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-#install.packages("telescope")
-#installed.packages("extraDistr")
-#
-rm(list = ls())
-#
-library("telescope")
-library("stats")
-library("extraDistr")       # Package for the "dbnbinom" probability mass function
-
-source("C:/Users/Anna SIMONI/CREST Dropbox/Anna Simoni/ANNA/Groupped_Panel_data/Simulations/Finite_Mixture/Code_for_our_paper/sampleUniNormMixture.R")
-source("C:/Users/Anna SIMONI/CREST Dropbox/Anna Simoni/ANNA/Groupped_Panel_data/Simulations/Finite_Mixture/Code_for_our_paper/priorOnK_spec.R")
-source("C:/Users/Anna SIMONI/CREST Dropbox/Anna Simoni/ANNA/Groupped_Panel_data/Simulations/Finite_Mixture/Code_for_our_paper/prior_alpha_e0.R")
-source("C:/Users/Anna SIMONI/CREST Dropbox/Anna Simoni/ANNA/Groupped_Panel_data/Simulations/Finite_Mixture/Code_for_our_paper/sample_e0_alpha.R")
-source("C:/Users/Anna SIMONI/CREST Dropbox/Anna Simoni/ANNA/Groupped_Panel_data/Simulations/Finite_Mixture/Code_for_our_paper/sampleK.R")
-source("C:/Users/Anna SIMONI/CREST Dropbox/Anna Simoni/ANNA/Groupped_Panel_data/Simulations/Finite_Mixture/Code_for_our_paper/identifyMixture.R")
-
-############################
-## (1) Set the Parameters ##
-############################
-set.seed(123)
-TT <- 3             # Number of time series observations
-N <- 100            # Number of cross-section observations
-MC <- 2               # Number of Monte Carlo iterations
-#
-sigma_Z <- 1        # standard deviation of the covariates Z
-beta <- 0           # value of the common parameter
-## Parameters of the mixture:
-K <- 3              # Number of the mixture components
-w_true <- c(1/3,1/3,1/3) # Vector of weights of each component of the mixture
-alpha_true <- c(0,5,-5)  # True values of the incidental parameter for the K mixture components
-var_u <- c(1,1,1)          # True values of the variance of the model error for the K mixture components
-## Parameters of the MCMC sampler:
-Mmax <- 150   # the maximum number of iterations
-thin <- 1       # the thinning imposed to reduce auto-correlation in the chain by only recording every `thined' observation
-burnin <- 100   # the number of burn-in iterations not recorded
-M <- Mmax/thin  # the number of recorded observations.
-
-Kmax <- 50      # the maximum number of components possible during sampling
-Kinit <- 10     # the initial number of filled components
-
-## Initial values:
-# we use initial values corresponding to equal component sizes and half of the value of `C0` for the variances.
-
-eta_0 <- rep(1/Kinit, Kinit)
-r <- 1    # dimension
-c0 <- 2 + (r-1)/2
-g0 <- 0.2 + (r-1) / 2
-
-# Initial value for beta
-beta_0 <- beta
-
-###############
-## (2) Prior ##
-###############
-# Static specification for the weights with a fixed prior on $e_0$ where the value is set to 1.
-G <- "MixStatic" 
-
-priorOn_n0 <- priorOnE0_spec("e0const", 1)        # Prior on the hyperparameter of the Dirichlet prior
-
-priorOn_K <- priorOnK_spec("BNB_143", 30)          # Prior on K. This gives the function to compute log(PMF)-log(.5).
-
-# Prior on the common coefficient beta: N(beta_prior,Omega_prior)
-beta_prior <- 0
-Omega_prior <- 1
-
-# Initialization of matrices to store the results
-theta_alpha <- array(,dim=c(M,Kmax,MC))
-theta_Sigma2 <- array(,dim=c(M,Kmax,MC))
-Beta <- matrix(,M,MC)
-Eta <- array(,dim=c(M,Kmax,MC))
-S_label <- array(,dim=c(M,N,MC))
-Nk <- array(,dim=c(M,Kmax,MC))
-K_MC <- matrix(,M,MC)
-#
-Kplus <- matrix(,M,MC) 
-Kplus_stat <- matrix(,MC,4)
-K_stat <- matrix(,MC,5)
-#
-# Estimators
-theta_alpha_hat <- matrix(,MC,Kmax)
-theta_Sigma2_hat <- matrix(,MC,Kmax)
-Eta_hat <- matrix(,MC,Kmax)
-Beta_hat <- matrix(,MC,3)
-
-#####################
-## (3) Monte Carlo ##
-#####################
-for (mc in 1:MC){
-  #print(mc)
-  cat("mc = ",mc)
-  ## (3.1) Simulate the Z and Y 
-  #
-  ZZ <- rnorm(N*TT, mean = 0, sd = sigma_Z)
-  Z <- matrix(ZZ,TT,N)
-  
-  rm(ZZ)
-  S_unif <- runif(N, min = 0, max = 1)
-  S <- matrix(,N,1)               # S is the latent allocation variable 
-  Y <- matrix(,TT,N)
-  
-  for (ii in 1:N){
-    # Fill the latent allocation variable S
-    if (S_unif[ii] >= 0 & S_unif[ii] < w_true[1]) {
-      S[ii,1] <- 1
-    } else if (S_unif[ii] >= w_true[1] & S_unif[ii] < (w_true[1] + w_true[2])) {
-      S[ii,1] <- 2
-    } else {
-      S[ii,1] <- 3
-    }
-    u <- rnorm(TT, mean = 0, sd = sqrt(var_u[S[ii]]))
-    Y[,ii] <- alpha_true[S[ii]] + beta*Z[,ii] + u
-  }
-  
-  rm(S,S_unif,u)
-  Y_mean <- colMeans(Y)
-  
-  ## (3.2) Initial values of the other parameters (not specified outside the loop):
-  # Component-specific priors on \alpha_k (i.e. N(b0,B0)) and \sigma_k^2 (i.e. IG(c_0,C_0)) following Richardson and Green (1997).
