R/Zmix_univ_tempered.R
In zoevanhavre/Zmix_devVersion2: What the package does (short line)
#' Run Gibbs sampler with prior tempering
#'
#' This function ...
#' @param y, k,iter, isSim=TRUE, alphas=
#' @keywords Gibbs sampler, univariate, normal, gaussian, mixture, parallel tempering, order estimation
#' @export
#' @examples
#' #... you know...
Zmix_univ_tempered<-function(y, k,iter=5000, tau=1, isSim=TRUE, alphas= c(30, 20, 10, 5, 3, 1, 0.5, 1/2^(c(2,3,4,5,6, 8, 10, 15, 20, 30)))){
if(isSim==TRUE) {Y<-y$Y
}else{ Y<-y}
parallelAccept<-function(w1, w2, a1, a2){
w1[w1< 1e-200]<-1e-200 # truncate so super small values dont crash everyting
w2[w2< 1e-200]<-1e-200
T1<-dDirichlet(w2, a1, log=TRUE)
T2<-dDirichlet(w1, a2, log=TRUE)
B1<-dDirichlet(w1, a1, log=TRUE)
B2<-dDirichlet(w2, a2, log=TRUE)
MH<-min(1, exp(T1+T2-B1-B2))
Ax<-sample(c(1,0), 1, prob=c(MH,1-MH))
return(Ax)}
nCh<-length(alphas)
TrackParallelTemp<-matrix(nrow=iter, ncol=nCh)
TrackParallelTemp[1,]<-c(1:nCh)
#tau=1
n <-length(Y)
a=2.5; b=2/var(Y)
d<-2
lambda=sum(Y)/n
# 1. set up priors
mux<-list(mu=seq(from=min(Y), to=max(Y),length.out=k),sigma=rep(1, k),p=rep(1/k,k), k=k)
n <-length(Y)
a=2.5; b<-0.5*var(Y);d<-2
lambda<-sum(Y)/n ;
pb <- txtProgressBar(min = 0, max = iter, style = 3)
# 2. set up matrices for parameters which will be saved
map = matrix(0,nrow = iter, ncol = 1)
Loglike = matrix(0,nrow = iter, ncol = 1)
Bigmu = replicate(nCh, matrix(0,nrow = iter, ncol = k) , simplify=F)
Bigsigma=replicate(nCh, matrix(0,nrow = iter, ncol = k) , simplify=F)
Bigp = replicate(nCh, matrix(0,nrow = iter, ncol = k) , simplify=F)
Pzs = replicate(nCh, matrix(0,nrow = n, ncol = k) , simplify=F)
ZSaved= replicate(nCh, matrix(0,nrow = n, ncol = iter) , simplify=F)
SteadyScore<-data.frame("Iteration"=c(1:iter), "K0"=k)
# start chains and create inits needed for i=1
for (.ch in 1:nCh){
Bigmu[[.ch]][1,] <- mux$mu # initial value of mu's
mu0=mux$mu
Bigp[[.ch]][1,] = mux$p # inital value of p's
p0=mux$p
Bigsigma[[.ch]][1,] = mux$sigma # inits for sigma
sig0=mux$sigma }
# initialize chains: ie. iteration 1:
j<-1
# 1. Generate P(Z[i](t)=j) for j=1,...,known
# from paper, computing pzs
for (.ch in 1:nCh){
for (i in 1:n) {
Pzs[[.ch]][i,]<-(p0/sqrt(sig0))*exp(-((Y[i]-mu0)^2)/(2*sig0))
Pzs[[.ch]][i,]<-Pzs[[.ch]][i,]/sum(Pzs[[.ch]][i,]) }} # Scale to equal 1?
