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MCMC-BASS-SC
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###################################
# 1. Continuous MCMC with Limit ###
###################################
MCMC_Limit <- function(i, t_mean, t_sd, initial, p_sd, lower, upper) {
current <- initial; sample <- vector()
if (i <= 0) {return(NA)}
for (i in 1:i) {
proposal <- current + rnorm(1, 0, p_sd)
p_density <- dnorm(proposal, t_mean, t_sd); c_density <- dnorm(current, t_mean, t_sd)
accept <- (p_density/c_density) > runif(1, min = 0, max = 1)
if(accept && proposal <= upper && proposal >= lower)
{current <- proposal; sample <- append(sample, current)}
}
if (length(sample) == 0) {return(NA)}
else {return(sample)}
}
##############################
# 2. MCMC-BASS-SC Function ###
##############################
MCMC_BASS_SC <- function(lower, upper, sample_size = 10000, anchor, t_SD, p_SD) {
standard_prob <- vector(); estimation <- matrix(0, sample_size, upper-lower+1)
for (k in lower:upper) {
total_estimate <- vector()
for (n in 1:sample_size) {
i = 0; j = 0; continue <- TRUE
sample = vector(); sample[1] <- anchor
while (continue == TRUE) {
initial <- sample[length(sample)]
temp <- MCMC_Limit(1, t_mean = k, t_sd = t_SD, initial, p_SD, lower, upper)
if (is.na(temp) == FALSE) {
sample <- append(sample, temp)
if (temp > anchor) {j = j + 1} # binomial sampling
else {i = i + 1}
continue <- decision[i+1,j+1] # cointinue<-FALSE when (i,j) hit the boundary
}
}
if (j > i) {estimate <- mean(sample[which(sample > anchor)])}
if (i > j) {estimate <- mean(sample[which(sample < anchor)])}
total_estimate <- append(total_estimate, estimate)
}
estimation[,(k-lower+1)] <- total_estimate # averge of each sample
}
colnames(estimation) <- lower:upper
estimation <- as.data.frame(estimation)
return(estimation)
}
###################
# 3. Simulation ###
###################
lower <- 31; upper <- 70; anchor <- 50.5
data_MBS.10.5 <- MCMC_BASS_SC(lower = lower, upper = upper, anchor = anchor, t_SD = 10, p_SD = 5)
data_MBS.20.5 <- MCMC_BASS_SC(lower = lower, upper = upper, anchor = anchor, t_SD = 20, p_SD = 5)
data_MBS.30.5 <- MCMC_BASS_SC(lower = lower, upper = upper, anchor = anchor, t_SD = 30, p_SD = 5)
data_MBS.10.10 <- MCMC_BASS_SC(lower = lower, upper = upper, anchor = anchor, t_SD = 10, p_SD = 10)
data_MBS.20.10 <- MCMC_BASS_SC(lower = lower, upper = upper, anchor = anchor, t_SD = 20, p_SD = 10)
data_MBS.30.10 <- MCMC_BASS_SC(lower = lower, upper = upper, anchor = anchor, t_SD = 30, p_SD = 10)
data_MBS.10.15 <- MCMC_BASS_SC(lower = lower, upper = upper, anchor = anchor, t_SD = 10, p_SD = 15)
data_MBS.20.15 <- MCMC_BASS_SC(lower = lower, upper = upper, anchor = anchor, t_SD = 20, p_SD = 15)
data_MBS.30.15 <- MCMC_BASS_SC(lower = lower, upper = upper, anchor = anchor, t_SD = 30, p_SD = 15)
################
# 4. Heatmap ###
################
MBS.10.5 <- ggplot(data_MBS.10.5, aes(x = data_MBS.10.5$"50")) + geom_density() + xlim(30,70) +
