FSD_approximation.R

#require foreach and doparallel for parallel processing
if (!require("foreach")) install.packages("foreach")
library(foreach)
if (!require("doParallel")) install.packages("doParallel")
library(doParallel)

numCores <- detectCores()
#registering clusters, can set a smaller number using numCores-1 

registerDoParallel(numCores)

#require randtoolbox for random number generations
if (!require("randtoolbox")) install.packages("randtoolbox")
library(randtoolbox)
#require Rfast for faster computation
if (!require("Rfast")) install.packages("Rfast")
library(Rfast)
if (!require("gnorm")) install.packages("gnorm")
library(gnorm)
if (!require("stats")) install.packages("stats")
library(stats)
if (!require("np")) install.packages("np")
library(np)
if (!require("nloptr")) install.packages("nloptr")
library(nloptr)

factdivide<-function(n1,n2){
  decin1<-n1-floor(n1)
  if(decin1==0){decin1=1}
  decin2<-n2-floor(n2)
  if(decin2==0){decin2=1}
  n1seq<-seq(decin1,n1, by=1)
  n2seq<-seq(decin2,n2,by=1)
  all<-list(n1seq,n2seq)
  maxlen <- max(lengths(all))
  all2 <- as.data.frame(lapply(all, function(lst) c(lst, rep(1, maxlen - length(lst)))))
  division<-all2[,1]/all2[,2]
  answer<-exp(sum(log(division)))*(gamma(decin1)/gamma(decin2))
  return(answer)
}
unbiasedsd<-function (x){
  n<-length(x)
  if(n==1){
    return(1000000)
  }
  sd1<-sd(x)
  if(n==2){
    c1<-0.7978845608
  }else if (n==3){
    c1<-0.8862269255 
  }else if (n==4){
    c1<-0.9213177319 
  }else{
    c1<-sqrt(2/(n-1))*factdivide(n1=((n/2)-1),n2=(((n-1)/2)-1))
  }
  listall<-sd1*c1
  (listall)
}
correctfactor<-function (n){
  if(n==2){
    c1<-0.7978845608
  }else if (n==3){
    c1<-0.8862269255 
  }else if (n==4){
    c1<-0.9213177319 
  }else{
    c1<-sqrt(2/(n-1))*factdivide(n1=((n/2)-1),n2=(((n-1)/2)-1))
  }
  listall<-c1
  (listall)
}
#load the deterministic simulation functions of 9 common unimodal distributions
dsexp<-function (uni,scale=1) {
  sample1<-qexp(uni,rate=scale)
  sample1
}
dsRayleigh<-function (uni,scale=1) {
  sample1 <- scale * sqrt(-2 * log((uni)))
  sample1[scale <= 0] <- NaN
  rev(sample1)
}
dsnorm<-function (uni,location=0,scale=1) {
  sample1<-qnorm(uni,mean =location,sd=scale)
  sample1
}
dsLaplace<-function (uni,location=0,scale=1) {
  sample1<-location - sign(uni - 0.5) * scale * (log(2) + ifelse(uni < 0.5, log(uni), log1p(-uni)))
  sample1
}
dslogis<-function (uni,location=0,scale=1) {
  sample1<-qlogis(uni,location=location,scale=scale)
  sample1
}
dsPareto<-function (uni,shape,scale=1) {
  sample1 <- scale*((uni))^(-1/shape)
  sample1[scale <= 0] <- NaN
  sample1[shape <= 0] <- NaN
  rev(sample1)
}
dslnorm<-function (uni,location=0,scale) {
  sample1 <- qlnorm(uni,meanlog=location,sdlog = scale)
  sample1
}
dsgamma<-function (uni,shape,scale = 1) {
  sample1<-qgamma(uni,shape=shape,scale=scale)
  sample1
}
dsWeibull<-function (uni,shape, scale = 1){
  sample1<-qweibull(uni,shape=shape, scale = scale)
  sample1
}
dsgnorm<-function (uni,location,shape, scale = 1){
  sample1<-qgnorm(p=uni, mu = location, alpha = scale, beta = shape)
  sample1
}

dsbeta<-function (uni,shape1,shape2) {
  sample1<-qbeta(uni,shape1=shape1,shape2=shape2)
  sample1
}

#moments for checking the accuracy of bootstrap 
moments<-function (x){
  n<-length(x)
  m1<-mean(x)
  var1<-(sum((x - m1)^2)/n)
  tm1<-(sum((x - m1)^3)/n)
  fm1<-(sum((x - m1)^4)/(n))
  listall<-c(mean=m1,variance=var1,tm=tm1,fm=fm1)
  (listall)
}
unbiasedmoments<-function (x){
  n<-length(x)
  m1<-mean(x)
  var1<-sd(x)^2
  var2<-(sum((x - m1)^2)/n)
  tm1<-(sum((x - m1)^3)/n)*(n^2/((n-1)*(n-2)))
  fm1<-(sum((x - m1)^4)/n)
  ufm1<--3*var2^2*(2*n-3)*n/((n-1)*(n-2)*(n-3))+(n^2-2*n+3)*fm1*n/((n-1)*(n-2)*(n-3))
  listall<-c(mean=m1,variance=var1,tm=tm1,fm=ufm1)
  (listall)
}

standardizedmoments<-function (x){
  n<-length(x)
  m1<-mean(x)
  var1<-sd(x)^2
  var2<-(sum((x - m1)^2)/n)
  tm1<-(sum((x - m1)^3)/n)*(n^2/((n-1)*(n-2)))
  fm1<-(sum((x - m1)^4)/n)
  ufm1<--3*var2^2*(2*n-3)*n/((n-1)*(n-2)*(n-3))+(n^2-2*n+3)*fm1*n/((n-1)*(n-2)*(n-3))
  listall<-c(mean=m1,variance=var1,skewness=tm1/((var1)^(3/2)),kurtosis=ufm1/((var1)^(2)))
  (listall)
}

biasedmoments_expected<-function (n,targetm,targetvar,targettm,targetfm){
  m1<-targetm
  var1=targetvar/(n/(n-1))
  
  tm1<-targettm/(n^2/((n-1)*(n-2)))
  
  fm1<-(targetfm+3*(targetvar/(n/(n-1)))^2*(2*n-3)*n/((n-1)*(n-2)*(n-3)))/((n^2-2*n+3)*n/((n-1)*(n-2)*(n-3)))
  
  listall<-c(mean=m1,variance=var1,tm=tm1,fm=fm1)
  (listall)
}
round_sum_preserved<-function(x,digits=0){
  x<-x*(10^digits)
  floorx<-floor(x)
  order1<-tail(order(x-floorx),round(sum(x))-sum(floorx))
  floorx[order1]<-floorx[order1]+1
  results1<-floorx/(10^digits)
  return(results1)
}
weighted_SE<-function(x,weights){
  i <- !is.na(x)
  weights <- weights[i]
  x <- x[i]
  n_eff1 <- (sum(weights))^2/(sum(weights^2))
  wSE1<-sqrt((n_eff1/(n_eff1-1) * (sum(weights*(x-weighted.mean(x,weights))^2)/sum(weights)))/n_eff1)
  return(wSE1)
}
weighted_SD<-function(x,weights){
  i <- !is.na(x)
  weights <- weights[i]
  x <- x[i]
  SD1 =sqrt(sum(weights*(x-weighted.mean(x,weights))^2)/sum(weights))
  return(SD1)
}

beta_arithematic_sequences_process<-function(function1=NULL,expect1=NULL,expect2=NULL,samplesize=NULL,seed1=NULL,weight1=NULL){
  beta_U1<-0.547
  
  beta_n1=20.108
  beta_n2=46.761
  
  beta_ari1<-0.328
  
  beta_L1<-0.478
  beta_L2<-38.53
  
  beta_beta1<-0.369
  beta_beta2<-18.933
  
  dataframe_Gaussian<-c()

  #first calculate the expected bias based on unbiased moments
  Results1<-c(0,expect1)
  
  dataframe_Gaussian<-rbind(dataframe_Gaussian,Results1)
  
  dataframe_exp<-c()
  
  Results1<-c(0,expect2)
  
  dataframe_exp<-rbind(dataframe_exp,Results1)
  
  #the first distribution used to approximate the finite sample distribution is based on the arithmetic sequence
  
  length2<-samplesize
  uniran1<-seq(from=1/(length2+1), to=1-1/(length2+1), by=1/(length2+1))
  
  x<-dsnorm(uni=uniran1,location = 0,scale =1)
  
  sortedx<-Sort(x,descending=FALSE,partial=NULL,stable=FALSE,na.last=NULL)
  Results1<-c(1,function1(x=sortedx))
  
  dataframe_Gaussian<-rbind(dataframe_Gaussian,Results1)
  
  x<-dsexp(uni=uniran1,scale =1)
  
  sortedx<-Sort(x,descending=FALSE,partial=NULL,stable=FALSE,na.last=NULL)
  Results1<-c(1,function1(x=sortedx))
  
  dataframe_exp<-rbind(dataframe_exp,Results1)
  
  #then, beta distribution with U-shape
  
  length2<-samplesize
  uniran1<-seq(from=1/(length2+1), to=1-1/(length2+1), by=1/(length2+1))
  uniran_beta_U1<-dsbeta(uni=uniran1,shape1 =beta_U1,shape2=beta_U1)
  
  x<-dsnorm(uni=uniran_beta_U1,location = 0,scale =1)
  
  sortedx<-Sort(x,descending=FALSE,partial=NULL,stable=FALSE,na.last=NULL)
  Results1<-c(2,function1(x=sortedx))
  
  dataframe_Gaussian<-rbind(dataframe_Gaussian,Results1)
  
  x<-dsexp(uni=uniran_beta_U1,scale =1)
  
  sortedx<-Sort(x,descending=FALSE,partial=NULL,stable=FALSE,na.last=NULL)
  Results1<-c(2,function1(x=sortedx))
  
  dataframe_exp<-rbind(dataframe_exp,Results1)
  
  #then, beta distribution with n-shape, left skewed
  
  length2<-samplesize
  uniran1<-seq(from=1/(length2+1), to=1-1/(length2+1), by=1/(length2+1))
  uniran_n1<-dsbeta(uni=uniran1,shape1 =beta_n1,shape2=beta_n2)
  
  x<-dsnorm(uni=uniran_n1,location = 0,scale =1)
  
  sortedx<-Sort(x,descending=FALSE,partial=NULL,stable=FALSE,na.last=NULL)
  Results1<-c(3,function1(x=sortedx))
  
  dataframe_Gaussian<-rbind(dataframe_Gaussian,Results1)
  
  x<-dsexp(uni=uniran_n1,scale =1)
  
  sortedx<-Sort(x,descending=FALSE,partial=NULL,stable=FALSE,na.last=NULL)
  Results1<-c(3,function1(x=sortedx))
  
  dataframe_exp<-rbind(dataframe_exp,Results1)
  
  #beta distribution with n-shape, right skewed
  
  length2<-samplesize
  uniran1<-seq(from=1/(length2+1), to=1-1/(length2+1), by=1/(length2+1))
  uniran_n2<-dsbeta(uni=uniran1,shape1 =beta_n2,shape2=beta_n1)
  
  x<-dsnorm(uni=uniran_n2,location = 0,scale =1)
  
  sortedx<-Sort(x,descending=FALSE,partial=NULL,stable=FALSE,na.last=NULL)
  Results1<-c(4,function1(x=sortedx))
  
  dataframe_Gaussian<-rbind(dataframe_Gaussian,Results1)
  
  x<-dsexp(uni=uniran_n2,scale =1)
  
  sortedx<-Sort(x,descending=FALSE,partial=NULL,stable=FALSE,na.last=NULL)
  Results1<-c(4,function1(x=sortedx))
  
  dataframe_exp<-rbind(dataframe_exp,Results1)
  
  #beta distribution mixed with arithematic sequence (right)
  
  length2<-samplesize
  
  uniran1<-seq(from=1/(length2+1), to=1-1/(length2+1), by=1/(length2+1))
  
  uniran_ari1<-dsbeta(uni=uniran1,shape1 =beta_ari1,shape2=beta_ari1)
  
  uniran_beta_ari1<-c(uniran1[1:(samplesize/2)],uniran_ari1[(samplesize/2+1):samplesize])

  x<-dsnorm(uni=uniran_beta_ari1,location = 0,scale =1)
  
  sortedx<-Sort(x,descending=FALSE,partial=NULL,stable=FALSE,na.last=NULL)
  Results1<-c(5,function1(x=sortedx))
  
  dataframe_Gaussian<-rbind(dataframe_Gaussian,Results1)
  
  x<-dsexp(uni=uniran_beta_ari1,scale =1)
  
  sortedx<-Sort(x,descending=FALSE,partial=NULL,stable=FALSE,na.last=NULL)
  Results1<-c(5,function1(x=sortedx))
  
  dataframe_exp<-rbind(dataframe_exp,Results1)
  
