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#' Caculate the estimator on the ISR method
#'
#' @param data is the orignal data set
#' @param data0 is the missing data set
#' @param real is to judge whether the data set is a real missing data set
#' @param example is to judge whether the data set is a simulation example.
#'
#' @return XISR, MSEISR, MAEISR, REISR, GCVISR,timeISR
#' @export
#'
#' @examples
#' library(MASS)
#' n=100;p=10;per=0.1
#' X0=data=matrix(mvrnorm(n*p,0,1),n,p)
#' m=round(per*n*p,digits=0)
#' mr=sample(1:(n*p),m,replace=FALSE)
#' X0[mr]=NA;data0=X0
#' ISR(data=data,data0=data0,real=FALSE,example=FALSE)
#the ISR method
ISR=function(data=0,data0,real=TRUE,example=FALSE)
#It defaults that the data set is a real data set
{#1
if (real||example){#2
etatol=0.7
}else{#2
etatol=0.9
}#2
lll=0
time=system.time(#2
while(lll==0){#3
X0=data0
n=nrow(X0);p=ncol(X0)
mr=which(is.na(X0)==TRUE)
m=nrow(as.matrix(mr))
cm0=colMeans(X0,na.rm=T)
ina=as.matrix(mr%%n)
jna=as.matrix(floor((mr+n-1)/n))
data0[is.na(data0)]=cm0[ceiling(which(is.na(X0))/n)]
X=as.matrix(data0)
Z=scale(X,center=TRUE,scale=FALSE)
niter=0;d=1;tol=1e-5;nb=10
while((d>=tol) & (niter<=nb)){#4
niter=niter+1
Zold=Z
lambda=svd(cor(Z))$d
l=lambda/sum(lambda)
J=rep(l,times=p);dim(J)=c(p,p)
upper.tri(J,diag=T);J[lower.tri(J)]=0
eta=matrix(colSums(J),nrow = 1,ncol = p,byrow = FALSE)
k=which(eta>=etatol)[1]
Ak=matrix(svd(Z)$v[,1:k],p,k)
Lambdak=diag(sqrt(lambda[1:k]),k,k)
for( i in 1:n){#5
M=is.na(X0[i,])
job=which(M==FALSE);jna=which(M==TRUE)
piob=nrow(as.matrix(job));pina=nrow(as.matrix(jna))
while((piob>0)&(pina>0)){#6
Qi=matrix(0,p,p)
for( u in 1:piob){#7
Qi[job[u],u]=1
}#7
for( v in 1:pina){#7
Qi[jna[v],v+piob]=1
}#7
zQi=Z[i,]%*%Qi
ZQi=Z%*%Qi#
AQi=t(t(Ak)%*%Qi)
ziob=matrix(zQi[,1:piob],1,piob)
zina=matrix(zQi[,piob+(1:pina)],1,pina)
Ziob=matrix(ZQi[,1:piob],n,piob,byrow=FALSE)
Zina=matrix(ZQi[,piob+(1:pina)],n,pina,byrow=FALSE)
Aiob=matrix(AQi[1:piob,],piob,k,byrow=FALSE)
Aina=matrix(AQi[piob+(1:pina),],pina,k,byrow=FALSE)
Ti=Ziob%*%Aiob;Ti
betaihat=ginv(t(Ti)%*%Ti)%*%t(Ti)%*%Zina;betaihat
zinahat=ziob%*%Aiob%*%betaihat;zinahat
ZQi[i,piob+(1:pina)]=zinahat
Z=Zi=ZQi%*%t(Qi)
pina=0
}#6
}#5
ZISR=Znew=Z
d=sqrt(sum(diag((t(Zold-Znew)%*%(Zold-Znew)))))
}#4
XISR=Xnew=Znew+matrix(rep(1,n*p),ncol=p)%*%diag(cm0)
for (j in 1:p){
Mj=is.na(X0[,j])
iob=which(Mj==FALSE)
chj=sum(abs(round(X0[iob,j])-X0[iob,j]))
if (chj==0){
XISR[,j]=round(XISR[,j])
}else{
XISR[,j]= XISR[,j]
}
}
lll=1
}#3
)#2
if(real){#2
MSEISR= MAEISR= REISR='NULL'
}else{#2
MSEISR=(1/m)*t(Xnew[mr]-data[mr])%*%(Xnew[mr]-data[mr])
MAEISR=(1/m)*sum(abs(Xnew[mr]-data[mr]))
REISR=(sum(abs(data[mr]-Xnew[mr])))/(sum(data[mr]))
}#2
lambdaISR=svd(cor(XISR))$d;lambdaISR
lISR=lambdaISR/sum(lambdaISR);J=rep(lISR,times=p);dim(J)=c(p,p)
upper.tri(J,diag=T);J[lower.tri(J)]=0;J;dim(J)=c(p,p)
etaISR=matrix(colSums(J),nrow = 1,ncol = p,byrow = FALSE)
wwISR=which(etaISR>=etatol);kISR=wwISR[1]
lambdaISRpk=lambdaISR[(kISR+1):p]
GCVISR=sum(lambdaISRpk)*p/(p-kISR)^2
return(list(XISR=XISR,MSEISR=MSEISR,MAEISR=MAEISR,REISR=REISR,GCVISR=GCVISR,timeISR=time))
}#1
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