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Main_Monte_Carlo.R
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Main_Monte_Carlo.R
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#install.packages("telescope")
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#installed.packages("extraDistr")
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#
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rm(list = ls())
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#
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library("telescope")
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library("stats")
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library("extraDistr") # Package for the "dbnbinom" probability mass function
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source("C:/Users/Anna SIMONI/CREST Dropbox/Anna Simoni/ANNA/Groupped_Panel_data/Simulations/Finite_Mixture/Code_for_our_paper/sampleUniNormMixture.R")
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source("C:/Users/Anna SIMONI/CREST Dropbox/Anna Simoni/ANNA/Groupped_Panel_data/Simulations/Finite_Mixture/Code_for_our_paper/priorOnK_spec.R")
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source("C:/Users/Anna SIMONI/CREST Dropbox/Anna Simoni/ANNA/Groupped_Panel_data/Simulations/Finite_Mixture/Code_for_our_paper/prior_alpha_e0.R")
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source("C:/Users/Anna SIMONI/CREST Dropbox/Anna Simoni/ANNA/Groupped_Panel_data/Simulations/Finite_Mixture/Code_for_our_paper/sample_e0_alpha.R")
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source("C:/Users/Anna SIMONI/CREST Dropbox/Anna Simoni/ANNA/Groupped_Panel_data/Simulations/Finite_Mixture/Code_for_our_paper/sampleK.R")
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source("C:/Users/Anna SIMONI/CREST Dropbox/Anna Simoni/ANNA/Groupped_Panel_data/Simulations/Finite_Mixture/Code_for_our_paper/identifyMixture.R")
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############################
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## (1) Set the Parameters ##
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############################
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set.seed(123)
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TT <- 50 # Number of time series observations
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N <- 50 # Number of cross-section observations
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MC <- 50 # Number of Monte Carlo iterations
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#
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sigma_Z <- 1 # standard deviation of the covariates Z
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beta <- 0 # value of the common parameter
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## Parameters of the mixture:
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K <- 3 # Number of the mixture components
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w_true <- c(1/3,1/3,1/3) # Vector of weights of each component of the mixture
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alpha_true <- c(0,5,-5) # True values of the incidental parameter for the K mixture components
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var_u <- c(1,1,1) # True values of the variance of the model error for the K mixture components
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## Parameters of the MCMC sampler:
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Mmax <- 10000 # the maximum number of iterations
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thin <- 1 # the thinning imposed to reduce auto-correlation in the chain by only recording every `thined' observation
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burnin <- 100 # the number of burn-in iterations not recorded
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M <- Mmax/thin # the number of recorded observations.
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Kmax <- 50 # the maximum number of components possible during sampling
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Kinit <- 10 # the initial number of filled components
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## Initial values:
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# we use initial values corresponding to equal component sizes and half of the value of `C0` for the variances.
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eta_0 <- rep(1/Kinit, Kinit)
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r <- 1 # dimension
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c0 <- 2 + (r-1)/2
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g0 <- 0.2 + (r-1) / 2
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# Initial value for beta
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beta_0 <- beta
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###############
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## (2) Prior ##
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###############
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# Static specification for the weights with a fixed prior on $e_0$ where the value is set to 1.
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G <- "MixStatic"
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priorOn_n0 <- priorOnE0_spec("e0const", 1) # Prior on the hyperparameter of the Dirichlet prior
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priorOn_K <- priorOnK_spec("BNB_143", 30) # Prior on K. This gives the function to compute log(PMF)-log(.5).
