elements")
# Estimation of beta on the given fixed basis
basis.opt=do.call(what=paste("create.",type.basis,".basis",sep=""),args=list(rangeval=X.fdata$rangeval,nbasis=p,...))
mod.basis=fregre.basis(fdataobj=X.fdata,y=Y,basis.x=basis.opt,basis.b=basis.opt)
p.opt=p
ord.opt=1:p.opt
# Express X.fdata and beta.est in the basis
X.est=mod.basis$x.fd
beta.est=mod.basis$beta.est
norm.beta.est=norm.fd(beta.est)
# Compute the residuals
e=mod.basis$residuals
}
}else{
stop(paste("Estimation method",est.method,"not implemented."))
}
## SIMPLE HYPOTHESIS ##
}else{
## 1. Optimal representation of X and beta0 ##
# Choose the number of basis elements
if(type.basis!="pc" & type.basis!="pls"){
# Basis method: select the number of elements of the basis is it is not given
if(is.null(p)){
# Method
meth=paste("PCvM test for the simple hypothesis in a functional linear model, using a representation in an optimal ",type.basis,"basis")
p.opt=optim.basis(X.fdata,type.basis=type.basis,numbasis=seq(31,71,by=2),verbose=FALSE)$numbasis.opt
if(p.opt==31){
cat("Readapting interval...\n")
p.opt=optim.basis(X.fdata,type.basis=type.basis,numbasis=seq(5,31,by=2),verbose=FALSE)$numbasis.opt
}else if(p.opt==71){
cat("Readapting interval...\n")
p.opt=optim.basis(X.fdata,type.basis=type.basis,numbasis=seq(71,101,by=2),verbose=FALSE)$numbasis.opt
}
ord.opt=1:p.opt
}else{
# Method
meth=paste("PCvM test for the simple hypothesis in a functional linear model, using a representation in a ",type.basis," basis of ",p,"elements")
p.opt=p
ord.opt=1:p.opt
}
# Express X.fdata in the basis
X.est=fdata2fd(X.fdata,type.basis=type.basis,nbasis=p)
beta.est=fdata2fd(beta0.fdata,type.basis=type.basis,nbasis=p)
# Compute the residuals
e=Y-inprod(X.est,beta.est)
}else if (type.basis=="pc"){
if(is.null(p)) stop("Simple hypothesis with type.basis=\"pc\" need the number of components p")
# Method
meth=paste("PCvM test for the simple hypothesis in a functional linear model, using a representation in a PC basis of ",p,"elements")
# Express X.fdata in a PC basis
fd2pc=fdata2pc(X.fdata,ncomp=p)
if(length(fd2pc$l)!=1){
X.est=fdata(mdata=fd2pc$coefs[,fd2pc$l]%*%fd2pc$basis$data[fd2pc$l,],argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)
}else{
X.est=fdata(mdata=fd2pc$coefs[,fd2pc$l]%*%t(fd2pc$basis$data[fd2pc$l,]),argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)
}
beta.est=beta0.fdata
p.opt=p
ord.opt=1:p.opt
# Compute the residuals
e=Y-inprod.fdata(X.est,beta.est)
}else if(type.basis=="pls"){
if(is.null(p)) stop("Simple hypothesis with type.basis=\"pls\" need the number of components p")
# Method
meth=paste("PCvM test for the simple hypothesis in a functional linear model, using a representation in a PLS basis of ",p,"elements")
# Express X.fdata in a PLS basis
fd2pls=fdata2pls(X.fdata,Y,ncomp=p)
if(length(fd2pls$l)!=1){
X.est=fdata(mdata=fd2pls$coefs[,fd2pls$l]%*%fd2pls$basis$data[fd2pls$l,],argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)
}else{
X.est=fdata(mdata=fd2pls$coefs[,fd2pls$l]%*%t(fd2pls$basis$data[fd2pls$l,]),argvals=X.fdata$argvals,rangeval=X.fdata$rangeval)
}
beta.est=beta0.fdata
p.opt=p
ord.opt=1:p.opt
# Compute the residuals
e=Y-inprod.fdata(X.est,beta.est)
}else{
stop(paste("Type of basis method",type.basis,"not implemented."))