  #beta distribution mixed with arithematic sequence (left)
  
  length2<-samplesize
  
  uniran1<-seq(from=1/(length2+1), to=1-1/(length2+1), by=1/(length2+1))
  
  uniran_ari1<-dsbeta(uni=uniran1,shape1 =beta_ari1,shape2=beta_ari1)
  
  uniran_beta_ari2<-c(uniran_ari1[1:(samplesize/2)],uniran1[(samplesize/2+1):samplesize])

  x<-dsnorm(uni=uniran_beta_ari2,location = 0,scale =1)
  
  sortedx<-Sort(x,descending=FALSE,partial=NULL,stable=FALSE,na.last=NULL)
  Results1<-c(6,function1(x=sortedx))
  
  dataframe_Gaussian<-rbind(dataframe_Gaussian,Results1)
  
  x<-dsexp(uni=uniran_beta_ari2,scale =1)
  
  sortedx<-Sort(x,descending=FALSE,partial=NULL,stable=FALSE,na.last=NULL)
  Results1<-c(6,function1(x=sortedx))
  
  dataframe_exp<-rbind(dataframe_exp,Results1)
  
  #beta distribution right skewed
  
  length2<-samplesize
  
  uniran1<-seq(from=1/(length2+1), to=1-1/(length2+1), by=1/(length2+1))
  
  uniran_beta_L1<-dsbeta(uni=uniran1,shape1 =beta_L1,shape2=beta_L2)
  
  x<-dsnorm(uni=uniran_beta_L1,location = 0,scale =1)
  
  sortedx<-Sort(x,descending=FALSE,partial=NULL,stable=FALSE,na.last=NULL)
  Results1<-c(7,function1(x=sortedx))
  
  dataframe_Gaussian<-rbind(dataframe_Gaussian,Results1)
  
  x<-dsexp(uni=uniran_beta_L1,scale =1)
  
  sortedx<-Sort(x,descending=FALSE,partial=NULL,stable=FALSE,na.last=NULL)
  Results1<-c(7,function1(x=sortedx))
  
  dataframe_exp<-rbind(dataframe_exp,Results1)
  
  #beta distribution left skewed
  
  length2<-samplesize
  
  uniran1<-seq(from=1/(length2+1), to=1-1/(length2+1), by=1/(length2+1))
  
  uniran_beta_L2<-dsbeta(uni=uniran1,shape1 =beta_L2,shape2=beta_L1)
  
  x<-dsnorm(uni=uniran_beta_L2,location = 0,scale =1)
  
  sortedx<-Sort(x,descending=FALSE,partial=NULL,stable=FALSE,na.last=NULL)
  Results1<-c(8,function1(x=sortedx))
  
  dataframe_Gaussian<-rbind(dataframe_Gaussian,Results1)
  
  x<-dsexp(uni=uniran_beta_L2,scale =1)
  
  sortedx<-Sort(x,descending=FALSE,partial=NULL,stable=FALSE,na.last=NULL)
  Results1<-c(8,function1(x=sortedx))
  
  dataframe_exp<-rbind(dataframe_exp,Results1)
  
  #beta distribution left skewed and right skewed mixed
  
  length2<-samplesize
  
  uniran1<-seq(from=1/(length2+1), to=1-1/(length2+1), by=1/(length2+1))
  
  uniran_beta_beta1<-dsbeta(uni=uniran1,shape1 =beta_beta1,shape2=beta_beta1)
  
  uniran_beta_beta2<-dsbeta(uni=uniran1,shape1 =beta_beta2,shape2=beta_beta2)
  
  uniran_beta_betaa<-c(uniran_beta_beta1[1:(samplesize/2)],uniran_beta_beta2[(samplesize/2+1):samplesize])

  x<-dsnorm(uni=uniran_beta_betaa,location = 0,scale =1)
  
  sortedx<-Sort(x,descending=FALSE,partial=NULL,stable=FALSE,na.last=NULL)
  Results1<-c(9,function1(x=sortedx))
  
  dataframe_Gaussian<-rbind(dataframe_Gaussian,Results1)
  
  x<-dsexp(uni=uniran_beta_betaa,scale =1)
  
  sortedx<-Sort(x,descending=FALSE,partial=NULL,stable=FALSE,na.last=NULL)
  Results1<-c(9,function1(x=sortedx))
  
  dataframe_exp<-rbind(dataframe_exp,Results1)
  
  #beta distribution left skewed and right skewed mixed
  
  length2<-samplesize
  
  uniran1<-seq(from=1/(length2+1), to=1-1/(length2+1), by=1/(length2+1))
  
  uniran_beta_beta1<-dsbeta(uni=uniran1,shape1 =beta_beta1,shape2=beta_beta1)
  
  uniran_beta_beta2<-dsbeta(uni=uniran1,shape1 =beta_beta2,shape2=beta_beta2)
  
  uniran_beta_betab<-c(uniran_beta_beta2[1:(samplesize/2)],uniran_beta_beta1[(samplesize/2+1):samplesize])

  x<-dsnorm(uni=uniran_beta_betab,location = 0,scale =1)
  
  sortedx<-Sort(x,descending=FALSE,partial=NULL,stable=FALSE,na.last=NULL)
  Results1<-c(10,function1(x=sortedx))
  
  dataframe_Gaussian<-rbind(dataframe_Gaussian,Results1)
  
  x<-dsexp(uni=uniran_beta_betab,scale =1)
  
  sortedx<-Sort(x,descending=FALSE,partial=NULL,stable=FALSE,na.last=NULL)
  Results1<-c(10,function1(x=sortedx))
  
  dataframe_exp<-rbind(dataframe_exp,Results1)
  
  #random sequence with antivariate
  
  length2<-samplesize
  set.seed(seed1)
  ran1<-runif(length2/2)
  uniran_anti<-c(ran1,1-ran1)
  
  x<-dsnorm(uni=uniran_anti,location = 0,scale =1)
  
  sortedx<-Sort(x,descending=FALSE,partial=NULL,stable=FALSE,na.last=NULL)
  Results1<-c(11,function1(x=sortedx))
  
  dataframe_Gaussian<-rbind(dataframe_Gaussian,Results1)
  
  x<-dsexp(uni=uniran_anti,scale =1)
  
  sortedx<-Sort(x,descending=FALSE,partial=NULL,stable=FALSE,na.last=NULL)
  Results1<-c(11,function1(x=sortedx))
  
  dataframe_exp<-rbind(dataframe_exp,Results1)
  
  #a final complement sequence that making the overall sequences a uniform shape
  
  length2<-samplesize
  
  factor1<-(1/(weight1[1:11]))
  
  if (sum(factor1[c(factor1>1 & factor1<10000)])>0){
    weight2<-round(weight1*(max(factor1[c(factor1>1 & factor1<10000)])))
  }else{
    weight2<-rep(1,12)
  }
  
  uniranall_weighted<-c(rep(uniran1,weight2[1]),rep(uniran_beta_U1,weight2[2]),rep(uniran_n1,weight2[3]),rep(uniran_n2,weight2[4]),
                        rep(uniran_beta_ari1,weight2[5]),rep(uniran_beta_ari2,weight2[6]),rep(uniran_beta_L1,weight2[7]),rep(uniran_beta_L2,weight2[8]),rep(uniran_beta_betaa,weight2[9]),rep(uniran_beta_betab,weight2[10]),rep(uniran_anti,weight2[11]))
  
  uniranlong<-seq(from=1/(2*length2+1), to=1-1/(2*length2+1), by=1/(2*length2+1))
  
  uniranall_Table <- as.data.frame(table(cut(uniranall_weighted,seq(0,1,1/(2*length2)))))
  
  hist(uniranall_weighted)
  if (weight2[12]>0){
    length3<-(length(uniranall_weighted)+length2*weight2[12])
  }else{
    length3<-length(uniranall_weighted)*((1.1))
  }
  uniranall_Table$Freq2 <- length3/((2*length2))-uniranall_Table$Freq
  
  uniranall_Table$Freq3<-uniranall_Table$Freq2/sum(uniranall_Table$Freq2)
  
  uniranall_Table$Levels<-uniranlong
  uniranall_Table$Freq4<-uniranall_Table$Freq3*length2
  
  if (min(uniranall_Table$Freq4)<0){
    uniranall_Table$Freq5 <- (uniranall_Table$Freq4)-min(uniranall_Table$Freq4)
  }else{
    uniranall_Table$Freq5<-uniranall_Table$Freq4
  }
  
  uniranall_Table$Freq6<-uniranall_Table$Freq5/(sum(uniranall_Table$Freq5)/sum(uniranall_Table$Freq4))
  uniranall_Table$Freq7<-round_sum_preserved(uniranall_Table$Freq6)
  
  uniran_complement <- rep(uniranall_Table$Levels, uniranall_Table$Freq7)
  
  x<-dsnorm(uni=uniran_complement,location = 0,scale =1)
  
  sortedx<-Sort(x,descending=FALSE,partial=NULL,stable=FALSE,na.last=NULL)
  Results1<-c(12,function1(x=sortedx))
  
  dataframe_Gaussian<-rbind(dataframe_Gaussian,Results1)
  
  x<-dsexp(uni=uniran_complement,scale =1)
  
  sortedx<-Sort(x,descending=FALSE,partial=NULL,stable=FALSE,na.last=NULL)
  Results1<-c(12,function1(x=sortedx))
  
  dataframe_exp<-rbind(dataframe_exp,Results1)
  
  
  dataframe1<-rbind(dataframe_Gaussian,dataframe_exp)
  