ggtitle("A: Posterior SD = 5") + geom_vline(xintercept=anchor, linetype = "dotted") +
theme_test() + xlab("Estimated Number of Dots") + ylab("Density")
MBS.20.5 <- ggplot(data_MBS.20.5, aes(x = data_MBS.20.5$"50")) + geom_density() + xlim(30,70) +
ggtitle("B: Posterior SD = 10") + geom_vline(xintercept=anchor, linetype = "dotted") +
theme_test() + xlab("Estimated Number of Dots") + ylab("Density")
MBS.30.5 <- ggplot(data_MBS.30.5, aes(x = data_MBS.30.5$"50")) + geom_density() + xlim(30,70) +
ggtitle("C: Posterior SD = 15") + geom_vline(xintercept=anchor, linetype = "dotted") +
theme_test() + xlab("Estimated Number of Dots") + ylab("Density")
grid.arrange(MBS.10.5, MBS.20.5, MBS.30.5, nrow = 1, left = "Proposal SD = 5")
MBS.10.10 <- ggplot(data_MBS.10.10, aes(x = data_MBS.10.10$"50")) + geom_density() + xlim(30,70) +
ggtitle("D: Posterior SD = 5") + geom_vline(xintercept=anchor, linetype = "dotted") +
theme_test() + xlab("Estimated Number of Dots") + ylab("Density")
MBS.20.10 <- ggplot(data_MBS.20.10, aes(x = data_MBS.20.10$"50")) + geom_density() + xlim(30,70) +
ggtitle("E: Posterior SD = 10") + geom_vline(xintercept=anchor, linetype = "dotted") +
theme_test() + xlab("Estimated Number of Dots") + ylab("Density")
MBS.30.10 <- ggplot(data_MBS.30.10, aes(x = data_MBS.30.10$"50")) + geom_density() + xlim(30,70) +
ggtitle("F: Posterior SD = 15") + geom_vline(xintercept=anchor, linetype = "dotted") +
theme_test() + xlab("Estimated Number of Dots") + ylab("Density")
grid.arrange(MBS.10.10, MBS.20.10, MBS.30.10, nrow = 1, left = "Proposal SD = 10")
MBS.10.15 <- ggplot(data_MBS.10.15, aes(x = data_MBS.10.15$"50")) + geom_density() + xlim(30,70) +
ggtitle("A: Posterior SD = 5") + geom_vline(xintercept=anchor, linetype = "dotted") +
theme_test() + xlab("Estimated Number of Dots") + ylab("Density")
MBS.20.15 <- ggplot(data_MBS.20.15, aes(x = data_MBS.20.15$"50")) + geom_density() + xlim(30,70) +
ggtitle("B: Posterior SD = 10") + geom_vline(xintercept=anchor, linetype = "dotted") +
theme_test() + xlab("Estimated Number of Dots") + ylab("Density")
MBS.30.15 <- ggplot(data_MBS.30.15, aes(x = data_MBS.30.15$"50")) + geom_density() + xlim(30,70) +
ggtitle("C: Posterior SD = 15") + geom_vline(xintercept=anchor, linetype = "dotted") +
theme_test() + xlab("Estimated Number of Dots") + ylab("Density")
grid.arrange(MBS.10.15, MBS.20.15, MBS.30.15, nrow = 1, left = "Proposal SD = 15")
###################
# 4. Additional ###
###################
data_MBS.5.3 <- MCMC_BASS_SC(lower = lower, upper = upper, anchor = anchor, t_SD = 5, p_SD = 3)
data_MBS.5.5 <- MCMC_BASS_SC(lower = lower, upper = upper, anchor = anchor, t_SD = 5, p_SD = 5)
data_MBS.5.7 <- MCMC_BASS_SC(lower = lower, upper = upper, anchor = anchor, t_SD = 5, p_SD = 7)
MBS.5.3 <- ggplot(data_MBS.5.3, aes(x = data_MBS.5.3$"50")) + geom_density() + xlim(30,70) +
ggtitle("A: Posterior SD = 3") + geom_vline(xintercept=anchor, linetype = "dotted") +
theme_test() + xlab("Estimated Number of Dots") + ylab("Density")