}
}
## 2. Bootstrap calibration ##
# Start up
pcvm.star=numeric(B)
e.hat.star=matrix(ncol=n,nrow=B)
# Calculus of the Adot.vec
Adot.vec=Adot(X.est)
# REAL WORLD
pcvm=PCvM.statistic(X=X.est,residuals=e,p=p.opt,Adot.vec=Adot.vec)
# BOOTSTRAP WORLD
if(verbose) cat("Done.\nBootstrap calibration...\n ")
if(verbose) pb=txtProgressBar(style=3)
## COMPOSITE HYPOTHESIS ##
if(is.null(beta0.fdata)){
# Calculate the design matrix of the linear model
# This allows to resample efficiently the residuals without estimating again the beta
if(est.method=="pc"){
if(is.null(p)){
# Design matrix for the PC estimation
X.matrix=mod.pc$fregre.pc$lm$x
}else{
# Design matrix for the PC estimation
X.matrix=mod.pc$lm$x
}
}else if(est.method=="pls"){
if(is.null(p)){
# Design matrix for the PLS estimation
X.matrix=mod.pls$fregre.pls$lm$x
}else{
# Design matrix for the PLS estimation
X.matrix=mod.pls$lm$x
}
}else if(est.method=="basis"){
if(is.null(p)){
# Design matrix for the basis estimation
X.matrix=mod.basis$lm$x
}else{
# Design matrix for the basis estimation
X.matrix=mod.basis$lm$x
}
}
# Projection matrix
P=(diag(rep(1,n))-X.matrix%*%solve(t(X.matrix)%*%X.matrix)%*%t(X.matrix))
# Bootstrap resampling
for(i in 1:B){
# Generate bootstrap errors
e.hat=rwild(e,"golden")
# Calculate Y.star
Y.star=Y-e+e.hat
# Residuals from the bootstrap estimated model
e.hat.star[i,]=P%*%Y.star
# Calculate PCVM.star
pcvm.star[i]=PCvM.statistic(X=X.est,residuals=e.hat.star[i,],p=p.opt,Adot.vec=Adot.vec)
if(verbose) setTxtProgressBar(pb,i/B)
}
## SIMPLE HYPOTHEIS ##
}else{
# Bootstrap resampling
for(i in 1:B){
# Generate bootstrap errors
e.hat.star[i,]=rwild(e,"golden")
# Calculate PCVM.star
pcvm.star[i]=PCvM.statistic(X=X.est,residuals=e.hat.star[i,],p=p.opt,Adot.vec=Adot.vec)
if(verbose) setTxtProgressBar(pb,i/B)
}
}
## 3. MC estimation of the p-value and order the result ##
# Compute the p-value
pvalue=sum(pcvm.star>pcvm)/B
## 4. Graphical representation of the integrated process ##
if(verbose) cat("\nDone.\nComputing graphical representation... ")
if(is.null(beta0.fdata) & plot.it){
gamma=rproc2fdata(n=G,t=X.fdata$argvals,sigma="OU",par.list=list(theta=2/diff(range(X.fdata$argvals))))
gamma=gamma/drop(norm.fdata(gamma))
ind=drop(inprod.fdata(X.fdata,gamma))
r=0.9*max(max(ind),-min(ind))
u=seq(-r,r,l=200)
mean.proc=numeric(length(u))
mean.boot.proc=matrix(ncol=length(u),nrow=B.plot)
res=apply(ind,2,function(iind){
iind.sort=sort(iind,index.return=TRUE)
stepfun(x=iind.sort$x,y=c(0,cumsum(e[iind.sort$ix])))(u)/sqrt(n)
}
)
mean.proc=apply(res,1,mean)
for(i in 1:B.plot){
res=apply(ind,2,function(iind){
iind.sort=sort(iind,index.return=TRUE)
stepfun(x=iind.sort$x,y=c(0,cumsum(e.hat.star[i,iind.sort$ix])))(u)/sqrt(n)
}
)
mean.boot.proc[i,]=apply(res,1,mean)
}
# Plot
dev.new()
plot(u,mean.proc,ylim=c(min(mean.proc,mean.boot.proc),max(mean.proc,mean.boot.proc))*1.05,type="l",xlab=expression(paste(symbol("\341"),list(X, gamma),symbol("\361"))),ylab=expression(R[n](u)),main="")
for(i in 1:B.plot) lines(u,mean.boot.proc[i,],lty=2,col=gray(0.8))
lines(u,mean.proc)
text(x=0.75*u[1],y=0.75*min(mean.proc,mean.boot.proc),labels=sprintf("p-value=%.3f",pvalue))
}
if(verbose) cat("Done.\n")
# Result: class htest
names(pcvm)="PCvM statistic"
result=structure(list(statistic=pcvm,boot.statistics=pcvm.star,p.value=pvalue,method=meth,B=B,type.basis=type.basis,beta.est=beta.est,p=p.opt,ord=ord.opt,data.name="Y=<X,b>+e"))
class(result) <- "htest"
return(result)
}
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