# 2 Make indicator matrix of assignments based on Pzs
# sample 1 of the k classes for each row by Pzs (prob)
for (.ch in 1:nCh){
Z<-matrix()
for (i in 1:n){Z[i]=sample((1:k),1, prob=Pzs[[.ch]][i,])}
matk = matrix((1:k), nrow = n, ncol = k, byrow = T)
IndiZ = (Z == matk)
ZSaved[[.ch]][,1]<-Z
# 3 compute ns and sx
ns = apply(IndiZ,2,sum)
for (i in 1:length(ns)){ if ( is.na(ns[i])) ns[i]<-0 }
sx = apply(IndiZ*Y, 2, sum)
# 4 Generate P[j](t) from dirichlet (and save)
Bigp[[.ch]][j,] = rdirichlet(m=1,par= ns+alphas[.ch])
# 5 Generate Mu's (and save)
Bigmu[[.ch]][j,]<-rnorm(k, mean=(lambda*tau+sx)/(tau+ns), sd=sqrt(Bigsigma[[.ch]][1,]/(tau+ns))) # must be sqrt as r takes in sd not var
# ybar<-sx/ns
for (i in 1:length(Bigmu[[.ch]][j,])){ if ( is.na(Bigmu[[.ch]][j,i])) Bigmu[[.ch]][j,i]<-0 }
# 6 Compute sv[j](t)
.bmu<- matrix((1:k), nrow = n, ncol = k, byrow = T)
for (t in 1:n) {.bmu[t,]<-Bigmu[[.ch]][j,]}
sv<-apply((Y*IndiZ-.bmu*IndiZ)^2, 2, sum) # changes, added /ns
# 7 Generate Sigma's (and save)
Bigsigma[[.ch]][j,]<- rinvgamma(k, a+(ns+1)/2, b+0.5*tau*(Bigmu[[.ch]][j,]-lambda)^2+0.5*sv)
}
# Log Likelihood: # Sum-n (log Sum-K ( weights x dnorm (y,thetas)))
for (i in 1:n){
non0id<-c(1:k)[ns > 0]
Loglike[j]<-Loglike[j]+ log(
sum( Bigp[[nCh]][j,non0id]*dnorm(Y[i], mean=Bigmu[[nCh]][j,non0id], sd=sqrt(Bigsigma[[nCh]][j,non0id]))))}
### now finish loop for j>2
for (j in 2:iter){
Sys.sleep(0.01)
setTxtProgressBar(pb, j)
if(j %% 10==0){
par(mfrow=c(1,3))
plot(SteadyScore$K0~SteadyScore$Iteration, main='#non-empty groups', type='l')
ts.plot(Bigp[[nCh]], main='Weights from target posterior', col=rainbow(k))
ts.plot(TrackParallelTemp[,c(nCh:1)], main='Track Parallel Tempering', col=rainbow(nCh))
Sys.sleep(0)}
for (.ch in 1:nCh){
#################
# 1. Generate P(Z[i](t)=j)
for (i in 1:n) {
Pzs[[.ch]][i,]<-(Bigp[[.ch]][j-1,]/sqrt(Bigsigma[[.ch]][j-1,]))*exp(-((Y[i]-Bigmu[[.ch]][j-1,])^2)/(2*Bigsigma[[.ch]][j-1,]))
# Scale to equal 1
Pzs[[.ch]][i,]<-Pzs[[.ch]][i,]/sum(Pzs[[.ch]][i,])
}
# 2 Make indicator matrix of assignments based on Pzs
# sample 1 of the k classes for each row by Pzs (prob)
for (i in 1:n){Z[i]=sample((1:k),1,replace=T, prob=Pzs[[.ch]][i,])}
matk = matrix((1:k), nrow = n, ncol = k, byrow = T)
#indicator
IndiZ = (Z == matk)
ZSaved[[.ch]][,j]<-Z
# 3 # compute ns and sx
ns = apply(IndiZ,2,sum)
# fix if ns=NA to =0
for (i in 1:length(ns)){
if ( is.na(ns[i])) ns[i]<-0 }
sx = apply(IndiZ*Y, 2, sum)
# 4 Generate P[j](t) from dirichlet (and save)
Bigp[[.ch]][j,] = rdirichlet(m=1,par= ns+alphas[.ch])
# 5 Generate Mu's (and save)
Bigmu[[.ch]][j,]<-rnorm(k, mean=(lambda*tau+sx)/(tau+ns), sd=sqrt(Bigsigma[[.ch]][j-1,]/(tau+ns)))
for (i in 1:length(Bigmu[[.ch]][j,])){ if ( is.na(Bigmu[[.ch]][j,i])) Bigmu[[.ch]][j,i]<-0 }
# 6 Compute sv[j](t)
.bmu<- matrix((1:k), nrow = n, ncol = k, byrow = T)
for (t in 1:n) {.bmu[t,]<-Bigmu[[.ch]][j,]}
sv<-apply((Y*IndiZ-.bmu*IndiZ)^2, 2, sum) # changes, added /ns
# 7 Generate Sigma's (and save)
Bigsigma[[.ch]][j,]<- rinvgamma(k, a+(ns+1)/2, b+0.5*tau*(Bigmu[[.ch]][j,]-lambda)^2+0.5*sv)
#calculate MAP estimates at this iteration:
#if (.ch==nCh){
#mapy[j,]<- prod(apply(Pzs[[.ch]], 2,sum))
#mapmu<-prod(dnorm(Bigmu[[.ch]][j,], mean=lambda, sd=(Bigsigma[[.ch]][j,]/tau) ))
#mapsig<- prod(dinvgamma(Bigsigma[[.ch]][j,], a,b))
#map[j,]<-prod(mapy[j,], mapmu, mapsig)}
}
#new #### PARALLEL TEMPERING MOVES ###
if(j>1) {TrackParallelTemp[j,]<-TrackParallelTemp[j-1,]} # SET para chains to previous values # how often??? lets go with probability 10% of switching at any time
if(j>20){
if( sample(c(1,0),1, 0.9)==1){ # FREQ OF TEMPERING!