  return(dataframe1)
}

eval_f <- function(x){
  return ((x[1]*dataframe_Process[1,1]+x[2]*dataframe_Process[1,2]+x[3]*dataframe_Process[1,3]+x[4]*dataframe_Process[1,4]+
             x[5]*dataframe_Process[1,5]+x[6]*dataframe_Process[1,6]+x[7]*dataframe_Process[1,7]+x[8]*dataframe_Process[1,8]
           +x[9]*dataframe_Process[1,9]+x[10]*dataframe_Process[1,10]+x[11]*dataframe_Process[1,11]+x[12]*dataframe_Process[1,12]
           -dataframe1[1,2])^2+(x[1]*dataframe_Process[2,1]+x[2]*dataframe_Process[2,2]+x[3]*dataframe_Process[2,3]+x[4]*dataframe_Process[2,4]+
                                  x[5]*dataframe_Process[2,5]+x[6]*dataframe_Process[2,6]+x[7]*dataframe_Process[2,7]+x[8]*dataframe_Process[2,8]
                                +x[9]*dataframe_Process[2,9]+x[10]*dataframe_Process[2,10]+x[11]*dataframe_Process[2,11]+x[12]*dataframe_Process[2,12]
                                -dataframe1[1,3])^2+(x[1]*dataframe_Process[3,1]+x[2]*dataframe_Process[3,2]+x[3]*dataframe_Process[3,3]+x[4]*dataframe_Process[3,4]+
                                                       x[5]*dataframe_Process[3,5]+x[6]*dataframe_Process[3,6]+x[7]*dataframe_Process[3,7]+x[8]*dataframe_Process[3,8]
                                                     +x[9]*dataframe_Process[3,9]+x[10]*dataframe_Process[3,10]+x[11]*dataframe_Process[3,11]+x[12]*dataframe_Process[3,12]
                                                     -dataframe1[1,4])^2+(x[1]*dataframe_Process[4,1]+x[2]*dataframe_Process[4,2]+x[3]*dataframe_Process[4,3]+x[4]*dataframe_Process[4,4]+
                                                                            x[5]*dataframe_Process[4,5]+x[6]*dataframe_Process[4,6]+x[7]*dataframe_Process[4,7]+x[8]*dataframe_Process[4,8]
                                                                          +x[9]*dataframe_Process[4,9]+x[10]*dataframe_Process[4,10]+x[11]*dataframe_Process[4,11]+x[12]*dataframe_Process[4,12]
                                                                          -dataframe1[1,5])^2+(x[1]*dataframe_Process[5,1]+x[2]*dataframe_Process[5,2]+x[3]*dataframe_Process[5,3]+x[4]*dataframe_Process[5,4]+
                                                                                                 x[5]*dataframe_Process[5,5]+x[6]*dataframe_Process[5,6]+x[7]*dataframe_Process[5,7]+x[8]*dataframe_Process[5,8]
                                                                                               +x[9]*dataframe_Process[5,9]+x[10]*dataframe_Process[5,10]+x[11]*dataframe_Process[5,11]+x[12]*dataframe_Process[5,12]
                                                                                               -dataframe1[14,2])^2+(x[1]*dataframe_Process[6,1]+x[2]*dataframe_Process[6,2]+x[3]*dataframe_Process[6,3]+x[4]*dataframe_Process[6,4]+
                                                                                                                       x[5]*dataframe_Process[6,5]+x[6]*dataframe_Process[6,6]+x[7]*dataframe_Process[6,7]+x[8]*dataframe_Process[6,8]
                                                                                                                     +x[9]*dataframe_Process[6,9]+x[10]*dataframe_Process[6,10]+x[11]*dataframe_Process[6,11]+x[12]*dataframe_Process[6,12]
                                                                                                                     -dataframe1[14,3])^2+(x[1]*dataframe_Process[7,1]+x[2]*dataframe_Process[7,2]+x[3]*dataframe_Process[7,3]+x[4]*dataframe_Process[7,4]+
                                                                                                                                             x[5]*dataframe_Process[7,5]+x[6]*dataframe_Process[7,6]+x[7]*dataframe_Process[7,7]+x[8]*dataframe_Process[7,8]
                                                                                                                                           +x[9]*dataframe_Process[7,9]+x[10]*dataframe_Process[7,10]+x[11]*dataframe_Process[7,11]+x[12]*dataframe_Process[7,12]
                                                                                                                                           -dataframe1[14,4])^2+(x[1]*dataframe_Process[8,1]+x[2]*dataframe_Process[8,2]+x[3]*dataframe_Process[8,3]+x[4]*dataframe_Process[8,4]+
                                                                                                                                                                   x[5]*dataframe_Process[8,5]+x[6]*dataframe_Process[8,6]+x[7]*dataframe_Process[8,7]+x[8]*dataframe_Process[8,8]
                                                                                                                                                                 +x[9]*dataframe_Process[8,9]+x[10]*dataframe_Process[8,10]+x[11]*dataframe_Process[8,11]+x[12]*dataframe_Process[8,12]
                                                                                                                                                                 -dataframe1[14,5])^2)
}
eval_grad_f <-function(x){
  return(c(2*dataframe_Process[1,1]*(x[1]*dataframe_Process[1,1]+x[2]*dataframe_Process[1,2]+x[3]*dataframe_Process[1,3]+x[4]*dataframe_Process[1,4]+
                                       x[5]*dataframe_Process[1,5]+x[6]*dataframe_Process[1,6]+x[7]*dataframe_Process[1,7]+x[8]*dataframe_Process[1,8]
                                     +x[9]*dataframe_Process[1,9]+x[10]*dataframe_Process[1,10]+x[11]*dataframe_Process[1,11]+x[12]*dataframe_Process[1,12]
                                     -dataframe1[1,2])+2*dataframe_Process[2,1]*(x[1]*dataframe_Process[2,1]+x[2]*dataframe_Process[2,2]+x[3]*dataframe_Process[2,3]+x[4]*dataframe_Process[2,4]+
                                                                                   x[5]*dataframe_Process[2,5]+x[6]*dataframe_Process[2,6]+x[7]*dataframe_Process[2,7]+x[8]*dataframe_Process[2,8]
                                                                                 +x[9]*dataframe_Process[2,9]+x[10]*dataframe_Process[2,10]+x[11]*dataframe_Process[2,11]+x[12]*dataframe_Process[2,12]
                                                                                 -dataframe1[1,3])+2*dataframe_Process[3,1]*(x[1]*dataframe_Process[3,1]+x[2]*dataframe_Process[3,2]+x[3]*dataframe_Process[3,3]+x[4]*dataframe_Process[3,4]+
                                                                                                                               x[5]*dataframe_Process[3,5]+x[6]*dataframe_Process[3,6]+x[7]*dataframe_Process[3,7]+x[8]*dataframe_Process[3,8]
                                                                                                                             +x[9]*dataframe_Process[3,9]+x[10]*dataframe_Process[3,10]+x[11]*dataframe_Process[3,11]+x[12]*dataframe_Process[3,12]
                                                                                                                             -dataframe1[1,4])+2*dataframe_Process[4,1]*(x[1]*dataframe_Process[4,1]+x[2]*dataframe_Process[4,2]+x[3]*dataframe_Process[4,3]+x[4]*dataframe_Process[4,4]+
                                                                                                                                                                           x[5]*dataframe_Process[4,5]+x[6]*dataframe_Process[4,6]+x[7]*dataframe_Process[4,7]+x[8]*dataframe_Process[4,8]
                                                                                                                                                                         +x[9]*dataframe_Process[4,9]+x[10]*dataframe_Process[4,10]+x[11]*dataframe_Process[4,11]+x[12]*dataframe_Process[4,12]
                                                                                                                                                                         -dataframe1[1,5])+2*dataframe_Process[5,1]*(x[1]*dataframe_Process[5,1]+x[2]*dataframe_Process[5,2]+x[3]*dataframe_Process[5,3]+x[4]*dataframe_Process[5,4]+
                                                                                                                                                                                                                       x[5]*dataframe_Process[5,5]+x[6]*dataframe_Process[5,6]+x[7]*dataframe_Process[5,7]+x[8]*dataframe_Process[5,8]
                                                                                                                                                                                                                     +x[9]*dataframe_Process[5,9]+x[10]*dataframe_Process[5,10]+x[11]*dataframe_Process[5,11]+x[12]*dataframe_Process[5,12]
                                                                                                                                                                                                                     -dataframe1[14,2])+2*dataframe_Process[6,1]*(x[1]*dataframe_Process[6,1]+x[2]*dataframe_Process[6,2]+x[3]*dataframe_Process[6,3]+x[4]*dataframe_Process[6,4]+
                                                                                                                                                                                                                                                                    x[5]*dataframe_Process[6,5]+x[6]*dataframe_Process[6,6]+x[7]*dataframe_Process[6,7]+x[8]*dataframe_Process[6,8]
                                                                                                                                                                                                                                                                  +x[9]*dataframe_Process[6,9]+x[10]*dataframe_Process[6,10]+x[11]*dataframe_Process[6,11]+x[12]*dataframe_Process[6,12]
                                                                                                                                                                                                                                                                  -dataframe1[14,3])+2*dataframe_Process[7,1]*(x[1]*dataframe_Process[7,1]+x[2]*dataframe_Process[7,2]+x[3]*dataframe_Process[7,3]+x[4]*dataframe_Process[7,4]+
                                                                                                                                                                                                                                                                                                                 x[5]*dataframe_Process[7,5]+x[6]*dataframe_Process[7,6]+x[7]*dataframe_Process[7,7]+x[8]*dataframe_Process[7,8]
                                                                                                                                                                                                                                                                                                               +x[9]*dataframe_Process[7,9]+x[10]*dataframe_Process[7,10]+x[11]*dataframe_Process[7,11]+x[12]*dataframe_Process[7,12]
                                                                                                                                                                                                                                                                                                               -dataframe1[14,4])+2*dataframe_Process[8,1]*(x[1]*dataframe_Process[8,1]+x[2]*dataframe_Process[8,2]+x[3]*dataframe_Process[8,3]+x[4]*dataframe_Process[8,4]+
                                                                                                                                                                                                                                                                                                                                                              x[5]*dataframe_Process[8,5]+x[6]*dataframe_Process[8,6]+x[7]*dataframe_Process[8,7]+x[8]*dataframe_Process[8,8]
                                                                                                                                                                                                                                                                                                                                                            +x[9]*dataframe_Process[8,9]+x[10]*dataframe_Process[8,10]+x[11]*dataframe_Process[8,11]+x[12]*dataframe_Process[8,12]
                                                                                                                                                                                                                                                                                                                                                            -dataframe1[14,5]),
           2*dataframe_Process[1,2]*(x[1]*dataframe_Process[1,1]+x[2]*dataframe_Process[1,2]+x[3]*dataframe_Process[1,3]+x[4]*dataframe_Process[1,4]+
                                       x[5]*dataframe_Process[1,5]+x[6]*dataframe_Process[1,6]+x[7]*dataframe_Process[1,7]+x[8]*dataframe_Process[1,8]
                                     +x[9]*dataframe_Process[1,9]+x[10]*dataframe_Process[1,10]+x[11]*dataframe_Process[1,11]+x[12]*dataframe_Process[1,12]
                                     -dataframe1[1,2])+2*dataframe_Process[2,2]*(x[1]*dataframe_Process[2,1]+x[2]*dataframe_Process[2,2]+x[3]*dataframe_Process[2,3]+x[4]*dataframe_Process[2,4]+
                                                                                   x[5]*dataframe_Process[2,5]+x[6]*dataframe_Process[2,6]+x[7]*dataframe_Process[2,7]+x[8]*dataframe_Process[2,8]
                                                                                 +x[9]*dataframe_Process[2,9]+x[10]*dataframe_Process[2,10]+x[11]*dataframe_Process[2,11]+x[12]*dataframe_Process[2,12]
                                                                                 -dataframe1[1,3])+2*dataframe_Process[3,2]*(x[1]*dataframe_Process[3,1]+x[2]*dataframe_Process[3,2]+x[3]*dataframe_Process[3,3]+x[4]*dataframe_Process[3,4]+
                                                                                                                               x[5]*dataframe_Process[3,5]+x[6]*dataframe_Process[3,6]+x[7]*dataframe_Process[3,7]+x[8]*dataframe_Process[3,8]
                                                                                                                             +x[9]*dataframe_Process[3,9]+x[10]*dataframe_Process[3,10]+x[11]*dataframe_Process[3,11]+x[12]*dataframe_Process[3,12]
                                                                                                                             -dataframe1[1,4])+2*dataframe_Process[4,2]*(x[1]*dataframe_Process[4,1]+x[2]*dataframe_Process[4,2]+x[3]*dataframe_Process[4,3]+x[4]*dataframe_Process[4,4]+
                                                                                                                                                                           x[5]*dataframe_Process[4,5]+x[6]*dataframe_Process[4,6]+x[7]*dataframe_Process[4,7]+x[8]*dataframe_Process[4,8]
                                                                                                                                                                         +x[9]*dataframe_Process[4,9]+x[10]*dataframe_Process[4,10]+x[11]*dataframe_Process[4,11]+x[12]*dataframe_Process[4,12]
                                                                                                                                                                         -dataframe1[1,5])+2*dataframe_Process[5,2]*(x[1]*dataframe_Process[5,1]+x[2]*dataframe_Process[5,2]+x[3]*dataframe_Process[5,3]+x[4]*dataframe_Process[5,4]+
                                                                                                                                                                                                                       x[5]*dataframe_Process[5,5]+x[6]*dataframe_Process[5,6]+x[7]*dataframe_Process[5,7]+x[8]*dataframe_Process[5,8]
                                                                                                                                                                                                                     +x[9]*dataframe_Process[5,9]+x[10]*dataframe_Process[5,10]+x[11]*dataframe_Process[5,11]+x[12]*dataframe_Process[5,12]
                                                                                                                                                                                                                     -dataframe1[14,2])+2*dataframe_Process[6,2]*(x[1]*dataframe_Process[6,1]+x[2]*dataframe_Process[6,2]+x[3]*dataframe_Process[6,3]+x[4]*dataframe_Process[6,4]+
                                                                                                                                                                                                                                                                    x[5]*dataframe_Process[6,5]+x[6]*dataframe_Process[6,6]+x[7]*dataframe_Process[6,7]+x[8]*dataframe_Process[6,8]
                                                                                                                                                                                                                                                                  +x[9]*dataframe_Process[6,9]+x[10]*dataframe_Process[6,10]+x[11]*dataframe_Process[6,11]+x[12]*dataframe_Process[6,12]
                                                                                                                                                                                                                                                                  -dataframe1[14,3])+2*dataframe_Process[7,2]*(x[1]*dataframe_Process[7,1]+x[2]*dataframe_Process[7,2]+x[3]*dataframe_Process[7,3]+x[4]*dataframe_Process[7,4]+
                                                                                                                                                                                                                                                                                                                 x[5]*dataframe_Process[7,5]+x[6]*dataframe_Process[7,6]+x[7]*dataframe_Process[7,7]+x[8]*dataframe_Process[7,8]
                                                                                                                                                                                                                                                                                                               +x[9]*dataframe_Process[7,9]+x[10]*dataframe_Process[7,10]+x[11]*dataframe_Process[7,11]+x[12]*dataframe_Process[7,12]
                                                                                                                                                                                                                                                                                                               -dataframe1[14,4])+2*dataframe_Process[8,2]*(x[1]*dataframe_Process[8,1]+x[2]*dataframe_Process[8,2]+x[3]*dataframe_Process[8,3]+x[4]*dataframe_Process[8,4]+
                                                                                                                                                                                                                                                                                                                                                              x[5]*dataframe_Process[8,5]+x[6]*dataframe_Process[8,6]+x[7]*dataframe_Process[8,7]+x[8]*dataframe_Process[8,8]
                                                                                                                                                                                                                                                                                                                                                            +x[9]*dataframe_Process[8,9]+x[10]*dataframe_Process[8,10]+x[11]*dataframe_Process[8,11]+x[12]*dataframe_Process[8,12]
