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prez1.r
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prez1.r
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# map references removed, and all graphing and plotting
# is redirected to write to files. other than that,
# this file is very close to the original
source("utility.r")
## Get Data
prez88 <- read.table("prez48to88.txt", header = T)
prez92 <- read.table("prez92.txt", header = T)
## Get y, X
y <- prez88[,1]
X <- as.matrix(prez88[, -(1:4)])
n <- nrow(X)
p <- ncol(X)
## Least square calculations
beta.ls <- lm(y ~ 0 + X)$coeff
V <- crossprod(X)
resid.ls <- y - c(X %*% beta.ls)
s2 <- sum(resid.ls^2) / (n - p)
## Xnew from 1992 data
Xnew <- as.matrix(prez92[, -(1:4)])
nnew <- nrow(Xnew)
## Spread and mean of p(ynew | y)
mean.ynew <- c(Xnew %*% beta.ls)
## Quantiles -- note: median = mean.
pr <- c(0.025, 0.25, 0.5, 0.75, 0.975)
## Map plots of predictive mean ( = median)
## Map of Pr(Dvot share > 0.5 | data)
###################################
# Changing the prior distribution #
###################################
#
# Shall use a non-conjugate specification
# p(beta, sigmaSq) = N(beta | 0, tauSq0 * diag(p)) x InvChiSq(sigmaSq | nu0, sigmaSq0)
#
## First get samples from the posterior with the default prior: p(beta, sigmaSq) = 1 / sigmaSq
## Notice that p(beta, InvsigmaSq) = 1 / InvsigmaSq
nsamp <- 1e4
inv_sigmaSq <- rgamma(nsamp, (n - p) / 2, (n - p) * s2 / 2)
Vchol <- chol(V)
z <- scale(matrix(rnorm(nsamp * p), ncol = nsamp), center = FALSE, scale = sqrt(inv_sigmaSq))
beta <- beta.ls + backsolve(Vchol, z) ## each column is a sample from the posterior
ynew <- Xnew %*% beta + scale(matrix(rnorm(nnew * nsamp), ncol = nsamp), center = FALSE, scale = sqrt(inv_sigmaSq))
## function to get prediction for a new prior
## by reweighting the sample from the default analysis
pred <- function(tauSq0, nu0, sigmaSq0){
log.wt <- (dgamma(inv_sigmaSq, nu0 / 2, nu0 * sigmaSq0 / 2, log = TRUE)
+ log(inv_sigmaSq)
- 0.5 * colSums(beta^2) / tauSq0)
wt <- exp(log.wt - max(log.wt))
wt <- wt / sum(wt)
pDemWin <- apply(ynew > 0.5, 1, weighted.mean, w = wt)
cat("ESS = ", 1 / sum(wt^2), "\n")
return(pDemWin)
}
## Informal graphical checks of residuals
year <- prez88$year
x11()
par(mfrow = c(2,2))
nsamp <- 4
sig2 <- 1 / rgamma(nsamp, (n - p) / 2, (n - p) * s2 / 2)
beta.z <- solve(chol(crossprod(X)), matrix(rnorm(nsamp * p), nrow = p))
beta <- beta.ls + beta.z %*% diag(sqrt(sig2))
resid <- y - X %*% beta
# If Partisan effects were present, then errors from the same
# election year will tend to have same sign:
for(i in 1:4){
sres <- split(resid[,i], as.factor(year))
if (i == 1) png(file="out1-1.png")
if (i == 2) png(file="out1-2.png")
if (i == 3) png(file="out1-3.png")
if (i == 4) png(file="out1-4.png")
boxplot(sres)
abline(h = 0)
dev.off()
}
#title(main = "residuals grouped by year - samples from posterior", outer = T, line = -2)
# Check normality assumption
x11()
par(mfrow = c(2,2))
for(i in 1:4){
if (i == 1) png(file="out2-1.png")
if (i == 2) png(file="out2-2.png")
if (i == 3) png(file="out2-3.png")
if (i == 4) png(file="out2-4.png")
qqnorm(c(resid[,i]) / sqrt(sig2[i]), main = "")
abline(0, 1, col = "gray")
dev.off()
}
#title(main = "residuals: normal QQplot - samples from posterior", outer = T, line = -2)
################################
## Including partisan effects ##
################################