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# Prior on the common coefficient beta: N(beta_prior,Omega_prior)
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beta_prior <- 0
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Omega_prior <- 1
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# Initialization of matrices to store the results
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theta_alpha <- array(,dim=c(M,Kmax,MC))
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theta_Sigma2 <- array(,dim=c(M,Kmax,MC))
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Beta <- matrix(,M,MC)
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Eta <- array(,dim=c(M,Kmax,MC))
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S_label <- array(,dim=c(M,N,MC))
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Nk <- array(,dim=c(M,Kmax,MC))
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K_MC <- matrix(,M,MC)
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#
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Kplus <- matrix(,M,MC)
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Kplus_stat <- matrix(,MC,4)
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K_stat <- matrix(,MC,5)
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#
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# Estimators
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theta_alpha_hat <- matrix(,MC,Kmax)
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theta_Sigma2_hat <- matrix(,MC,Kmax)
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Eta_hat <- matrix(,MC,Kmax)
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Beta_hat <- matrix(,MC,3)
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#####################
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## (3) Monte Carlo ##
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#####################
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for (mc in 1:MC){
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#print(mc)
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cat("mc = ",mc)
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## (3.1) Simulate the Z and Y
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#
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ZZ <- rnorm(N*TT, mean = 0, sd = sigma_Z)
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Z <- matrix(ZZ,TT,N)
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rm(ZZ)
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S_unif <- runif(N, min = 0, max = 1)
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S <- matrix(,N,1) # S is the latent allocation variable
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Y <- matrix(,TT,N)
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for (ii in 1:N){
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# Fill the latent allocation variable S
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if (S_unif[ii] >= 0 & S_unif[ii] < w_true[1]) {
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S[ii,1] <- 1
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} else if (S_unif[ii] >= w_true[1] & S_unif[ii] < (w_true[1] + w_true[2])) {
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S[ii,1] <- 2
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} else {
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S[ii,1] <- 3
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}
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u <- rnorm(TT, mean = 0, sd = sqrt(var_u[S[ii]]))
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Y[,ii] <- alpha_true[S[ii]] + beta*Z[,ii] + u
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}
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rm(S,S_unif,u)
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Y_mean <- colMeans(Y)
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## (3.2) Initial values of the other parameters (not specified outside the loop):
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# 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).
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R <- diff(range(Y))
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C0 <- diag(c(0.02*(R^2)), nrow = r)
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sigma2_0 <- array(0, dim = c(1, 1, Kinit))
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sigma2_0[1, 1, ] <- 0.5 * C0
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G0 <- diag(10/(R^2), nrow = r)
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B0 <- diag((R^2), nrow = r) # prior variance of alpha_k
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b0 <- as.matrix((max(Y) + min(Y))/2, ncol = 1) # prior mean of alpha_k
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## (3.3) Initial partition of the data and initial parameter values to start the MCMC.
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# We use `kmeans()` to determine the initial partition $S_0$ and the initial component-specific means \mu_0.
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cl_y <- kmeans(Y_mean, centers = Kinit, nstart = 30)
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S_0 <- cl_y$cluster
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alpha_0 <- t(cl_y$centers)
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## (3.3) MCMC sampling
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# We draw samples from the posterior using the telescoping sampler of Fruhwirth-Schnatter.
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estGibbs <- sampleUniNormMixture(
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Y, Z, S_0, alpha_0, sigma2_0, eta_0, beta_0,
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c0, g0, G0, C0, b0, B0, beta_prior, Omega_prior,
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M, burnin, thin, Kmax,
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G, priorOn_K, priorOn_n0)
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#The sampling function returns a named list where the sampled parameters and latent variables are contained.
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theta_alpha[,,mc] <- estGibbs$Mu
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theta_Sigma2[,,mc] <- estGibbs$Sigma2
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Beta[,mc] <- estGibbs$Beta
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Eta[,,mc] <- estGibbs$Eta
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S_label[,,mc] <- estGibbs$S
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Nk[,,mc] <- estGibbs$Nk
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K_MC[,mc] <- estGibbs$K
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Kplus[,mc] <- estGibbs$Kp
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nonnormpost_mode_list <- estGibbs$nonnormpost_mode_list
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acc <- estGibbs$acc
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e0 <- estGibbs$e0
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alpha_Dir <- estGibbs$alpha
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#########################################
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## Identification of the mixture model ##
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#########################################
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## Step 1: Estimating $K_+$ and $K$
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## K_+ ##
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Nk_mc <- Nk[,,mc]
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Kplus_mc <- rowSums(Nk_mc != 0, na.rm = TRUE)
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p_Kplus <- tabulate(Kplus_mc, nbins = max(Kplus_mc))/M
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# The distribution of $K_+$ is characterized using the 1st and 3rd quartile as well as the median.
