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#' Calculate the H-representation of the marginal polytope
#'
#' Computes the minimal halfspace representation of the marginal polytope
#' defined, for example, in \insertCite{BS2022;textual}{MCARtest}.
#'
#' @param bS A binary matrix specifying the set of observation patterns. Each row encodes a single pattern.
#' @param M A vector of positive integers giving the alphabet sizes of the discrete variables.
#' @param round A logical value indicating whether or not to round coefficients to 15 significant figures.
#' The function \code{RoundErrors} can be used separately to substitute other values for 15. Defaults to FALSE.
#'
#' @return A halfspace representation object as used by the \code{rcdd} package. See \insertCite{RCDD;textual}{MCARtest} for more detail.
#' @export
#'
#' @importFrom rcdd d2q
#' @importFrom rcdd makeV
#' @importFrom rcdd redundant
#' @importFrom rcdd scdd
#'
#' @references
#' \insertRef{BS2022}{MCARtest}
#'
#' \insertRef{RCDD}{MCARtest}
#'
#' @examples
#' bS=matrix(c(1,1,0, 1,0,1, 0,1,1),byrow=TRUE,ncol=3)
#' MargPolyHrep(bS,c(2,2,2))
MargPolyHrep=function(bS,M,round=FALSE){
A=Amatrix(bS,M)
v=nrow(A); cardS=nrow(bS)
Vrep=d2q(makeV(points=t(A)))
Hrep=scdd(Vrep)$output
Hrep=redundant(Hrep)$output
Names=rep(0,v); Names[1:2]=c("l","b"); current=2
for(S in 1:cardS){
MS=M[bS[S,]==1]
for(i in 1:prod(MS)){
current=current+1
xS=as.vector(arrayInd(i,MS))
Name=bS[S,]; Name[Name==0]="d"; Name[Name==1]=xS
Name=paste(Name,collapse="")
Names[current]=paste("p",Name,sep="")
}
}
colnames(Hrep)=Names
if(round==FALSE){
return(Hrep)
}
if(round==TRUE){
return(RoundErrors(Hrep))
}
}
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