#Pick chains, just chose one and the one next to it
if( j%%2==0){chainset<- c(1:(nCh-1))[c(1:(nCh-1))%%2==0] #evens
} else {chainset<- c(1:(nCh-1))[c(1:(nCh-1))%%2!=0] } #odds
for( eachChain in 1:length(chainset)){
Chain1<-chainset[eachChain]
Chain2<-Chain1+1
#Chain1<-sample( 1:(nCh-1), 1)
#Chain2<-Chain1+1
## allow non-adjacent chains
#Chain1<-sample( c(1:nCh), 1)
#Chain2<-sample(c(1:nCh)[c(1:nCh)!=Chain1],1)
# check ratio
MHratio<- parallelAccept(Bigp[[Chain1]][j,], Bigp[[Chain2]][j,], rep(alphas[Chain1],k), rep(alphas[Chain2],k))
if (MHratio==1){ # switch the chains (weights, mean, sigma, Zs)
#new
.tpt1<- TrackParallelTemp[j,Chain1 ]
.tpt2<- TrackParallelTemp[j,Chain2 ]
TrackParallelTemp[j,Chain1 ]<-.tpt2
TrackParallelTemp[j,Chain2 ]<-.tpt1
# Weights
.p1<- Bigp[[Chain1]][j,]
.p2<- Bigp[[Chain2]][j,]
Bigp[[Chain1]][j,]<-.p2
Bigp[[Chain2]][j,]<-.p1
# Means
.m1<- Bigmu[[Chain1]][j,]
.m2<- Bigmu[[Chain2]][j,]
Bigmu[[Chain1]][j,]<-.m2
Bigmu[[Chain2]][j,]<-.m1
# SD
.s1<- Bigsigma[[Chain1]][j,]
.s2<- Bigsigma[[Chain2]][j,]
Bigsigma[[Chain1]][j,]<-.s2
Bigsigma[[Chain2]][j,]<-.s1
# Zs
.z1<- ZSaved[[Chain1]][,j]
.z2<- ZSaved[[Chain2]][,j]
ZSaved[[Chain1]][,j]<-.z2
ZSaved[[Chain2]][,j]<-.z1
} } }
}
#logLikelihood
for (i in 1:n){
non0id<-c(1:k)[ns > 0]
Loglike[j]<-Loglike[j]+ log( sum( Bigp[[nCh]][j,non0id]*dnorm(Y[i], mean=Bigmu[[nCh]][j,non0id], sd=sqrt(Bigsigma[[nCh]][j,non0id]))))}
SteadyScore$K0[j]<-sum(table(ZSaved[[nCh]][,j])>0)
}
close(pb)
#SteadyScore<-unlist(lapply(apply(ZSaved[[nCh]], 2, table), length))
BigRes<-list(Bigmu = Bigmu, Bigsigma=Bigsigma, Bigp = Bigp, Loglike=Loglike, Zs=ZSaved, YZ=y, SteadyScore=SteadyScore,TrackParallelTemp=TrackParallelTemp)
#SmallRes<-trimit(BigRes, nEnd=EndSize)
#return(SmallRes)
return(BigRes)
}
zoevanhavre/Zmix_devVersion2 documentation built on May 4, 2019, 11:25 p.m.