                                                                                                                                                                                                                                                                                                                                                            -dataframe1[14,5]),
           2*dataframe_Process[1,3]*(x[1]*dataframe_Process[1,1]+x[2]*dataframe_Process[1,2]+x[3]*dataframe_Process[1,3]+x[4]*dataframe_Process[1,4]+
                                       x[5]*dataframe_Process[1,5]+x[6]*dataframe_Process[1,6]+x[7]*dataframe_Process[1,7]+x[8]*dataframe_Process[1,8]
                                     +x[9]*dataframe_Process[1,9]+x[10]*dataframe_Process[1,10]+x[11]*dataframe_Process[1,11]+x[12]*dataframe_Process[1,12]
                                     -dataframe1[1,2])+2*dataframe_Process[2,3]*(x[1]*dataframe_Process[2,1]+x[2]*dataframe_Process[2,2]+x[3]*dataframe_Process[2,3]+x[4]*dataframe_Process[2,4]+
                                                                                   x[5]*dataframe_Process[2,5]+x[6]*dataframe_Process[2,6]+x[7]*dataframe_Process[2,7]+x[8]*dataframe_Process[2,8]
                                                                                 +x[9]*dataframe_Process[2,9]+x[10]*dataframe_Process[2,10]+x[11]*dataframe_Process[2,11]+x[12]*dataframe_Process[2,12]
                                                                                 -dataframe1[1,3])+2*dataframe_Process[3,3]*(x[1]*dataframe_Process[3,1]+x[2]*dataframe_Process[3,2]+x[3]*dataframe_Process[3,3]+x[4]*dataframe_Process[3,4]+
                                                                                                                               x[5]*dataframe_Process[3,5]+x[6]*dataframe_Process[3,6]+x[7]*dataframe_Process[3,7]+x[8]*dataframe_Process[3,8]
                                                                                                                             +x[9]*dataframe_Process[3,9]+x[10]*dataframe_Process[3,10]+x[11]*dataframe_Process[3,11]+x[12]*dataframe_Process[3,12]
                                                                                                                             -dataframe1[1,4])+2*dataframe_Process[4,3]*(x[1]*dataframe_Process[4,1]+x[2]*dataframe_Process[4,2]+x[3]*dataframe_Process[4,3]+x[4]*dataframe_Process[4,4]+
                                                                                                                                                                           x[5]*dataframe_Process[4,5]+x[6]*dataframe_Process[4,6]+x[7]*dataframe_Process[4,7]+x[8]*dataframe_Process[4,8]
                                                                                                                                                                         +x[9]*dataframe_Process[4,9]+x[10]*dataframe_Process[4,10]+x[11]*dataframe_Process[4,11]+x[12]*dataframe_Process[4,12]
                                                                                                                                                                         -dataframe1[1,5])+2*dataframe_Process[5,3]*(x[1]*dataframe_Process[5,1]+x[2]*dataframe_Process[5,2]+x[3]*dataframe_Process[5,3]+x[4]*dataframe_Process[5,4]+
                                                                                                                                                                                                                       x[5]*dataframe_Process[5,5]+x[6]*dataframe_Process[5,6]+x[7]*dataframe_Process[5,7]+x[8]*dataframe_Process[5,8]
                                                                                                                                                                                                                     +x[9]*dataframe_Process[5,9]+x[10]*dataframe_Process[5,10]+x[11]*dataframe_Process[5,11]+x[12]*dataframe_Process[5,12]
                                                                                                                                                                                                                     -dataframe1[14,2])+2*dataframe_Process[6,3]*(x[1]*dataframe_Process[6,1]+x[2]*dataframe_Process[6,2]+x[3]*dataframe_Process[6,3]+x[4]*dataframe_Process[6,4]+
                                                                                                                                                                                                                                                                    x[5]*dataframe_Process[6,5]+x[6]*dataframe_Process[6,6]+x[7]*dataframe_Process[6,7]+x[8]*dataframe_Process[6,8]
                                                                                                                                                                                                                                                                  +x[9]*dataframe_Process[6,9]+x[10]*dataframe_Process[6,10]+x[11]*dataframe_Process[6,11]+x[12]*dataframe_Process[6,12]
                                                                                                                                                                                                                                                                  -dataframe1[14,3])+2*dataframe_Process[7,3]*(x[1]*dataframe_Process[7,1]+x[2]*dataframe_Process[7,2]+x[3]*dataframe_Process[7,3]+x[4]*dataframe_Process[7,4]+
                                                                                                                                                                                                                                                                                                                 x[5]*dataframe_Process[7,5]+x[6]*dataframe_Process[7,6]+x[7]*dataframe_Process[7,7]+x[8]*dataframe_Process[7,8]
                                                                                                                                                                                                                                                                                                               +x[9]*dataframe_Process[7,9]+x[10]*dataframe_Process[7,10]+x[11]*dataframe_Process[7,11]+x[12]*dataframe_Process[7,12]
                                                                                                                                                                                                                                                                                                               -dataframe1[14,4])+2*dataframe_Process[8,3]*(x[1]*dataframe_Process[8,1]+x[2]*dataframe_Process[8,2]+x[3]*dataframe_Process[8,3]+x[4]*dataframe_Process[8,4]+
                                                                                                                                                                                                                                                                                                                                                              x[5]*dataframe_Process[8,5]+x[6]*dataframe_Process[8,6]+x[7]*dataframe_Process[8,7]+x[8]*dataframe_Process[8,8]
                                                                                                                                                                                                                                                                                                                                                            +x[9]*dataframe_Process[8,9]+x[10]*dataframe_Process[8,10]+x[11]*dataframe_Process[8,11]+x[12]*dataframe_Process[8,12]
                                                                                                                                                                                                                                                                                                                                                            -dataframe1[14,5]),
           2*dataframe_Process[1,4]*(x[1]*dataframe_Process[1,1]+x[2]*dataframe_Process[1,2]+x[3]*dataframe_Process[1,3]+x[4]*dataframe_Process[1,4]+
                                       x[5]*dataframe_Process[1,5]+x[6]*dataframe_Process[1,6]+x[7]*dataframe_Process[1,7]+x[8]*dataframe_Process[1,8]
                                     +x[9]*dataframe_Process[1,9]+x[10]*dataframe_Process[1,10]+x[11]*dataframe_Process[1,11]+x[12]*dataframe_Process[1,12]
                                     -dataframe1[1,2])+2*dataframe_Process[2,4]*(x[1]*dataframe_Process[2,1]+x[2]*dataframe_Process[2,2]+x[3]*dataframe_Process[2,3]+x[4]*dataframe_Process[2,4]+
                                                                                   x[5]*dataframe_Process[2,5]+x[6]*dataframe_Process[2,6]+x[7]*dataframe_Process[2,7]+x[8]*dataframe_Process[2,8]
                                                                                 +x[9]*dataframe_Process[2,9]+x[10]*dataframe_Process[2,10]+x[11]*dataframe_Process[2,11]+x[12]*dataframe_Process[2,12]
                                                                                 -dataframe1[1,3])+2*dataframe_Process[3,4]*(x[1]*dataframe_Process[3,1]+x[2]*dataframe_Process[3,2]+x[3]*dataframe_Process[3,3]+x[4]*dataframe_Process[3,4]+
                                                                                                                               x[5]*dataframe_Process[3,5]+x[6]*dataframe_Process[3,6]+x[7]*dataframe_Process[3,7]+x[8]*dataframe_Process[3,8]
                                                                                                                             +x[9]*dataframe_Process[3,9]+x[10]*dataframe_Process[3,10]+x[11]*dataframe_Process[3,11]+x[12]*dataframe_Process[3,12]
                                                                                                                             -dataframe1[1,4])+2*dataframe_Process[4,4]*(x[1]*dataframe_Process[4,1]+x[2]*dataframe_Process[4,2]+x[3]*dataframe_Process[4,3]+x[4]*dataframe_Process[4,4]+
                                                                                                                                                                           x[5]*dataframe_Process[4,5]+x[6]*dataframe_Process[4,6]+x[7]*dataframe_Process[4,7]+x[8]*dataframe_Process[4,8]
                                                                                                                                                                         +x[9]*dataframe_Process[4,9]+x[10]*dataframe_Process[4,10]+x[11]*dataframe_Process[4,11]+x[12]*dataframe_Process[4,12]
                                                                                                                                                                         -dataframe1[1,5])+2*dataframe_Process[5,4]*(x[1]*dataframe_Process[5,1]+x[2]*dataframe_Process[5,2]+x[3]*dataframe_Process[5,3]+x[4]*dataframe_Process[5,4]+
                                                                                                                                                                                                                       x[5]*dataframe_Process[5,5]+x[6]*dataframe_Process[5,6]+x[7]*dataframe_Process[5,7]+x[8]*dataframe_Process[5,8]
                                                                                                                                                                                                                     +x[9]*dataframe_Process[5,9]+x[10]*dataframe_Process[5,10]+x[11]*dataframe_Process[5,11]+x[12]*dataframe_Process[5,12]
                                                                                                                                                                                                                     -dataframe1[14,2])+2*dataframe_Process[6,4]*(x[1]*dataframe_Process[6,1]+x[2]*dataframe_Process[6,2]+x[3]*dataframe_Process[6,3]+x[4]*dataframe_Process[6,4]+
                                                                                                                                                                                                                                                                    x[5]*dataframe_Process[6,5]+x[6]*dataframe_Process[6,6]+x[7]*dataframe_Process[6,7]+x[8]*dataframe_Process[6,8]
                                                                                                                                                                                                                                                                  +x[9]*dataframe_Process[6,9]+x[10]*dataframe_Process[6,10]+x[11]*dataframe_Process[6,11]+x[12]*dataframe_Process[6,12]
                                                                                                                                                                                                                                                                  -dataframe1[14,3])+2*dataframe_Process[7,4]*(x[1]*dataframe_Process[7,1]+x[2]*dataframe_Process[7,2]+x[3]*dataframe_Process[7,3]+x[4]*dataframe_Process[7,4]+
                                                                                                                                                                                                                                                                                                                 x[5]*dataframe_Process[7,5]+x[6]*dataframe_Process[7,6]+x[7]*dataframe_Process[7,7]+x[8]*dataframe_Process[7,8]
                                                                                                                                                                                                                                                                                                               +x[9]*dataframe_Process[7,9]+x[10]*dataframe_Process[7,10]+x[11]*dataframe_Process[7,11]+x[12]*dataframe_Process[7,12]
                                                                                                                                                                                                                                                                                                               -dataframe1[14,4])+2*dataframe_Process[8,4]*(x[1]*dataframe_Process[8,1]+x[2]*dataframe_Process[8,2]+x[3]*dataframe_Process[8,3]+x[4]*dataframe_Process[8,4]+
                                                                                                                                                                                                                                                                                                                                                              x[5]*dataframe_Process[8,5]+x[6]*dataframe_Process[8,6]+x[7]*dataframe_Process[8,7]+x[8]*dataframe_Process[8,8]
                                                                                                                                                                                                                                                                                                                                                            +x[9]*dataframe_Process[8,9]+x[10]*dataframe_Process[8,10]+x[11]*dataframe_Process[8,11]+x[12]*dataframe_Process[8,12]
                                                                                                                                                                                                                                                                                                                                                            -dataframe1[14,5]),
           2*dataframe_Process[1,5]*(x[1]*dataframe_Process[1,1]+x[2]*dataframe_Process[1,2]+x[3]*dataframe_Process[1,3]+x[4]*dataframe_Process[1,4]+
                                       x[5]*dataframe_Process[1,5]+x[6]*dataframe_Process[1,6]+x[7]*dataframe_Process[1,7]+x[8]*dataframe_Process[1,8]
                                     +x[9]*dataframe_Process[1,9]+x[10]*dataframe_Process[1,10]+x[11]*dataframe_Process[1,11]+x[12]*dataframe_Process[1,12]
                                     -dataframe1[1,2])+2*dataframe_Process[2,5]*(x[1]*dataframe_Process[2,1]+x[2]*dataframe_Process[2,2]+x[3]*dataframe_Process[2,3]+x[4]*dataframe_Process[2,4]+
                                                                                   x[5]*dataframe_Process[2,5]+x[6]*dataframe_Process[2,6]+x[7]*dataframe_Process[2,7]+x[8]*dataframe_Process[2,8]
                                                                                 +x[9]*dataframe_Process[2,9]+x[10]*dataframe_Process[2,10]+x[11]*dataframe_Process[2,11]+x[12]*dataframe_Process[2,12]
                                                                                 -dataframe1[1,3])+2*dataframe_Process[3,5]*(x[1]*dataframe_Process[3,1]+x[2]*dataframe_Process[3,2]+x[3]*dataframe_Process[3,3]+x[4]*dataframe_Process[3,4]+
                                                                                                                               x[5]*dataframe_Process[3,5]+x[6]*dataframe_Process[3,6]+x[7]*dataframe_Process[3,7]+x[8]*dataframe_Process[3,8]
                                                                                                                             +x[9]*dataframe_Process[3,9]+x[10]*dataframe_Process[3,10]+x[11]*dataframe_Process[3,11]+x[12]*dataframe_Process[3,12]
                                                                                                                             -dataframe1[1,4])+2*dataframe_Process[4,5]*(x[1]*dataframe_Process[4,1]+x[2]*dataframe_Process[4,2]+x[3]*dataframe_Process[4,3]+x[4]*dataframe_Process[4,4]+
                                                                                                                                                                           x[5]*dataframe_Process[4,5]+x[6]*dataframe_Process[4,6]+x[7]*dataframe_Process[4,7]+x[8]*dataframe_Process[4,8]
                                                                                                                                                                         +x[9]*dataframe_Process[4,9]+x[10]*dataframe_Process[4,10]+x[11]*dataframe_Process[4,11]+x[12]*dataframe_Process[4,12]
                                                                                                                                                                         -dataframe1[1,5])+2*dataframe_Process[5,5]*(x[1]*dataframe_Process[5,1]+x[2]*dataframe_Process[5,2]+x[3]*dataframe_Process[5,3]+x[4]*dataframe_Process[5,4]+
                                                                                                                                                                                                                       x[5]*dataframe_Process[5,5]+x[6]*dataframe_Process[5,6]+x[7]*dataframe_Process[5,7]+x[8]*dataframe_Process[5,8]
                                                                                                                                                                                                                     +x[9]*dataframe_Process[5,9]+x[10]*dataframe_Process[5,10]+x[11]*dataframe_Process[5,11]+x[12]*dataframe_Process[5,12]
                                                                                                                                                                                                                     -dataframe1[14,2])+2*dataframe_Process[6,5]*(x[1]*dataframe_Process[6,1]+x[2]*dataframe_Process[6,2]+x[3]*dataframe_Process[6,3]+x[4]*dataframe_Process[6,4]+