regions <- list(Northeast = c(7, 8, 19, 20, 21, 29, 30, 32, 38, 39, 45, 48),
South = c(1, 4, 9, 10, 17, 18, 24, 33, 36, 40, 42, 43, 46),
Midwest = c(13, 14, 15, 16, 22, 23, 25, 27, 34, 35, 41, 49),
West = c(2, 3, 5, 6, 11, 12, 26, 28, 31, 37, 44, 47, 50))
st2regn <- rep(NA, 50)
for(i in 1:4)
st2regn[regions[[i]]] <- i
year <- prez88$year
state <- prez88$state
t <- (year - 1948)/4 + 1 ## It is not recommended to use "t" as a variable
r <- st2regn[state] ## but I'd do it here to be compatible with
rt <- 11*(r - 1) + t ## notations used in class
X.del <- diag(11)[t, ] ## The X_delta design matrix
n.t <- apply(X.del, 2, sum)
X.gam <- diag(44)[rt, ]
n.rt <- apply(X.gam, 2, sum) ## The X_gamma design matrix
## QR decomposition of X -- useful for repeated use of least squares
X.qr <- qr(X) ## X = Q %*% R, where Q is orthonormal and R is upper triang
R <- qr.R(X.qr) ## t(X) %*% X = t(R) %*% R -- so to generate from N(0, (X'X)^{-1})
## we can use backsolve(R, rnorm(p))
##################
# Gibbs sampler ##
##################
prez.gibbs <- function(pars = NULL, nu.d, tau2.0d, nu.g, tau2.0g, n.sweep = 1e3){
if(!is.null(pars)){
beta <- pars[1:p]
resid <- y - c(X %*% beta)
sig2 <- pars[p + 1]
del <- pars[p + 1 + (1:11)]
gam <- pars[p + 12 + (1:44)]
tau2.d <- pars[p + 57]
tau2.g <- pars[p + 57 + (1:4)]
} else{
beta <- as.numeric(lm(y ~ 0 + X)$coeff)
resid <- y - c(X %*% beta)
sig2 <- sum(resid^2) / (n - p)
del <- rep(0, 11)
gam <- rep(0, 44)
tau2.d <- 1
tau2.g <- rep(1, 4)
}
par.store <- matrix(NA, nrow = n.sweep, ncol = p + 1 + 11 + 44 + 5)
for(iter in 1:n.sweep){
#----------------------
# Update sig2 and beta
#----------------------
y.b <- y - del[t] - gam[rt] ## modified response for beta
beta.ls <- as.numeric(qr.coef(X.qr, y.b))
res.b <- qr.resid(X.qr, y.b)
s2.b <- sum(res.b^2) / (n - p)
sig2 <- 1 / rgamma(1, (n - p)/2, (n - p) * s2.b / 2)
beta <- beta.ls + sqrt(sig2) * backsolve(R, rnorm(p))
fit.b <- c(X %*% beta)
#--------------
# Update delta
#--------------
y.d <- y - fit.b - gam[rt] ## modified response for delta
bar.y.d <- apply(y.d * X.del, 2, sum) / n.t
del.mean <- (n.t * bar.y.d / sig2) / (n.t / sig2 + 1 / tau2.d)
del.var <- 1 / (n.t / sig2 + 1 /tau2.d)
del <- rnorm(11, del.mean, sqrt(del.var) )
#--------------
# Update tau2.d
#--------------
tau2.d <- 1 / rgamma(1, (11 + nu.d) / 2, (nu.d * tau2.0d + sum(del^2))/2)
#--------------
# Update gamma
#--------------
y.g <- y - fit.b - del[t] ## modified response fo gamma
bar.y.g <- apply(y.g * X.gam, 2, sum) / n.rt
gam.mean <- (n.rt * bar.y.g / sig2) / (n.rt / sig2 + 1 / rep(tau2.g, each = 11))
gam.var <- 1 / (n.rt / sig2 + 1 / rep(tau2.g, each = 11))
gam <- rnorm(44, gam.mean, sqrt(gam.var) )
#--------------
# Update tau.g
#--------------
tau2.g <- 1 / rgamma(4, (11 + nu.g) / 2, (nu.g * tau2.0g + apply(matrix(gam, 11, 4)^2, 2, sum)) / 2)
#--------
# Store
#--------
par.store[iter, ] <- c(beta, sig2, del, gam, tau2.d, tau2.g)
}
return(par.store)
}
## run sampler
prez.mc <- prez.gibbs(nu.d = -1, tau2.0d = 0, nu.g = -1, tau2.0g = 0, n.sweep = 4e3)
# Retain last half
last <- seq(2e3 + 10, 4e3, 10)
pr <- c(0.025, 0.25, 0.5, 0.75, 0.975)
par.samp <- prez.mc[last, ]
## Plot credible intervals for regional partisan shifts = del[t] + gam[rt]
del <- par.samp[, p + 1 + (1:11)]
gam <- par.samp[, p + 1 + 11 + (1:44)]
shifts <- kronecker(t(rep(1,4)), del) + gam
shifts.CI <- apply(shifts, 2, quantile, p = pr)
x11()
par(mfrow = c(2,2), mar = c(5, 4, 6, 2) + 0.1)
start <- 0
for(i in 1:4){