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#' Run Gibbs sampler with prior tempering
#'
#' This function ...
#' @param y, k,iter, isSim=TRUE, alphas=
#' @keywords Gibbs sampler, univariate, normal, gaussian, mixture, parallel tempering, order estimation
#' @export
#' @examples
#' #... you know...
Zmix_univ_tempered<-function(y, k,iter=5000, tau=1, isSim=TRUE, alphas= c(30, 20, 10, 5, 3, 1, 0.5, 1/2^(c(2,3,4,5,6, 8, 10, 15, 20, 30)))){
if(isSim==TRUE) {Y<-y$Y
}else{ Y<-y}
parallelAccept<-function(w1, w2, a1, a2){
w1[w1< 1e-200]<-1e-200 # truncate so super small values dont crash everyting
w2[w2< 1e-200]<-1e-200
T1<-dDirichlet(w2, a1, log=TRUE)
T2<-dDirichlet(w1, a2, log=TRUE)
B1<-dDirichlet(w1, a1, log=TRUE)
B2<-dDirichlet(w2, a2, log=TRUE)
MH<-min(1, exp(T1+T2-B1-B2))
Ax<-sample(c(1,0), 1, prob=c(MH,1-MH))
return(Ax)}
nCh<-length(alphas)
TrackParallelTemp<-matrix(nrow=iter, ncol=nCh)
TrackParallelTemp[1,]<-c(1:nCh)
#tau=1
n <-length(Y)
a=2.5; b=2/var(Y)
d<-2
lambda=sum(Y)/n
# 1. set up priors
mux<-list(mu=seq(from=min(Y), to=max(Y),length.out=k),sigma=rep(1, k),p=rep(1/k,k), k=k)
n <-length(Y)
a=2.5; b<-0.5*var(Y);d<-2
lambda<-sum(Y)/n ;
pb <- txtProgressBar(min = 0, max = iter, style = 3)
# 2. set up matrices for parameters which will be saved
map = matrix(0,nrow = iter, ncol = 1)
Loglike = matrix(0,nrow = iter, ncol = 1)
Bigmu = replicate(nCh, matrix(0,nrow = iter, ncol = k) , simplify=F)
Bigsigma=replicate(nCh, matrix(0,nrow = iter, ncol = k) , simplify=F)
Bigp = replicate(nCh, matrix(0,nrow = iter, ncol = k) , simplify=F)
Pzs = replicate(nCh, matrix(0,nrow = n, ncol = k) , simplify=F)
ZSaved= replicate(nCh, matrix(0,nrow = n, ncol = iter) , simplify=F)
SteadyScore<-data.frame("Iteration"=c(1:iter), "K0"=k)
# start chains and create inits needed for i=1
for (.ch in 1:nCh){
Bigmu[[.ch]][1,] <- mux$mu # initial value of mu's
mu0=mux$mu
Bigp[[.ch]][1,] = mux$p # inital value of p's
p0=mux$p
Bigsigma[[.ch]][1,] = mux$sigma # inits for sigma
sig0=mux$sigma }
# initialize chains: ie. iteration 1:
j<-1
# 1. Generate P(Z[i](t)=j) for j=1,...,known
# from paper, computing pzs
for (.ch in 1:nCh){
for (i in 1:n) {
Pzs[[.ch]][i,]<-(p0/sqrt(sig0))*exp(-((Y[i]-mu0)^2)/(2*sig0))
Pzs[[.ch]][i,]<-Pzs[[.ch]][i,]/sum(Pzs[[.ch]][i,]) }} # Scale to equal 1?