                                                                                                                                                                                                                                                                    x[5]*dataframe_Process[6,5]+x[6]*dataframe_Process[6,6]+x[7]*dataframe_Process[6,7]+x[8]*dataframe_Process[6,8]
                                                                                                                                                                                                                                                                  +x[9]*dataframe_Process[6,9]+x[10]*dataframe_Process[6,10]+x[11]*dataframe_Process[6,11]+x[12]*dataframe_Process[6,12]
                                                                                                                                                                                                                                                                  -dataframe1[14,3])+2*dataframe_Process[7,5]*(x[1]*dataframe_Process[7,1]+x[2]*dataframe_Process[7,2]+x[3]*dataframe_Process[7,3]+x[4]*dataframe_Process[7,4]+
                                                                                                                                                                                                                                                                                                                 x[5]*dataframe_Process[7,5]+x[6]*dataframe_Process[7,6]+x[7]*dataframe_Process[7,7]+x[8]*dataframe_Process[7,8]
                                                                                                                                                                                                                                                                                                               +x[9]*dataframe_Process[7,9]+x[10]*dataframe_Process[7,10]+x[11]*dataframe_Process[7,11]+x[12]*dataframe_Process[7,12]
                                                                                                                                                                                                                                                                                                               -dataframe1[14,4])+2*dataframe_Process[8,5]*(x[1]*dataframe_Process[8,1]+x[2]*dataframe_Process[8,2]+x[3]*dataframe_Process[8,3]+x[4]*dataframe_Process[8,4]+
                                                                                                                                                                                                                                                                                                                                                              x[5]*dataframe_Process[8,5]+x[6]*dataframe_Process[8,6]+x[7]*dataframe_Process[8,7]+x[8]*dataframe_Process[8,8]
                                                                                                                                                                                                                                                                                                                                                            +x[9]*dataframe_Process[8,9]+x[10]*dataframe_Process[8,10]+x[11]*dataframe_Process[8,11]+x[12]*dataframe_Process[8,12]
                                                                                                                                                                                                                                                                                                                                                            -dataframe1[14,5]),
           2*dataframe_Process[1,6]*(x[1]*dataframe_Process[1,1]+x[2]*dataframe_Process[1,2]+x[3]*dataframe_Process[1,3]+x[4]*dataframe_Process[1,4]+
                                       x[5]*dataframe_Process[1,5]+x[6]*dataframe_Process[1,6]+x[7]*dataframe_Process[1,7]+x[8]*dataframe_Process[1,8]
                                     +x[9]*dataframe_Process[1,9]+x[10]*dataframe_Process[1,10]+x[11]*dataframe_Process[1,11]+x[12]*dataframe_Process[1,12]
                                     -dataframe1[1,2])+2*dataframe_Process[2,6]*(x[1]*dataframe_Process[2,1]+x[2]*dataframe_Process[2,2]+x[3]*dataframe_Process[2,3]+x[4]*dataframe_Process[2,4]+
                                                                                   x[5]*dataframe_Process[2,5]+x[6]*dataframe_Process[2,6]+x[7]*dataframe_Process[2,7]+x[8]*dataframe_Process[2,8]
                                                                                 +x[9]*dataframe_Process[2,9]+x[10]*dataframe_Process[2,10]+x[11]*dataframe_Process[2,11]+x[12]*dataframe_Process[2,12]
                                                                                 -dataframe1[1,3])+2*dataframe_Process[3,6]*(x[1]*dataframe_Process[3,1]+x[2]*dataframe_Process[3,2]+x[3]*dataframe_Process[3,3]+x[4]*dataframe_Process[3,4]+
                                                                                                                               x[5]*dataframe_Process[3,5]+x[6]*dataframe_Process[3,6]+x[7]*dataframe_Process[3,7]+x[8]*dataframe_Process[3,8]
                                                                                                                             +x[9]*dataframe_Process[3,9]+x[10]*dataframe_Process[3,10]+x[11]*dataframe_Process[3,11]+x[12]*dataframe_Process[3,12]
                                                                                                                             -dataframe1[1,4])+2*dataframe_Process[4,6]*(x[1]*dataframe_Process[4,1]+x[2]*dataframe_Process[4,2]+x[3]*dataframe_Process[4,3]+x[4]*dataframe_Process[4,4]+
                                                                                                                                                                           x[5]*dataframe_Process[4,5]+x[6]*dataframe_Process[4,6]+x[7]*dataframe_Process[4,7]+x[8]*dataframe_Process[4,8]
                                                                                                                                                                         +x[9]*dataframe_Process[4,9]+x[10]*dataframe_Process[4,10]+x[11]*dataframe_Process[4,11]+x[12]*dataframe_Process[4,12]
                                                                                                                                                                         -dataframe1[1,5])+2*dataframe_Process[5,6]*(x[1]*dataframe_Process[5,1]+x[2]*dataframe_Process[5,2]+x[3]*dataframe_Process[5,3]+x[4]*dataframe_Process[5,4]+
                                                                                                                                                                                                                       x[5]*dataframe_Process[5,5]+x[6]*dataframe_Process[5,6]+x[7]*dataframe_Process[5,7]+x[8]*dataframe_Process[5,8]
                                                                                                                                                                                                                     +x[9]*dataframe_Process[5,9]+x[10]*dataframe_Process[5,10]+x[11]*dataframe_Process[5,11]+x[12]*dataframe_Process[5,12]
                                                                                                                                                                                                                     -dataframe1[14,2])+2*dataframe_Process[6,6]*(x[1]*dataframe_Process[6,1]+x[2]*dataframe_Process[6,2]+x[3]*dataframe_Process[6,3]+x[4]*dataframe_Process[6,4]+
                                                                                                                                                                                                                                                                    x[5]*dataframe_Process[6,5]+x[6]*dataframe_Process[6,6]+x[7]*dataframe_Process[6,7]+x[8]*dataframe_Process[6,8]
                                                                                                                                                                                                                                                                  +x[9]*dataframe_Process[6,9]+x[10]*dataframe_Process[6,10]+x[11]*dataframe_Process[6,11]+x[12]*dataframe_Process[6,12]
                                                                                                                                                                                                                                                                  -dataframe1[14,3])+2*dataframe_Process[7,6]*(x[1]*dataframe_Process[7,1]+x[2]*dataframe_Process[7,2]+x[3]*dataframe_Process[7,3]+x[4]*dataframe_Process[7,4]+
                                                                                                                                                                                                                                                                                                                 x[5]*dataframe_Process[7,5]+x[6]*dataframe_Process[7,6]+x[7]*dataframe_Process[7,7]+x[8]*dataframe_Process[7,8]
                                                                                                                                                                                                                                                                                                               +x[9]*dataframe_Process[7,9]+x[10]*dataframe_Process[7,10]+x[11]*dataframe_Process[7,11]+x[12]*dataframe_Process[7,12]
                                                                                                                                                                                                                                                                                                               -dataframe1[14,4])+2*dataframe_Process[8,6]*(x[1]*dataframe_Process[8,1]+x[2]*dataframe_Process[8,2]+x[3]*dataframe_Process[8,3]+x[4]*dataframe_Process[8,4]+
                                                                                                                                                                                                                                                                                                                                                              x[5]*dataframe_Process[8,5]+x[6]*dataframe_Process[8,6]+x[7]*dataframe_Process[8,7]+x[8]*dataframe_Process[8,8]
                                                                                                                                                                                                                                                                                                                                                            +x[9]*dataframe_Process[8,9]+x[10]*dataframe_Process[8,10]+x[11]*dataframe_Process[8,11]+x[12]*dataframe_Process[8,12]
                                                                                                                                                                                                                                                                                                                                                            -dataframe1[14,5]),
           2*dataframe_Process[1,7]*(x[1]*dataframe_Process[1,1]+x[2]*dataframe_Process[1,2]+x[3]*dataframe_Process[1,3]+x[4]*dataframe_Process[1,4]+
                                       x[5]*dataframe_Process[1,5]+x[6]*dataframe_Process[1,6]+x[7]*dataframe_Process[1,7]+x[8]*dataframe_Process[1,8]
                                     +x[9]*dataframe_Process[1,9]+x[10]*dataframe_Process[1,10]+x[11]*dataframe_Process[1,11]+x[12]*dataframe_Process[1,12]
                                     -dataframe1[1,2])+2*dataframe_Process[2,7]*(x[1]*dataframe_Process[2,1]+x[2]*dataframe_Process[2,2]+x[3]*dataframe_Process[2,3]+x[4]*dataframe_Process[2,4]+
                                                                                   x[5]*dataframe_Process[2,5]+x[6]*dataframe_Process[2,6]+x[7]*dataframe_Process[2,7]+x[8]*dataframe_Process[2,8]
                                                                                 +x[9]*dataframe_Process[2,9]+x[10]*dataframe_Process[2,10]+x[11]*dataframe_Process[2,11]+x[12]*dataframe_Process[2,12]
                                                                                 -dataframe1[1,3])+2*dataframe_Process[3,7]*(x[1]*dataframe_Process[3,1]+x[2]*dataframe_Process[3,2]+x[3]*dataframe_Process[3,3]+x[4]*dataframe_Process[3,4]+
                                                                                                                               x[5]*dataframe_Process[3,5]+x[6]*dataframe_Process[3,6]+x[7]*dataframe_Process[3,7]+x[8]*dataframe_Process[3,8]
                                                                                                                             +x[9]*dataframe_Process[3,9]+x[10]*dataframe_Process[3,10]+x[11]*dataframe_Process[3,11]+x[12]*dataframe_Process[3,12]
                                                                                                                             -dataframe1[1,4])+2*dataframe_Process[4,7]*(x[1]*dataframe_Process[4,1]+x[2]*dataframe_Process[4,2]+x[3]*dataframe_Process[4,3]+x[4]*dataframe_Process[4,4]+
                                                                                                                                                                           x[5]*dataframe_Process[4,5]+x[6]*dataframe_Process[4,6]+x[7]*dataframe_Process[4,7]+x[8]*dataframe_Process[4,8]
                                                                                                                                                                         +x[9]*dataframe_Process[4,9]+x[10]*dataframe_Process[4,10]+x[11]*dataframe_Process[4,11]+x[12]*dataframe_Process[4,12]
                                                                                                                                                                         -dataframe1[1,5])+2*dataframe_Process[5,7]*(x[1]*dataframe_Process[5,1]+x[2]*dataframe_Process[5,2]+x[3]*dataframe_Process[5,3]+x[4]*dataframe_Process[5,4]+
                                                                                                                                                                                                                       x[5]*dataframe_Process[5,5]+x[6]*dataframe_Process[5,6]+x[7]*dataframe_Process[5,7]+x[8]*dataframe_Process[5,8]
                                                                                                                                                                                                                     +x[9]*dataframe_Process[5,9]+x[10]*dataframe_Process[5,10]+x[11]*dataframe_Process[5,11]+x[12]*dataframe_Process[5,12]
                                                                                                                                                                                                                     -dataframe1[14,2])+2*dataframe_Process[6,7]*(x[1]*dataframe_Process[6,1]+x[2]*dataframe_Process[6,2]+x[3]*dataframe_Process[6,3]+x[4]*dataframe_Process[6,4]+
                                                                                                                                                                                                                                                                    x[5]*dataframe_Process[6,5]+x[6]*dataframe_Process[6,6]+x[7]*dataframe_Process[6,7]+x[8]*dataframe_Process[6,8]
                                                                                                                                                                                                                                                                  +x[9]*dataframe_Process[6,9]+x[10]*dataframe_Process[6,10]+x[11]*dataframe_Process[6,11]+x[12]*dataframe_Process[6,12]
                                                                                                                                                                                                                                                                  -dataframe1[14,3])+2*dataframe_Process[7,7]*(x[1]*dataframe_Process[7,1]+x[2]*dataframe_Process[7,2]+x[3]*dataframe_Process[7,3]+x[4]*dataframe_Process[7,4]+
                                                                                                                                                                                                                                                                                                                 x[5]*dataframe_Process[7,5]+x[6]*dataframe_Process[7,6]+x[7]*dataframe_Process[7,7]+x[8]*dataframe_Process[7,8]
                                                                                                                                                                                                                                                                                                               +x[9]*dataframe_Process[7,9]+x[10]*dataframe_Process[7,10]+x[11]*dataframe_Process[7,11]+x[12]*dataframe_Process[7,12]
                                                                                                                                                                                                                                                                                                               -dataframe1[14,4])+2*dataframe_Process[8,7]*(x[1]*dataframe_Process[8,1]+x[2]*dataframe_Process[8,2]+x[3]*dataframe_Process[8,3]+x[4]*dataframe_Process[8,4]+
                                                                                                                                                                                                                                                                                                                                                              x[5]*dataframe_Process[8,5]+x[6]*dataframe_Process[8,6]+x[7]*dataframe_Process[8,7]+x[8]*dataframe_Process[8,8]
                                                                                                                                                                                                                                                                                                                                                            +x[9]*dataframe_Process[8,9]+x[10]*dataframe_Process[8,10]+x[11]*dataframe_Process[8,11]+x[12]*dataframe_Process[8,12]
                                                                                                                                                                                                                                                                                                                                                            -dataframe1[14,5]),
           2*dataframe_Process[1,8]*(x[1]*dataframe_Process[1,1]+x[2]*dataframe_Process[1,2]+x[3]*dataframe_Process[1,3]+x[4]*dataframe_Process[1,4]+
                                       x[5]*dataframe_Process[1,5]+x[6]*dataframe_Process[1,6]+x[7]*dataframe_Process[1,7]+x[8]*dataframe_Process[1,8]
                                     +x[9]*dataframe_Process[1,9]+x[10]*dataframe_Process[1,10]+x[11]*dataframe_Process[1,11]+x[12]*dataframe_Process[1,12]
                                     -dataframe1[1,2])+2*dataframe_Process[2,8]*(x[1]*dataframe_Process[2,1]+x[2]*dataframe_Process[2,2]+x[3]*dataframe_Process[2,3]+x[4]*dataframe_Process[2,4]+