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Kplus_stat[mc,1:3] <- quantile(Kplus_mc, probs = c(0.25, 0.5, 0.75))
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# Point estimate for K_+ by taking the mode and determine the number of MCMC draws where exactly K_+
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# components were filled.
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Kplus_hat <- which.max(p_Kplus)
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Kplus_stat[mc,4] <- Kplus_hat
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M0 <- sum(Kplus_mc == Kplus_hat)
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## K ##
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# We also determine the posterior distribution of the number of components K directly drawn using the
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# telescoping sampler.
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K_mc <- K_MC[,mc]
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p_K <- tabulate(K_mc, nbins = max(K_mc, na.rm = TRUE))/M
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round(p_K[1:20], digits = 2)
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# The posterior mode can be determined as well as the posterior mean and quantiles of the posterior.
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K_hat <- which.max(tabulate(K_mc, nbins = max(K_mc)))
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K_stat[mc,4] <- K_hat
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K_stat[mc,5] <- mean(K_mc)
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K_stat[mc,1:3] <- quantile(K_mc, probs = c(0.25, 0.5, 0.75))
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## Step 2: Extracting the draws with exactly \hat{K}_+ non-empty components
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# Select those draws where the number of filled components was exactly \hat{K}_+:
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index <- Kplus_mc == Kplus_hat
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Nk_mc[is.na(Nk_mc)] <- 0
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Nk_Kplus <- (Nk_mc[index, ] > 0)
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# We extract the cluster means, data cluster sizes and cluster assignments for the draws where exactly
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# \hat{K}_+ components were filled.
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Mu_inter <- estGibbs$Mu[index, , , drop = FALSE]
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Mu_Kplus <- array(0, dim = c(M0, 1, Kplus_hat))
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Mu_Kplus[, 1, ] <- Mu_inter[, 1, ][Nk_Kplus] # Here, 'Nk_Kplus' is used to subset the extracted
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# elements from Mu_inter. It selects specific rows
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# from the first column of Mu_inter based on the
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# indices in Nk_Kplus.
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Sigma2_inter <- estGibbs$Sigma2[index, , , drop = FALSE]
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Sigma2_Kplus <- array(0, dim = c(M0, 1, Kplus_hat))
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Sigma2_Kplus[, 1, ] <- Sigma2_inter[, 1, ][Nk_Kplus]
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Eta_inter <- estGibbs$Eta[index, ]
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Eta_Kplus <- matrix(Eta_inter[Nk_Kplus], ncol = Kplus_hat)
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Eta_Kplus <- sweep(Eta_Kplus, 1, rowSums(Eta_Kplus), "/") # it normalizes the weights eta
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w <- which(index)
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S_Kplus <- matrix(0, M0, ncol(Y))
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for (i in seq_along(w)) {
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m <- w[i]
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perm_S <- rep(0, Kmax)
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perm_S[Nk_mc[m, ] != 0] <- 1:Kplus_hat # It assigns the sequence '1:Kplus_hat' to the positions in
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# perm_S where Nk[m, ] is not zero.
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S_Kplus[i, ] <- perm_S[S_label[m, ,mc]] # 'perm_S[S[m, ]]' indexes the vector 'perm_S' using the
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# values in S[m, ]. This permutes or reassigns the values in
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# 'S[m, ]' based on the mapping in 'perm_S'.