# 2 Make indicator matrix of assignments based on Pzs
# sample 1 of the k classes for each row by Pzs (prob)
for (.ch in 1:nCh){
Z<-matrix()
for (i in 1:n){Z[i]=sample((1:k),1, prob=Pzs[[.ch]][i,])}
matk = matrix((1:k), nrow = n, ncol = k, byrow = T)
IndiZ = (Z == matk)
ZSaved[[.ch]][,1]<-Z
# 3 compute ns and sx
ns = apply(IndiZ,2,sum)
for (i in 1:length(ns)){ if ( is.na(ns[i])) ns[i]<-0 }
sx = apply(IndiZ*Y, 2, sum)
# 4 Generate P[j](t) from dirichlet (and save)
Bigp[[.ch]][j,] = rdirichlet(m=1,par= ns+alphas[.ch])
# 5 Generate Mu's (and save)
Bigmu[[.ch]][j,]<-rnorm(k, mean=(lambda*tau+sx)/(tau+ns), sd=sqrt(Bigsigma[[.ch]][1,]/(tau+ns))) # must be sqrt as r takes in sd not var
# ybar<-sx/ns
for (i in 1:length(Bigmu[[.ch]][j,])){ if ( is.na(Bigmu[[.ch]][j,i])) Bigmu[[.ch]][j,i]<-0 }
# 6 Compute sv[j](t)
.bmu<- matrix((1:k), nrow = n, ncol = k, byrow = T)
for (t in 1:n) {.bmu[t,]<-Bigmu[[.ch]][j,]}
sv<-apply((Y*IndiZ-.bmu*IndiZ)^2, 2, sum) # changes, added /ns
# 7 Generate Sigma's (and save)
Bigsigma[[.ch]][j,]<- rinvgamma(k, a+(ns+1)/2, b+0.5*tau*(Bigmu[[.ch]][j,]-lambda)^2+0.5*sv)
}
# Log Likelihood: # Sum-n (log Sum-K ( weights x dnorm (y,thetas)))
for (i in 1:n){
non0id<-c(1:k)[ns > 0]
Loglike[j]<-Loglike[j]+ log(
sum( Bigp[[nCh]][j,non0id]*dnorm(Y[i], mean=Bigmu[[nCh]][j,non0id], sd=sqrt(Bigsigma[[nCh]][j,non0id]))))}
### now finish loop for j>2
for (j in 2:iter){
Sys.sleep(0.01)
setTxtProgressBar(pb, j)
if(j %% 10==0){
par(mfrow=c(1,3))
plot(SteadyScore$K0~SteadyScore$Iteration, main='#non-empty groups', type='l')
ts.plot(Bigp[[nCh]], main='Weights from target posterior', col=rainbow(k))
ts.plot(TrackParallelTemp[,c(nCh:1)], main='Track Parallel Tempering', col=rainbow(nCh))
Sys.sleep(0)}
for (.ch in 1:nCh){
#################
# 1. Generate P(Z[i](t)=j)
for (i in 1:n) {
Pzs[[.ch]][i,]<-(Bigp[[.ch]][j-1,]/sqrt(Bigsigma[[.ch]][j-1,]))*exp(-((Y[i]-Bigmu[[.ch]][j-1,])^2)/(2*Bigsigma[[.ch]][j-1,]))
# Scale to equal 1
Pzs[[.ch]][i,]<-Pzs[[.ch]][i,]/sum(Pzs[[.ch]][i,])
}
# 2 Make indicator matrix of assignments based on Pzs
# sample 1 of the k classes for each row by Pzs (prob)
for (i in 1:n){Z[i]=sample((1:k),1,replace=T, prob=Pzs[[.ch]][i,])}
matk = matrix((1:k), nrow = n, ncol = k, byrow = T)
#indicator
IndiZ = (Z == matk)
ZSaved[[.ch]][,j]<-Z
# 3 # compute ns and sx
ns = apply(IndiZ,2,sum)
# fix if ns=NA to =0
for (i in 1:length(ns)){
if ( is.na(ns[i])) ns[i]<-0 }
sx = apply(IndiZ*Y, 2, sum)
# 4 Generate P[j](t) from dirichlet (and save)
Bigp[[.ch]][j,] = rdirichlet(m=1,par= ns+alphas[.ch])
# 5 Generate Mu's (and save)
Bigmu[[.ch]][j,]<-rnorm(k, mean=(lambda*tau+sx)/(tau+ns), sd=sqrt(Bigsigma[[.ch]][j-1,]/(tau+ns)))
for (i in 1:length(Bigmu[[.ch]][j,])){ if ( is.na(Bigmu[[.ch]][j,i])) Bigmu[[.ch]][j,i]<-0 }
# 6 Compute sv[j](t)
.bmu<- matrix((1:k), nrow = n, ncol = k, byrow = T)
for (t in 1:n) {.bmu[t,]<-Bigmu[[.ch]][j,]}
sv<-apply((Y*IndiZ-.bmu*IndiZ)^2, 2, sum) # changes, added /ns
# 7 Generate Sigma's (and save)
Bigsigma[[.ch]][j,]<- rinvgamma(k, a+(ns+1)/2, b+0.5*tau*(Bigmu[[.ch]][j,]-lambda)^2+0.5*sv)
#calculate MAP estimates at this iteration:
#if (.ch==nCh){
#mapy[j,]<- prod(apply(Pzs[[.ch]], 2,sum))
#mapmu<-prod(dnorm(Bigmu[[.ch]][j,], mean=lambda, sd=(Bigsigma[[.ch]][j,]/tau) ))
#mapsig<- prod(dinvgamma(Bigsigma[[.ch]][j,], a,b))
#map[j,]<-prod(mapy[j,], mapmu, mapsig)}
}
#new #### PARALLEL TEMPERING MOVES ###
if(j>1) {TrackParallelTemp[j,]<-TrackParallelTemp[j-1,]} # SET para chains to previous values # how often??? lets go with probability 10% of switching at any time
if(j>20){
if( sample(c(1,0),1, 0.9)==1){ # FREQ OF TEMPERING!