                                                                                   x[5]*dataframe_Process[2,5]+x[6]*dataframe_Process[2,6]+x[7]*dataframe_Process[2,7]+x[8]*dataframe_Process[2,8]
                                                                                 +x[9]*dataframe_Process[2,9]+x[10]*dataframe_Process[2,10]+x[11]*dataframe_Process[2,11]+x[12]*dataframe_Process[2,12]
                                                                                 -dataframe1[1,3])+2*dataframe_Process[3,8]*(x[1]*dataframe_Process[3,1]+x[2]*dataframe_Process[3,2]+x[3]*dataframe_Process[3,3]+x[4]*dataframe_Process[3,4]+
                                                                                                                               x[5]*dataframe_Process[3,5]+x[6]*dataframe_Process[3,6]+x[7]*dataframe_Process[3,7]+x[8]*dataframe_Process[3,8]
                                                                                                                             +x[9]*dataframe_Process[3,9]+x[10]*dataframe_Process[3,10]+x[11]*dataframe_Process[3,11]+x[12]*dataframe_Process[3,12]
                                                                                                                             -dataframe1[1,4])+2*dataframe_Process[4,8]*(x[1]*dataframe_Process[4,1]+x[2]*dataframe_Process[4,2]+x[3]*dataframe_Process[4,3]+x[4]*dataframe_Process[4,4]+
                                                                                                                                                                           x[5]*dataframe_Process[4,5]+x[6]*dataframe_Process[4,6]+x[7]*dataframe_Process[4,7]+x[8]*dataframe_Process[4,8]
                                                                                                                                                                         +x[9]*dataframe_Process[4,9]+x[10]*dataframe_Process[4,10]+x[11]*dataframe_Process[4,11]+x[12]*dataframe_Process[4,12]
                                                                                                                                                                         -dataframe1[1,5])+2*dataframe_Process[5,8]*(x[1]*dataframe_Process[5,1]+x[2]*dataframe_Process[5,2]+x[3]*dataframe_Process[5,3]+x[4]*dataframe_Process[5,4]+
                                                                                                                                                                                                                       x[5]*dataframe_Process[5,5]+x[6]*dataframe_Process[5,6]+x[7]*dataframe_Process[5,7]+x[8]*dataframe_Process[5,8]
                                                                                                                                                                                                                     +x[9]*dataframe_Process[5,9]+x[10]*dataframe_Process[5,10]+x[11]*dataframe_Process[5,11]+x[12]*dataframe_Process[5,12]
                                                                                                                                                                                                                     -dataframe1[14,2])+2*dataframe_Process[6,8]*(x[1]*dataframe_Process[6,1]+x[2]*dataframe_Process[6,2]+x[3]*dataframe_Process[6,3]+x[4]*dataframe_Process[6,4]+
                                                                                                                                                                                                                                                                    x[5]*dataframe_Process[6,5]+x[6]*dataframe_Process[6,6]+x[7]*dataframe_Process[6,7]+x[8]*dataframe_Process[6,8]
                                                                                                                                                                                                                                                                  +x[9]*dataframe_Process[6,9]+x[10]*dataframe_Process[6,10]+x[11]*dataframe_Process[6,11]+x[12]*dataframe_Process[6,12]
                                                                                                                                                                                                                                                                  -dataframe1[14,3])+2*dataframe_Process[7,8]*(x[1]*dataframe_Process[7,1]+x[2]*dataframe_Process[7,2]+x[3]*dataframe_Process[7,3]+x[4]*dataframe_Process[7,4]+
                                                                                                                                                                                                                                                                                                                 x[5]*dataframe_Process[7,5]+x[6]*dataframe_Process[7,6]+x[7]*dataframe_Process[7,7]+x[8]*dataframe_Process[7,8]
                                                                                                                                                                                                                                                                                                               +x[9]*dataframe_Process[7,9]+x[10]*dataframe_Process[7,10]+x[11]*dataframe_Process[7,11]+x[12]*dataframe_Process[7,12]
                                                                                                                                                                                                                                                                                                               -dataframe1[14,4])+2*dataframe_Process[8,8]*(x[1]*dataframe_Process[8,1]+x[2]*dataframe_Process[8,2]+x[3]*dataframe_Process[8,3]+x[4]*dataframe_Process[8,4]+
                                                                                                                                                                                                                                                                                                                                                              x[5]*dataframe_Process[8,5]+x[6]*dataframe_Process[8,6]+x[7]*dataframe_Process[8,7]+x[8]*dataframe_Process[8,8]
                                                                                                                                                                                                                                                                                                                                                            +x[9]*dataframe_Process[8,9]+x[10]*dataframe_Process[8,10]+x[11]*dataframe_Process[8,11]+x[12]*dataframe_Process[8,12]
                                                                                                                                                                                                                                                                                                                                                            -dataframe1[14,5]),
           2*dataframe_Process[1,9]*(x[1]*dataframe_Process[1,1]+x[2]*dataframe_Process[1,2]+x[3]*dataframe_Process[1,3]+x[4]*dataframe_Process[1,4]+
                                       x[5]*dataframe_Process[1,5]+x[6]*dataframe_Process[1,6]+x[7]*dataframe_Process[1,7]+x[8]*dataframe_Process[1,8]
                                     +x[9]*dataframe_Process[1,9]+x[10]*dataframe_Process[1,10]+x[11]*dataframe_Process[1,11]+x[12]*dataframe_Process[1,12]
                                     -dataframe1[1,2])+2*dataframe_Process[2,9]*(x[1]*dataframe_Process[2,1]+x[2]*dataframe_Process[2,2]+x[3]*dataframe_Process[2,3]+x[4]*dataframe_Process[2,4]+
                                                                                   x[5]*dataframe_Process[2,5]+x[6]*dataframe_Process[2,6]+x[7]*dataframe_Process[2,7]+x[8]*dataframe_Process[2,8]
                                                                                 +x[9]*dataframe_Process[2,9]+x[10]*dataframe_Process[2,10]+x[11]*dataframe_Process[2,11]+x[12]*dataframe_Process[2,12]
                                                                                 -dataframe1[1,3])+2*dataframe_Process[3,9]*(x[1]*dataframe_Process[3,1]+x[2]*dataframe_Process[3,2]+x[3]*dataframe_Process[3,3]+x[4]*dataframe_Process[3,4]+
                                                                                                                               x[5]*dataframe_Process[3,5]+x[6]*dataframe_Process[3,6]+x[7]*dataframe_Process[3,7]+x[8]*dataframe_Process[3,8]
                                                                                                                             +x[9]*dataframe_Process[3,9]+x[10]*dataframe_Process[3,10]+x[11]*dataframe_Process[3,11]+x[12]*dataframe_Process[3,12]
                                                                                                                             -dataframe1[1,4])+2*dataframe_Process[4,9]*(x[1]*dataframe_Process[4,1]+x[2]*dataframe_Process[4,2]+x[3]*dataframe_Process[4,3]+x[4]*dataframe_Process[4,4]+
                                                                                                                                                                           x[5]*dataframe_Process[4,5]+x[6]*dataframe_Process[4,6]+x[7]*dataframe_Process[4,7]+x[8]*dataframe_Process[4,8]
                                                                                                                                                                         +x[9]*dataframe_Process[4,9]+x[10]*dataframe_Process[4,10]+x[11]*dataframe_Process[4,11]+x[12]*dataframe_Process[4,12]
                                                                                                                                                                         -dataframe1[1,5])+2*dataframe_Process[5,9]*(x[1]*dataframe_Process[5,1]+x[2]*dataframe_Process[5,2]+x[3]*dataframe_Process[5,3]+x[4]*dataframe_Process[5,4]+
                                                                                                                                                                                                                       x[5]*dataframe_Process[5,5]+x[6]*dataframe_Process[5,6]+x[7]*dataframe_Process[5,7]+x[8]*dataframe_Process[5,8]
                                                                                                                                                                                                                     +x[9]*dataframe_Process[5,9]+x[10]*dataframe_Process[5,10]+x[11]*dataframe_Process[5,11]+x[12]*dataframe_Process[5,12]
                                                                                                                                                                                                                     -dataframe1[14,2])+2*dataframe_Process[6,9]*(x[1]*dataframe_Process[6,1]+x[2]*dataframe_Process[6,2]+x[3]*dataframe_Process[6,3]+x[4]*dataframe_Process[6,4]+
                                                                                                                                                                                                                                                                    x[5]*dataframe_Process[6,5]+x[6]*dataframe_Process[6,6]+x[7]*dataframe_Process[6,7]+x[8]*dataframe_Process[6,8]
                                                                                                                                                                                                                                                                  +x[9]*dataframe_Process[6,9]+x[10]*dataframe_Process[6,10]+x[11]*dataframe_Process[6,11]+x[12]*dataframe_Process[6,12]
                                                                                                                                                                                                                                                                  -dataframe1[14,3])+2*dataframe_Process[7,9]*(x[1]*dataframe_Process[7,1]+x[2]*dataframe_Process[7,2]+x[3]*dataframe_Process[7,3]+x[4]*dataframe_Process[7,4]+
                                                                                                                                                                                                                                                                                                                 x[5]*dataframe_Process[7,5]+x[6]*dataframe_Process[7,6]+x[7]*dataframe_Process[7,7]+x[8]*dataframe_Process[7,8]
                                                                                                                                                                                                                                                                                                               +x[9]*dataframe_Process[7,9]+x[10]*dataframe_Process[7,10]+x[11]*dataframe_Process[7,11]+x[12]*dataframe_Process[7,12]
                                                                                                                                                                                                                                                                                                               -dataframe1[14,4])+2*dataframe_Process[8,9]*(x[1]*dataframe_Process[8,1]+x[2]*dataframe_Process[8,2]+x[3]*dataframe_Process[8,3]+x[4]*dataframe_Process[8,4]+
                                                                                                                                                                                                                                                                                                                                                              x[5]*dataframe_Process[8,5]+x[6]*dataframe_Process[8,6]+x[7]*dataframe_Process[8,7]+x[8]*dataframe_Process[8,8]
                                                                                                                                                                                                                                                                                                                                                            +x[9]*dataframe_Process[8,9]+x[10]*dataframe_Process[8,10]+x[11]*dataframe_Process[8,11]+x[12]*dataframe_Process[8,12]
                                                                                                                                                                                                                                                                                                                                                            -dataframe1[14,5]),
           2*dataframe_Process[1,10]*(x[1]*dataframe_Process[1,1]+x[2]*dataframe_Process[1,2]+x[3]*dataframe_Process[1,3]+x[4]*dataframe_Process[1,4]+
                                        x[5]*dataframe_Process[1,5]+x[6]*dataframe_Process[1,6]+x[7]*dataframe_Process[1,7]+x[8]*dataframe_Process[1,8]
                                      +x[9]*dataframe_Process[1,9]+x[10]*dataframe_Process[1,10]+x[11]*dataframe_Process[1,11]+x[12]*dataframe_Process[1,12]
                                      -dataframe1[1,2])+2*dataframe_Process[2,10]*(x[1]*dataframe_Process[2,1]+x[2]*dataframe_Process[2,2]+x[3]*dataframe_Process[2,3]+x[4]*dataframe_Process[2,4]+
                                                                                     x[5]*dataframe_Process[2,5]+x[6]*dataframe_Process[2,6]+x[7]*dataframe_Process[2,7]+x[8]*dataframe_Process[2,8]
                                                                                   +x[9]*dataframe_Process[2,9]+x[10]*dataframe_Process[2,10]+x[11]*dataframe_Process[2,11]+x[12]*dataframe_Process[2,12]
                                                                                   -dataframe1[1,3])+2*dataframe_Process[3,10]*(x[1]*dataframe_Process[3,1]+x[2]*dataframe_Process[3,2]+x[3]*dataframe_Process[3,3]+x[4]*dataframe_Process[3,4]+
                                                                                                                                  x[5]*dataframe_Process[3,5]+x[6]*dataframe_Process[3,6]+x[7]*dataframe_Process[3,7]+x[8]*dataframe_Process[3,8]
                                                                                                                                +x[9]*dataframe_Process[3,9]+x[10]*dataframe_Process[3,10]+x[11]*dataframe_Process[3,11]+x[12]*dataframe_Process[3,12]
                                                                                                                                -dataframe1[1,4])+2*dataframe_Process[4,10]*(x[1]*dataframe_Process[4,1]+x[2]*dataframe_Process[4,2]+x[3]*dataframe_Process[4,3]+x[4]*dataframe_Process[4,4]+
                                                                                                                                                                               x[5]*dataframe_Process[4,5]+x[6]*dataframe_Process[4,6]+x[7]*dataframe_Process[4,7]+x[8]*dataframe_Process[4,8]
                                                                                                                                                                             +x[9]*dataframe_Process[4,9]+x[10]*dataframe_Process[4,10]+x[11]*dataframe_Process[4,11]+x[12]*dataframe_Process[4,12]
                                                                                                                                                                             -dataframe1[1,5])+2*dataframe_Process[5,10]*(x[1]*dataframe_Process[5,1]+x[2]*dataframe_Process[5,2]+x[3]*dataframe_Process[5,3]+x[4]*dataframe_Process[5,4]+
                                                                                                                                                                                                                            x[5]*dataframe_Process[5,5]+x[6]*dataframe_Process[5,6]+x[7]*dataframe_Process[5,7]+x[8]*dataframe_Process[5,8]
                                                                                                                                                                                                                          +x[9]*dataframe_Process[5,9]+x[10]*dataframe_Process[5,10]+x[11]*dataframe_Process[5,11]+x[12]*dataframe_Process[5,12]
                                                                                                                                                                                                                          -dataframe1[14,2])+2*dataframe_Process[6,10]*(x[1]*dataframe_Process[6,1]+x[2]*dataframe_Process[6,2]+x[3]*dataframe_Process[6,3]+x[4]*dataframe_Process[6,4]+
                                                                                                                                                                                                                                                                          x[5]*dataframe_Process[6,5]+x[6]*dataframe_Process[6,6]+x[7]*dataframe_Process[6,7]+x[8]*dataframe_Process[6,8]
                                                                                                                                                                                                                                                                        +x[9]*dataframe_Process[6,9]+x[10]*dataframe_Process[6,10]+x[11]*dataframe_Process[6,11]+x[12]*dataframe_Process[6,12]
                                                                                                                                                                                                                                                                        -dataframe1[14,3])+2*dataframe_Process[7,10]*(x[1]*dataframe_Process[7,1]+x[2]*dataframe_Process[7,2]+x[3]*dataframe_Process[7,3]+x[4]*dataframe_Process[7,4]+