MBS.5.5 <- ggplot(data_MBS.5.5, aes(x = data_MBS.5.5$"50")) + geom_density() + xlim(30,70) +
ggtitle("B: Posterior SD = 5") + geom_vline(xintercept=anchor, linetype = "dotted") +
theme_test() + xlab("Estimated Number of Dots") + ylab("Density")
MBS.5.7 <- ggplot(data_MBS.5.7, aes(x = data_MBS.5.7$"50")) + geom_density() + xlim(30,70) +
ggtitle("C: Posterior SD = 7") + geom_vline(xintercept=anchor, linetype = "dotted") +
theme_test() + xlab("Estimated Number of Dots") + ylab("Density")
grid.arrange(MBS.5.3, MBS.5.5, MBS.5.7, nrow = 1, left = "Posterior SD = 5")
ave_estimate <- colMeans(data_MBC)
prob_accept <- vector()
for (i in 1:(upper-lower+1)) {
prob_accept <- append(prob_accept,length(which(data_MBC[,i]>=anchor))/length(data_MBC[,i]))
}
probability_MBC <- data.frame(Dots = 21:30, prob_accept, ave_estimate)
# Why not symmetric？
ggplot(probability_MBC, aes(Dots, prob_accept)) + geom_point() + geom_line() +
geom_hline(yintercept = 0.5, linetype="dotted") + ylim(0, 1) + ggtitle("MCMC, BASS & SC") +
geom_vline(xintercept = anchor, linetype="dotted") + xlim(lower, upper) +
xlab("True Number of Dots") + ylab("Proportion Judged More Than 25 Dots")
ggplot(probability_MBC, aes(Dots, ave_estimate)) + geom_point() + geom_line() +
xlab("True Number of Dots") + ylab("Estimated Number of Dots") + ggtitle("MCMC, BASS & SC") +
geom_abline(intercept = 0, slope = 1) + ylim(lower, upper) + xlim(lower, upper)
data_MBS.5.5 <- MCMC_BASS_SC(lower = lower, upper = upper, anchor = anchor, t_SD = 5, p_SD = 5)
data_MBS.10.5 <- MCMC_BASS_SC(lower = lower, upper = upper, anchor = anchor, t_SD = 10, p_SD = 5)
data_MBS.15.5 <- MCMC_BASS_SC(lower = lower, upper = upper, anchor = anchor, t_SD = 15, p_SD = 5)
data_MBS1 <- heat_map(data_MBS.5.5)
data_MBS2 <- heat_map(data_MBS.10.5)
data_MBS3 <- heat_map(data_MBS.15.5)
heatmap_MBS1 <- ggplot(data_MBS1, aes(x = X2, y = X1, fill = value)) + geom_tile(color="white")+
scale_fill_gradient(low = "white", high = "steelblue") + theme_test() +
xlim(lower,upper) + ylim(lower,upper) + geom_hline(yintercept = anchor, linetype = "dotted") +
xlab("True Number of Dots") + ylab("Estimated Number of Dots") + ggtitle("A: Posterior SD = 5")
heatmap_MBS2 <- ggplot(data_MBS2, aes(x = X2, y = X1, fill = value)) + geom_tile(color="white")+
scale_fill_gradient(low = "white", high = "steelblue") + theme_test() +
xlim(lower,upper) + ylim(lower,upper) + geom_hline(yintercept = anchor, linetype = "dotted") +
xlab("True Number of Dots") + ylab("Estimated Number of Dots") + ggtitle("B: Posterior SD = 10")
heatmap_MBS3 <- ggplot(data_MBS3, aes(x = X2, y = X1, fill = value)) + geom_tile(color="white")+
scale_fill_gradient(low = "white", high = "steelblue") + theme_test() +
xlim(lower,upper) + ylim(lower,upper) + geom_hline(yintercept = anchor, linetype = "dotted") +
xlab("True Number of Dots") + ylab("Estimated Number of Dots") + ggtitle("C: Posterior SD = 15")
grid.arrange(heatmap_MBS1, heatmap_MBS2, heatmap_MBS3, nrow = 1, left = "Proposal SD = 5")