-  R <- diff(range(Y))
-  C0 <- diag(c(0.02*(R^2)), nrow = r)
-  sigma2_0 <- array(0, dim = c(1, 1, Kinit))
-  sigma2_0[1, 1, ] <- 0.5 * C0
-  G0 <- diag(10/(R^2), nrow = r)
-  B0 <- diag((R^2), nrow = r)                       # prior variance of alpha_k
-  b0 <- as.matrix((max(Y) + min(Y))/2, ncol = 1)    # prior mean of alpha_k 
-
-  ## (3.3) Initial partition of the data and initial parameter values to start the MCMC. 
-  # We use `kmeans()` to determine the initial partition $S_0$ and the initial component-specific means \mu_0.
-  
-  cl_y <- kmeans(Y_mean, centers = Kinit, nstart = 30)
-  S_0 <- cl_y$cluster
-  alpha_0 <- t(cl_y$centers)
-  
-  ## (3.3) MCMC sampling 
-  # We draw samples from the posterior using the telescoping sampler of Fruhwirth-Schnatter. 
-  estGibbs <- sampleUniNormMixture(
-    Y, Z, S_0, alpha_0, sigma2_0, eta_0, beta_0,
-    c0, g0, G0, C0, b0, B0, beta_prior, Omega_prior,
-    M, burnin, thin, Kmax,
-    G, priorOn_K, priorOn_n0)
-  
-  #The sampling function returns a named list where the sampled parameters and latent variables are contained. 
-  theta_alpha[,,mc] <- estGibbs$Mu
-  theta_Sigma2[,,mc] <- estGibbs$Sigma2
-  Beta[,mc] <- estGibbs$Beta
-  Eta[,,mc] <- estGibbs$Eta
-  S_label[,,mc] <- estGibbs$S
-  Nk[,,mc] <- estGibbs$Nk
-  K_MC[,mc] <- estGibbs$K
-  Kplus[,mc] <- estGibbs$Kp
-  nonnormpost_mode_list <- estGibbs$nonnormpost_mode_list
-  acc <- estGibbs$acc
-  e0 <- estGibbs$e0
-  alpha_Dir <- estGibbs$alpha  
-  
-  #########################################
-  ## Identification of the mixture model ##
-  #########################################
-  ## Step 1: Estimating $K_+$ and $K$
-  ## K_+ ##
-  Nk_mc <- Nk[,,mc]
-  Kplus_mc <- rowSums(Nk_mc != 0, na.rm = TRUE)  
-  p_Kplus <- tabulate(Kplus_mc, nbins = max(Kplus_mc))/M  
-  # The distribution of $K_+$ is characterized using the 1st and 3rd quartile as well as the median.
-  Kplus_stat[mc,1:3] <- quantile(Kplus_mc, probs = c(0.25, 0.5, 0.75))
-  
-  # Point estimate for K_+ by taking the mode and determine the number of MCMC draws where exactly K_+ 
-  # components were filled.
-  Kplus_hat <- which.max(p_Kplus)
-  Kplus_stat[mc,4] <- Kplus_hat
-  M0 <- sum(Kplus_mc == Kplus_hat)
-  
-  ## K ##
-  # We also determine the posterior distribution of the number of components K directly drawn using the 
-  # telescoping sampler.
-  K_mc <- K_MC[,mc]
-  p_K <- tabulate(K_mc, nbins = max(K_mc, na.rm = TRUE))/M
-  round(p_K[1:20], digits = 2)
-  
-  # The posterior mode can be determined as well as the posterior mean and quantiles of the posterior.