                                                                                                                                                                                                                                                                                                                        x[5]*dataframe_Process[7,5]+x[6]*dataframe_Process[7,6]+x[7]*dataframe_Process[7,7]+x[8]*dataframe_Process[7,8]
                                                                                                                                                                                                                                                                                                                      +x[9]*dataframe_Process[7,9]+x[10]*dataframe_Process[7,10]+x[11]*dataframe_Process[7,11]+x[12]*dataframe_Process[7,12]
                                                                                                                                                                                                                                                                                                                      -dataframe1[14,4])+2*dataframe_Process[8,10]*(x[1]*dataframe_Process[8,1]+x[2]*dataframe_Process[8,2]+x[3]*dataframe_Process[8,3]+x[4]*dataframe_Process[8,4]+
                                                                                                                                                                                                                                                                                                                                                                      x[5]*dataframe_Process[8,5]+x[6]*dataframe_Process[8,6]+x[7]*dataframe_Process[8,7]+x[8]*dataframe_Process[8,8]
                                                                                                                                                                                                                                                                                                                                                                    +x[9]*dataframe_Process[8,9]+x[10]*dataframe_Process[8,10]+x[11]*dataframe_Process[8,11]+x[12]*dataframe_Process[8,12]
                                                                                                                                                                                                                                                                                                                                                                    -dataframe1[14,5]),
           2*dataframe_Process[1,11]*(x[1]*dataframe_Process[1,1]+x[2]*dataframe_Process[1,2]+x[3]*dataframe_Process[1,3]+x[4]*dataframe_Process[1,4]+
                                        x[5]*dataframe_Process[1,5]+x[6]*dataframe_Process[1,6]+x[7]*dataframe_Process[1,7]+x[8]*dataframe_Process[1,8]
                                      +x[9]*dataframe_Process[1,9]+x[10]*dataframe_Process[1,10]+x[11]*dataframe_Process[1,11]+x[12]*dataframe_Process[1,12]
                                      -dataframe1[1,2])+2*dataframe_Process[2,11]*(x[1]*dataframe_Process[2,1]+x[2]*dataframe_Process[2,2]+x[3]*dataframe_Process[2,3]+x[4]*dataframe_Process[2,4]+
                                                                                     x[5]*dataframe_Process[2,5]+x[6]*dataframe_Process[2,6]+x[7]*dataframe_Process[2,7]+x[8]*dataframe_Process[2,8]
                                                                                   +x[9]*dataframe_Process[2,9]+x[10]*dataframe_Process[2,10]+x[11]*dataframe_Process[2,11]+x[12]*dataframe_Process[2,12]
                                                                                   -dataframe1[1,3])+2*dataframe_Process[3,11]*(x[1]*dataframe_Process[3,1]+x[2]*dataframe_Process[3,2]+x[3]*dataframe_Process[3,3]+x[4]*dataframe_Process[3,4]+
                                                                                                                                  x[5]*dataframe_Process[3,5]+x[6]*dataframe_Process[3,6]+x[7]*dataframe_Process[3,7]+x[8]*dataframe_Process[3,8]
                                                                                                                                +x[9]*dataframe_Process[3,9]+x[10]*dataframe_Process[3,10]+x[11]*dataframe_Process[3,11]+x[12]*dataframe_Process[3,12]
                                                                                                                                -dataframe1[1,4])+2*dataframe_Process[4,11]*(x[1]*dataframe_Process[4,1]+x[2]*dataframe_Process[4,2]+x[3]*dataframe_Process[4,3]+x[4]*dataframe_Process[4,4]+
                                                                                                                                                                               x[5]*dataframe_Process[4,5]+x[6]*dataframe_Process[4,6]+x[7]*dataframe_Process[4,7]+x[8]*dataframe_Process[4,8]
                                                                                                                                                                             +x[9]*dataframe_Process[4,9]+x[10]*dataframe_Process[4,10]+x[11]*dataframe_Process[4,11]+x[12]*dataframe_Process[4,12]
                                                                                                                                                                             -dataframe1[1,5])+2*dataframe_Process[5,11]*(x[1]*dataframe_Process[5,1]+x[2]*dataframe_Process[5,2]+x[3]*dataframe_Process[5,3]+x[4]*dataframe_Process[5,4]+
                                                                                                                                                                                                                            x[5]*dataframe_Process[5,5]+x[6]*dataframe_Process[5,6]+x[7]*dataframe_Process[5,7]+x[8]*dataframe_Process[5,8]
                                                                                                                                                                                                                          +x[9]*dataframe_Process[5,9]+x[10]*dataframe_Process[5,10]+x[11]*dataframe_Process[5,11]+x[12]*dataframe_Process[5,12]
                                                                                                                                                                                                                          -dataframe1[14,2])+2*dataframe_Process[6,11]*(x[1]*dataframe_Process[6,1]+x[2]*dataframe_Process[6,2]+x[3]*dataframe_Process[6,3]+x[4]*dataframe_Process[6,4]+
                                                                                                                                                                                                                                                                          x[5]*dataframe_Process[6,5]+x[6]*dataframe_Process[6,6]+x[7]*dataframe_Process[6,7]+x[8]*dataframe_Process[6,8]
                                                                                                                                                                                                                                                                        +x[9]*dataframe_Process[6,9]+x[10]*dataframe_Process[6,10]+x[11]*dataframe_Process[6,11]+x[12]*dataframe_Process[6,12]
                                                                                                                                                                                                                                                                        -dataframe1[14,3])+2*dataframe_Process[7,11]*(x[1]*dataframe_Process[7,1]+x[2]*dataframe_Process[7,2]+x[3]*dataframe_Process[7,3]+x[4]*dataframe_Process[7,4]+
                                                                                                                                                                                                                                                                                                                        x[5]*dataframe_Process[7,5]+x[6]*dataframe_Process[7,6]+x[7]*dataframe_Process[7,7]+x[8]*dataframe_Process[7,8]
                                                                                                                                                                                                                                                                                                                      +x[9]*dataframe_Process[7,9]+x[10]*dataframe_Process[7,10]+x[11]*dataframe_Process[7,11]+x[12]*dataframe_Process[7,12]
                                                                                                                                                                                                                                                                                                                      -dataframe1[14,4])+2*dataframe_Process[8,11]*(x[1]*dataframe_Process[8,1]+x[2]*dataframe_Process[8,2]+x[3]*dataframe_Process[8,3]+x[4]*dataframe_Process[8,4]+
                                                                                                                                                                                                                                                                                                                                                                      x[5]*dataframe_Process[8,5]+x[6]*dataframe_Process[8,6]+x[7]*dataframe_Process[8,7]+x[8]*dataframe_Process[8,8]
                                                                                                                                                                                                                                                                                                                                                                    +x[9]*dataframe_Process[8,9]+x[10]*dataframe_Process[8,10]+x[11]*dataframe_Process[8,11]+x[12]*dataframe_Process[8,12]
                                                                                                                                                                                                                                                                                                                                                                    -dataframe1[14,5]),
           2*dataframe_Process[1,12]*(x[1]*dataframe_Process[1,1]+x[2]*dataframe_Process[1,2]+x[3]*dataframe_Process[1,3]+x[4]*dataframe_Process[1,4]+
                                        x[5]*dataframe_Process[1,5]+x[6]*dataframe_Process[1,6]+x[7]*dataframe_Process[1,7]+x[8]*dataframe_Process[1,8]
                                      +x[9]*dataframe_Process[1,9]+x[10]*dataframe_Process[1,10]+x[11]*dataframe_Process[1,11]+x[12]*dataframe_Process[1,12]
                                      -dataframe1[1,2])+2*dataframe_Process[2,12]*(x[1]*dataframe_Process[2,1]+x[2]*dataframe_Process[2,2]+x[3]*dataframe_Process[2,3]+x[4]*dataframe_Process[2,4]+
                                                                                     x[5]*dataframe_Process[2,5]+x[6]*dataframe_Process[2,6]+x[7]*dataframe_Process[2,7]+x[8]*dataframe_Process[2,8]
                                                                                   +x[9]*dataframe_Process[2,9]+x[10]*dataframe_Process[2,10]+x[11]*dataframe_Process[2,11]+x[12]*dataframe_Process[2,12]
                                                                                   -dataframe1[1,3])+2*dataframe_Process[3,12]*(x[1]*dataframe_Process[3,1]+x[2]*dataframe_Process[3,2]+x[3]*dataframe_Process[3,3]+x[4]*dataframe_Process[3,4]+
                                                                                                                                  x[5]*dataframe_Process[3,5]+x[6]*dataframe_Process[3,6]+x[7]*dataframe_Process[3,7]+x[8]*dataframe_Process[3,8]
                                                                                                                                +x[9]*dataframe_Process[3,9]+x[10]*dataframe_Process[3,10]+x[11]*dataframe_Process[3,11]+x[12]*dataframe_Process[3,12]
                                                                                                                                -dataframe1[1,4])+2*dataframe_Process[4,12]*(x[1]*dataframe_Process[4,1]+x[2]*dataframe_Process[4,2]+x[3]*dataframe_Process[4,3]+x[4]*dataframe_Process[4,4]+
                                                                                                                                                                               x[5]*dataframe_Process[4,5]+x[6]*dataframe_Process[4,6]+x[7]*dataframe_Process[4,7]+x[8]*dataframe_Process[4,8]
                                                                                                                                                                             +x[9]*dataframe_Process[4,9]+x[10]*dataframe_Process[4,10]+x[11]*dataframe_Process[4,11]+x[12]*dataframe_Process[4,12]
                                                                                                                                                                             -dataframe1[1,5])+2*dataframe_Process[5,12]*(x[1]*dataframe_Process[5,1]+x[2]*dataframe_Process[5,2]+x[3]*dataframe_Process[5,3]+x[4]*dataframe_Process[5,4]+
                                                                                                                                                                                                                            x[5]*dataframe_Process[5,5]+x[6]*dataframe_Process[5,6]+x[7]*dataframe_Process[5,7]+x[8]*dataframe_Process[5,8]
                                                                                                                                                                                                                          +x[9]*dataframe_Process[5,9]+x[10]*dataframe_Process[5,10]+x[11]*dataframe_Process[5,11]+x[12]*dataframe_Process[5,12]
                                                                                                                                                                                                                          -dataframe1[14,2])+2*dataframe_Process[6,12]*(x[1]*dataframe_Process[6,1]+x[2]*dataframe_Process[6,2]+x[3]*dataframe_Process[6,3]+x[4]*dataframe_Process[6,4]+
                                                                                                                                                                                                                                                                          x[5]*dataframe_Process[6,5]+x[6]*dataframe_Process[6,6]+x[7]*dataframe_Process[6,7]+x[8]*dataframe_Process[6,8]
                                                                                                                                                                                                                                                                        +x[9]*dataframe_Process[6,9]+x[10]*dataframe_Process[6,10]+x[11]*dataframe_Process[6,11]+x[12]*dataframe_Process[6,12]
                                                                                                                                                                                                                                                                        -dataframe1[14,3])+2*dataframe_Process[7,12]*(x[1]*dataframe_Process[7,1]+x[2]*dataframe_Process[7,2]+x[3]*dataframe_Process[7,3]+x[4]*dataframe_Process[7,4]+
                                                                                                                                                                                                                                                                                                                        x[5]*dataframe_Process[7,5]+x[6]*dataframe_Process[7,6]+x[7]*dataframe_Process[7,7]+x[8]*dataframe_Process[7,8]
                                                                                                                                                                                                                                                                                                                      +x[9]*dataframe_Process[7,9]+x[10]*dataframe_Process[7,10]+x[11]*dataframe_Process[7,11]+x[12]*dataframe_Process[7,12]
                                                                                                                                                                                                                                                                                                                      -dataframe1[14,4])+2*dataframe_Process[8,12]*(x[1]*dataframe_Process[8,1]+x[2]*dataframe_Process[8,2]+x[3]*dataframe_Process[8,3]+x[4]*dataframe_Process[8,4]+
                                                                                                                                                                                                                                                                                                                                                                      x[5]*dataframe_Process[8,5]+x[6]*dataframe_Process[8,6]+x[7]*dataframe_Process[8,7]+x[8]*dataframe_Process[8,8]
                                                                                                                                                                                                                                                                                                                                                                    +x[9]*dataframe_Process[8,9]+x[10]*dataframe_Process[8,10]+x[11]*dataframe_Process[8,11]+x[12]*dataframe_Process[8,12]
                                                                                                                                                                                                                                                                                                                                                                    -dataframe1[14,5])) )
}
eval_g_eq<-function(x){
  constr<-c((1-(x[12]+x[11]+x[1]+x[2]+x[3]+x[4]+x[5]+x[6]+x[7]+x[8]+x[9]+x[10]))^2+(x[3]-x[4])^2+(x[5]-x[6])^2+(x[7]-x[8])^2+(x[9]-x[10])^2)
  grad<-c(-2*(1-(x[12]+x[11]+x[1]+x[2]+x[3]+x[4]+x[5]+x[6]+x[7]+x[8]+x[9]+x[10])),
          -2*(1-(x[12]+x[11]+x[1]+x[2]+x[3]+x[4]+x[5]+x[6]+x[7]+x[8]+x[9]+x[10])),
          2*(x[3]-x[4])-2*(1-(x[12]+x[11]+x[1]+x[2]+x[3]+x[4]+x[5]+x[6]+x[7]+x[8]+x[9]+x[10])),
          -2*(x[3]-x[4])-2*(1-(x[12]+x[11]+x[1]+x[2]+x[3]+x[4]+x[5]+x[6]+x[7]+x[8]+x[9]+x[10])),
          2*(x[5]-x[6])-2*(1-(x[12]+x[11]+x[1]+x[2]+x[3]+x[4]+x[5]+x[6]+x[7]+x[8]+x[9]+x[10])),
          -2*(x[5]-x[6])-2*(1-(x[12]+x[11]+x[1]+x[2]+x[3]+x[4]+x[5]+x[6]+x[7]+x[8]+x[9]+x[10])),
          2*(x[7]-x[8])-2*(1-(x[12]+x[11]+x[1]+x[2]+x[3]+x[4]+x[5]+x[6]+x[7]+x[8]+x[9]+x[10])),
          -2*(x[7]-x[8])-2*(1-(x[12]+x[11]+x[1]+x[2]+x[3]+x[4]+x[5]+x[6]+x[7]+x[8]+x[9]+x[10])),
          2*(x[9]-x[10])-2*(1-(x[12]+x[11]+x[1]+x[2]+x[3]+x[4]+x[5]+x[6]+x[7]+x[8]+x[9]+x[10])),
          -2*(x[9]-x[10])-2*(1-(x[12]+x[11]+x[1]+x[2]+x[3]+x[4]+x[5]+x[6]+x[7]+x[8]+x[9]+x[10])),
          -2*(1-(x[12]+x[11]+x[1]+x[2]+x[3]+x[4]+x[5]+x[6]+x[7]+x[8]+x[9]+x[10])),
          -2*(1-(x[12]+x[11]+x[1]+x[2]+x[3]+x[4]+x[5]+x[6]+x[7]+x[8]+x[9]+x[10]))
  )
  return(list("constraints"=constr,"jacobian"=grad))
}
x0<-rep(1,12)
lb<-rep(0,12)
ub<-rep(1,12)
opts <- list("algorithm"="NLOPT_LD_SLSQP",
             "xtol_rel"=1.0e-10,"maxeval" = 1000)