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}
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## Step 3: Clustering and relabeling of the MCMC draws in the point process representation
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# For model identification, we cluster the draws of the means where exactly \hat{K}_+ components were
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# filled in the point process representation using k-means clustering.
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Func_init <- nonnormpost_mode_list[[Kplus_hat]]$mu
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identified_Kplus <- identifyMixture(
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Mu_Kplus, Mu_Kplus, Eta_Kplus, S_Kplus, Sigma2_Kplus, Func_init)
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#A named list is returned which contains the proportion of draws where the clustering did not result in a
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# permutation and hence no relabeling could be performed and the draws had to be omitted.
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identified_Kplus$non_perm_rate
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## Step 4: Estimating data cluster specific parameters and determining the final partition
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# The relabeled draws are also returned which can be used to determine posterior mean values for data
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# cluster specific parameters.
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Mu_mean <- colMeans(identified_Kplus$Mu)
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theta_alpha_hat[mc,1:length(Mu_mean)] <- colMeans(identified_Kplus$Mu)
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theta_Sigma2_hat[mc,1:length(Mu_mean)] <- colMeans(identified_Kplus$Sigma2)
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Eta_hat[mc,1:length(colMeans(identified_Kplus$Eta))] <- colMeans(identified_Kplus$Eta)
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Beta_hat[mc,1] <- mean(Beta[,mc])
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Beta_hat[mc,2:3] <- quantile(Beta[,mc], probs = c(0.025,0.975), na.rm =TRUE)
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rm(R, C0, sigma2_0, G0, B0, b0, cl_y, S_0, alpha_0, estGibbs, nonnormpost_mode_list, acc, e0, alpha_Dir,
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Kplus_mc, Nk_mc, p_Kplus, Kplus_hat, K_mc, index, Mu_inter, Mu_Kplus, Nk_Kplus, Sigma2_inter,
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Sigma2_Kplus, Eta_inter, Eta_Kplus, w, S_Kplus, perm_S, Func_init, identified_Kplus, Mu_mean)
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}
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theta_alpha_hat2<-t(apply(theta_alpha_hat[,1:3],1,sort))
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theta_alpha_hat_means <- colMeans(theta_alpha_hat2)
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Beta_hat_mean <- colMeans(Beta_hat)
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MSE_beta = mean((Beta_hat[,1] - beta)^2)
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MSE_alpha = colMeans((theta_alpha_hat2[,1:3] - matrix(rep(sort(alpha_true),MC), ncol=3, byrow=TRUE))^2)
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RMSE_alpha <- sqrt(MSE_alpha)
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Eta_hat2 <- t(matrix(Eta_hat[order(row(theta_alpha_hat[,1:3]), theta_alpha_hat[,1:3])], byrow = FALSE, ncol = ncol(Eta_hat)))
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Eta_hat_means <- colMeans(Eta_hat2)
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MSE_eta = colMeans((Eta_hat2[,1:3] - matrix(rep(w_true,MC), ncol=3, byrow=TRUE))^2)
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theta_Sigma2_hat2 <- t(matrix(theta_Sigma2_hat[order(row(theta_alpha_hat[,1:3]), theta_alpha_hat[,1:3])], byrow = FALSE, ncol = ncol(theta_Sigma2_hat)))
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theta_Sigma2_hat_means <- colMeans(theta_Sigma2_hat2)
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Sigma2_sorted <- var_u[order(sort(alpha_true))]
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MSE_Sigma2 = colMeans((theta_Sigma2_hat2[,1:3] - matrix(rep(Sigma2_sorted,MC), ncol=3, byrow=TRUE))^2)
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#MSE_beta = mean(Beta_hat[,1] - beta)^2
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#MSE_alpha = colMeans((theta_alpha_hat[,1:3] - matrix(rep(alpha_true,MC), ncol=3, byrow=TRUE))^2)
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#MSE_eta = colMeans((Eta_hat[,1:3] - matrix(rep(w_true,MC), ncol=3, byrow=TRUE))^2)
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