-  K_hat <- which.max(tabulate(K_mc, nbins = max(K_mc)))
-  K_stat[mc,4] <- K_hat
-  K_stat[mc,5] <- mean(K_mc)
-  K_stat[mc,1:3] <- quantile(K_mc, probs = c(0.25, 0.5, 0.75))
-  
-  ## Step 2: Extracting the draws with exactly \hat{K}_+ non-empty components
-  # Select those draws where the number of filled components was exactly \hat{K}_+:
-  index <- Kplus_mc == Kplus_hat
-  Nk_mc[is.na(Nk_mc)] <- 0
-  Nk_Kplus <- (Nk_mc[index, ] > 0)
-  
-  # We extract the cluster means, data cluster sizes and cluster assignments for the draws where exactly 
-  # \hat{K}_+ components were filled.
-  Mu_inter <- estGibbs$Mu[index, , , drop = FALSE]
-  Mu_Kplus <- array(0, dim = c(M0, 1, Kplus_hat)) 
-  Mu_Kplus[, 1, ] <- Mu_inter[, 1, ][Nk_Kplus]          # Here, 'Nk_Kplus' is used to subset the extracted 
-  # elements from Mu_inter. It selects specific rows  
-  # from the first column of Mu_inter based on the 
-  # indices in Nk_Kplus.
-  
-  Sigma2_inter <- estGibbs$Sigma2[index, , , drop = FALSE]
-  Sigma2_Kplus <- array(0, dim = c(M0, 1, Kplus_hat)) 
-  Sigma2_Kplus[, 1, ] <- Sigma2_inter[, 1, ][Nk_Kplus]
-  
-  Eta_inter <- estGibbs$Eta[index, ]
-  Eta_Kplus <- matrix(Eta_inter[Nk_Kplus], ncol = Kplus_hat)
-  Eta_Kplus <- sweep(Eta_Kplus, 1, rowSums(Eta_Kplus), "/")     # it normalizes the weights eta
-  
-  w <- which(index)
-  S_Kplus <- matrix(0, M0, ncol(Y))
-  for (i in seq_along(w)) {
-    m <- w[i]
-    perm_S <- rep(0, Kmax)
-    perm_S[Nk_mc[m, ] != 0] <- 1:Kplus_hat         # It assigns the sequence '1:Kplus_hat' to the positions in 
-    # perm_S where Nk[m, ] is not zero.
-    S_Kplus[i, ] <- perm_S[S_label[m, ,mc]]              # 'perm_S[S[m, ]]' indexes the vector 'perm_S' using the 
-    # values in S[m, ]. This permutes or reassigns the values in 
-    # 'S[m, ]' based on the mapping in 'perm_S'.
-  }
-  
-  ## Step 3: Clustering and relabeling of the MCMC draws in the point process representation
-  # For model identification, we cluster the draws of the means where exactly \hat{K}_+ components were 
-  # filled in the point process representation using k-means clustering. 
-  Func_init <- nonnormpost_mode_list[[Kplus_hat]]$mu  
-  identified_Kplus <- identifyMixture(
-    Mu_Kplus, Mu_Kplus, Eta_Kplus, S_Kplus, Sigma2_Kplus, Func_init)
-  
-  #A named list is returned which contains the proportion of draws where the clustering did not result in a 
-  # permutation and hence no relabeling could be performed and the draws had to be omitted.
-  identified_Kplus$non_perm_rate
-  
-  
-  ## Step 4: Estimating data cluster specific parameters and determining the final partition 
-  # The relabeled draws are also returned which can be used to determine posterior mean values for data 
-  # cluster specific parameters.
-  Mu_mean <- colMeans(identified_Kplus$Mu)
-  theta_alpha_hat[mc,1:length(Mu_mean)] <- colMeans(identified_Kplus$Mu)
-  theta_Sigma2_hat[mc,1:length(Mu_mean)] <- colMeans(identified_Kplus$Sigma2)
-  Eta_hat[mc,1:length(colMeans(identified_Kplus$Eta))] <- colMeans(identified_Kplus$Eta)
-  Beta_hat[mc,1] <- mean(Beta[,mc])
-  Beta_hat[mc,2:3] <- quantile(Beta[,mc], probs = c(0.025,0.975), na.rm =TRUE)
-  
-  rm(R, C0, sigma2_0, G0, B0, b0, cl_y, S_0, alpha_0, estGibbs, nonnormpost_mode_list, acc, e0, alpha_Dir, 
-     Kplus_mc, Nk_mc, p_Kplus, Kplus_hat, K_mc, index, Mu_inter, Mu_Kplus, Nk_Kplus, Sigma2_inter, 
-     Sigma2_Kplus, Eta_inter, Eta_Kplus, w, S_Kplus, perm_S, Func_init, identified_Kplus, Mu_mean)
-}
-
-MSE_beta = mean(Beta_hat[,1] - beta)^2
-MSE_alpha = colMeans((theta_alpha_hat[,1:3] - matrix(rep(alpha_true,MC), ncol=3, byrow=TRUE))^2)
-MSE_eta = colMeans((Eta_hat[,1:3] - matrix(rep(w_true,MC), ncol=3, byrow=TRUE))^2)
-
-