simulatedbatch_Finitesample<-foreach(batchnumber =4*c(1:1000), .combine = 'rbind') %dopar% {
  library(Rfast)
  if (!require("foreach")) install.packages("foreach")
  library(foreach)
  if (!require("doParallel")) install.packages("doParallel")
  library(doParallel)
  #registering clusters, can set a smaller number using numCores-1 
  
  #require randtoolbox for random number generations
  if (!require("randtoolbox")) install.packages("randtoolbox")
  library(randtoolbox)
  #require Rfast for faster computation
  if (!require("Rfast")) install.packages("Rfast")
  library(Rfast)
  if (!require("gnorm")) install.packages("gnorm")
  library(gnorm)
  library(nloptr)
  samplesize=16
  seed1=batchnumber
  #set.seed(batchnumber)
  
  weight1<-rep(1/12,12)
  weight2<-rep(1,12)
  step1<-1
  repeat{
    
    dataframe1<-beta_arithematic_sequences_process(function1 = moments,expect1=biasedmoments_expected(n=samplesize,targetm=0,targetvar=1,targettm=0,targetfm=3),
                                                   expect2=biasedmoments_expected(n=samplesize,targetm=1,targetvar=1,targettm=2,targetfm=9),
                                                   samplesize=samplesize,seed1=seed1,weight1=weight1)
    
    
    dataframe_Process<-t(cbind(dataframe1[2:13,2:5],dataframe1[15:26,2:5]))
    
    x0<-weight2
    lb<-rep(0,12)
    ub<-rep(1,12)
    opts <- list("algorithm"="NLOPT_LD_SLSQP",
                 "xtol_rel"=1.0e-10,"maxeval" = 1000)
    
    library(nloptr)
    results1 <- nloptr( x0=x0,
                        eval_f=eval_f,
                        lb=lb,
                        ub=ub,
                        eval_grad_f=eval_grad_f,
                        eval_g_eq=eval_g_eq,
                        opts=opts)
    
    step1<-step1+1
    weight1<-results1$solution
    
    diffweights<-sum(abs(weight1-weight2))
    if (step1==10||diffweights<0.01){
      break
    }
    weight2<-weight1
  }

  c(samplesize=samplesize,seed1=seed1,objective=results1$objective,weights=results1$solution)
}

#then use the calculated weight and seed number(corresponds to the randome sequences selected) to estimate finite sample bias
simulatedbatch_Finitesample<-as.data.frame(simulatedbatch_Finitesample)

#filter the error larger than 1e^-20
simulatedbatch_Finitesample2<-simulatedbatch_Finitesample[simulatedbatch_Finitesample$objective<1e-20,]

#about 20 tries can have one with error smaller than 1e^-20.

all_Gaussian_sd<-c()
all_exp_sd<-c()
dataframe2<-c()
for(batch1 in (1:nrow(simulatedbatch_Finitesample2))){
  seed2<-as.numeric(simulatedbatch_Finitesample2[batch1,2])
  weight2<-as.numeric(simulatedbatch_Finitesample2[batch1,4:15])
  dataframe2<-beta_arithematic_sequences_process(function1=sd,expect1=0,expect2=0.723422511,samplesize=16,seed1=seed2,weight1=weight2)
  dataframe_Process_Gaussian<-(cbind(dataframe2[2:13,2],weight2))
  dataframe_Process_exp<-(cbind(dataframe2[15:26,2],weight2))
  all_Gaussian_sd<-rbind(all_Gaussian_sd,dataframe_Process_Gaussian)
  all_exp_sd<-rbind(all_exp_sd,dataframe_Process_exp)
}

weighted.mean(all_Gaussian_sd[,1],all_Gaussian_sd[,2])
weighted_SD(all_Gaussian_sd[,1],all_Gaussian_sd[,2])

#compared to the true value
correctfactor(16)
#this value is just from 39 samples!

weighted.mean(all_exp_sd[,1],all_exp_sd[,2])
weighted_SD(all_exp_sd[,1],all_exp_sd[,2])

#A Monte Carlo study comparison
sdnorm<-c()
for (i in (1:10000)){
  x<-rnorm(16)
  sdnorm<-c(sdnorm,sd(x))
}
mean(sdnorm)
sd(sdnorm)

sdexp<-c()
for (i in (1:10000)){
  x<-rexp(16)
  sdexp<-c(sdexp,sd(x))
}
mean(sdexp)
sd(sdexp)

#just 39 samples, can not only estimate the finite sample bias ,but also standard deviation of the sampling distribution!


all_Gaussian_median<-c()
all_exp_median<-c()
for(batch1 in (1:nrow(simulatedbatch_Finitesample2))){
  seed2<-as.numeric(simulatedbatch_Finitesample2[batch1,2])
  weight2<-as.numeric(simulatedbatch_Finitesample2[batch1,4:15])
  dataframe2<-beta_arithematic_sequences_process(function1=median,expect1=0,expect2=0.723422511,samplesize=16,seed1=seed2,weight1=weight2)
  dataframe_Process_Gaussian<-(cbind(dataframe2[2:13,2],weight2))
  dataframe_Process_exp<-(cbind(dataframe2[15:26,2],weight2))
  all_Gaussian_median<-rbind(all_Gaussian_median,dataframe_Process_Gaussian)
  all_exp_median<-rbind(all_exp_median,dataframe_Process_exp)
}

weighted.mean(all_Gaussian_median[,1],all_Gaussian_median[,2])
weighted_SD(all_Gaussian_median[,1],all_Gaussian_median[,2])

weighted.mean(all_exp_median[,1],all_exp_median[,2])
weighted_SD(all_exp_median[,1],all_exp_median[,2])


mediannorm<-c()
for (i in (1:10000)){
  x<-rnorm(16)
  mediannorm<-c(mediannorm,median(x))
}
mean(mediannorm)
sd(mediannorm)

medianexp<-c()
for (i in (1:10000)){
  x<-rexp(16)
  medianexp<-c(medianexp,median(x))
}
mean(medianexp)
sd(medianexp)