if (i == 1) png(file="out3-1.png")
if (i == 2) png(file="out3-2.png")
if (i == 3) png(file="out3-3.png")
if (i == 4) png(file="out3-4.png")
plot(year, 0 * year, ty = "n", ylim = range(shifts.CI[, start + 1:11]), ann = F)
#title(main = names(regions)[i])
for(j in 1:11)
fivept(shifts.CI[,start + j], x = 1944 + 4 * j, max.wd = 4 * 0.33)
abline(h = 0, lty = 3)
start <- start + 11
}
#title(main = "Regional partisan effect by year", outer = T, line = -2)
## Get posterior predictive probability of
## Dshare > 0.5 in 1992 (separately for each state)
##
## Notice that
## p(ynew_s | y, beta, sig2, del, gam)
## = integral N(ynew_s | xnew_s'beta, sig2 + tau2.d + tau2.g[r(s)])
getprob <- function(pars){
beta <- pars[1:p]
sig2 <- pars[p + 1]
tau2.d <- pars[p + 57]
tau2.g <- pars[p + 57 + (1:4)]
return(pnorm(0.5, c(Xnew %*% beta), sqrt(sig2 + tau2.d + tau2.g[st2regn]), lower.tail = F))
}
prob.samp <- apply(par.samp, 1, getprob)
pred.Dwin <- apply(prob.samp, 1, mean)
st.col <- rgb(1 - pred.Dwin, 0, pred.Dwin)
x11()
#####################################
# #
# Convergence diagnostics via PSRF #
# #
#####################################
## Variables of interest?
## We will take phi = Prob of winning in 1992
## Other choices are Xnew %*% beta, or the parameters
## themselves
out <- list()
phi <- list()
## get overdispersed starting values --
## we will simply overdisperse on beta and hope that
## would automatically overdisperse the variable of
## interest.
## Look at the posterior distribution of beta in the
## the OLR version.
sig2.start <- 1 / rgamma(5, (n - p) / 2, (n - p) * s2 / 2)
sd.beta <- sqrt(diag(solve(V)))
beta.start <- beta.ls + 10 * sd.beta * matrix(rnorm(5 * p), ncol = 5) %*% diag(sqrt(sig2.start))
del.start <- matrix(0, 11, 5)
gam.start <- matrix(0, nrow = 44, ncol = 5)
pars.start <- rbind(beta.start, sig2.start, del.start, gam.start, matrix(1e-5, 5, 5))
for(i in 1:5){
prez.mc <- prez.gibbs(pars = pars.start[, i], nu.d = -1, tau2.0d = 0, nu.g = -1, tau2.0g = 0, n.sweep = 4e3)
par.samp <- prez.mc[last, ]
phi[[i]] <- apply(par.samp, 1, getprob)
out[[i]] <- par.samp
cat(i, " ")
}
cat("\n")
logit <- function(p)
return(log(p) - log(1 - p))
## plot of 4 randomly selected state pairs
x11()
st.ix <- replicate(4, sample(50, size = 2))
phi.start <- apply(pars.start, 2, getprob)
par(mfrow = c(2,2))
for(j in 1:4){
if (j == 1) png(file="out4-1.png")
if (j == 2) png(file="out4-2.png")
if (j == 3) png(file="out4-3.png")
if (j == 4) png(file="out4-4.png")
plot(0, 0, ty = "n", ylim = c(-10,10), xlim = c(-10,10), ann = F)
for(i in 1:5){
lines(logit(t(cbind(phi.start[st.ix[,j], i], phi[[i]][st.ix[,j], ]))), col = i)
points(logit(t(phi.start[st.ix[,j], i])), pch = 19, col = i)
}
dev.off()
}
## function to compute PSRF for state i
getPSRF <- function(i, n.chain, ch.len){
phi.i <- matrix(NA, n.chain, ch.len)
for(j in 1:n.chain)
phi.i[j, ] <- phi[[j]][i, ]
psi <- apply(phi.i, 1, mean)
s2 <- apply(phi.i, 1, var)
W <- mean(s2)
B <- ch.len * var(psi)
return(sqrt(((1 - 1 / ch.len) * W + ((n.chain + 1) / (n.chain * ch.len))* B) / W))
# 1 234 3 45 4 5 43 2 10
}
## Use above function to get the PSRF values
PSRF <- sapply(1:50, getPSRF, n.chain = 5, ch.len = 200)
print(PSRF) ## want each < 1.1
## pooling the chains
phi.pool <- matrix(NA, 50, 1000)
for(i in 1:5)
phi.pool[, 200 * (i - 1) + (1:200)] <- phi[[i]]
pred.Dwin <- apply(phi.pool, 1, mean)
st.col <- rgb(1 - pred.Dwin, 0, pred.Dwin)
print (pred.Dwin)
x11()