#Pick chains, just chose one and the one next to it
if( j%%2==0){chainset<- c(1:(nCh-1))[c(1:(nCh-1))%%2==0] #evens
} else {chainset<- c(1:(nCh-1))[c(1:(nCh-1))%%2!=0] } #odds
for( eachChain in 1:length(chainset)){
Chain1<-chainset[eachChain]
Chain2<-Chain1+1
#Chain1<-sample( 1:(nCh-1), 1)
#Chain2<-Chain1+1
## allow non-adjacent chains
#Chain1<-sample( c(1:nCh), 1)
#Chain2<-sample(c(1:nCh)[c(1:nCh)!=Chain1],1)
# check ratio
MHratio<- parallelAccept(Bigp[[Chain1]][j,], Bigp[[Chain2]][j,], rep(alphas[Chain1],k), rep(alphas[Chain2],k))
if (MHratio==1){ # switch the chains (weights, mean, sigma, Zs)
#new
.tpt1<- TrackParallelTemp[j,Chain1 ]
.tpt2<- TrackParallelTemp[j,Chain2 ]
TrackParallelTemp[j,Chain1 ]<-.tpt2
TrackParallelTemp[j,Chain2 ]<-.tpt1
# Weights
.p1<- Bigp[[Chain1]][j,]
.p2<- Bigp[[Chain2]][j,]
Bigp[[Chain1]][j,]<-.p2
Bigp[[Chain2]][j,]<-.p1
# Means
.m1<- Bigmu[[Chain1]][j,]
.m2<- Bigmu[[Chain2]][j,]
Bigmu[[Chain1]][j,]<-.m2
Bigmu[[Chain2]][j,]<-.m1
# SD
.s1<- Bigsigma[[Chain1]][j,]
.s2<- Bigsigma[[Chain2]][j,]
Bigsigma[[Chain1]][j,]<-.s2
Bigsigma[[Chain2]][j,]<-.s1
# Zs
.z1<- ZSaved[[Chain1]][,j]
.z2<- ZSaved[[Chain2]][,j]
ZSaved[[Chain1]][,j]<-.z2
ZSaved[[Chain2]][,j]<-.z1
} } }
}
#logLikelihood
for (i in 1:n){
non0id<-c(1:k)[ns > 0]
Loglike[j]<-Loglike[j]+ log( sum( Bigp[[nCh]][j,non0id]*dnorm(Y[i], mean=Bigmu[[nCh]][j,non0id], sd=sqrt(Bigsigma[[nCh]][j,non0id]))))}
SteadyScore$K0[j]<-sum(table(ZSaved[[nCh]][,j])>0)
}
close(pb)
#SteadyScore<-unlist(lapply(apply(ZSaved[[nCh]], 2, table), length))
BigRes<-list(Bigmu = Bigmu, Bigsigma=Bigsigma, Bigp = Bigp, Loglike=Loglike, Zs=ZSaved, YZ=y, SteadyScore=SteadyScore,TrackParallelTemp=TrackParallelTemp)
#SmallRes<-trimit(BigRes, nEnd=EndSize)
#return(SmallRes)
return(BigRes)
}
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