115 Homework - Calculate DIC for single AR model – M2L2

Caution

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--- date: 2025-07-02 title: "Homework - Calculate DIC for single AR model -- M2L2" subtitle: "Capstone Project: Bayesian Conjugate Analysis for Autogressive Time Series Models" description: "" categories: - Bayesian Statistics - Time Series keywords: - Time Series - R code fig-caption: Notes about ... Bayesian Statistics title-block-banner: images/banner_deep.jpg --- ::::: {.content-visible unless-profile="HC"} ::: {.callout-caution} Section omitted to comply with the Honor Code ::: ::::: ::::: {.content-hidden unless-profile="HC"} ## AR(3) 1. For the earthquake data from the previous analysis, you should conclude that the data should be fitted with an AR(3) model. Now fit the data using an AR(3) model, using prior $m_0=(0,0,0)^T,C_0=10I_3, n_0= d_0= 0.02$. Obtain 5000 posterior samples for all the model parameters and then calculate the DIC using these 5000 posterior sample. What is the DIC value you get? ```{r} #| label: lst-capstone-earthquake-data ## read data, you need to make sure the data file is in your current working directory earthquakes.dat <- read.delim("data/earthquakes.txt") earthquakes.dat$Quakes=as.numeric(earthquakes.dat$Quakes) y.dat=earthquakes.dat$Quakes[1:100] ## this is the training data y.new=earthquakes.dat$Quakes[101:103] ## this is the test data y.sample=y.dat ``` ```{r} #| label: fig-capstone-posterior-inference-earthquake-ar(3) #| fig-cap: "DIC for Earthquake Data" ## fit AR($p$) with p=3 and sample from posterior ## 1. setup library(mvtnorm) n.all=length(y.sample) # Total number of observations p=3 # AR order m0=matrix(rep(0,p),ncol=1) # Prior mean of AR coefficients C0=diag(p)*10 # Prior covariance (diffuse) n0=0.02; d0=0.02 # Prior parameters for innovation variance (Inverse-Gamma) ## 2. prepare data matrices Y = matrix(y.sample[(p+1):n.all], ncol=1) #Fmtx = matrix(c(y.sample[2:(n.all-1)],y.sample[1:(n.all-p)]),nrow=p,byrow=TRUE) Fmtx = t(embed(y.sample, p+1)[, 2:(p+1)]) n=length(Y) ## 3. Posterior update e = Y - t(Fmtx) %*% m0 # residuals under prior mean Q = t(Fmtx) %*% C0 %*% Fmtx + diag(n) # precision-like matrix Q.inv = chol2inv(chol(Q)) # inverse via Cholesky A = C0 %*% Fmtx %*% Q.inv # Kalman gain-like update m = m0 + A %*% e # posterior mean of phi C = C0 - A %*% Q %*% t(A) # posterior covariance of phi n.star = n + n0 # post parameters for innovation variance d.star = t(Y - t(Fmtx) %*% m0) %*% Q.inv %*% (Y - t(Fmtx) %*% m0) + d0 # post parameters for variance n.sample=5000 nu.sample=rep(0,n.sample) phi.sample=matrix(0,nrow=n.sample,ncol=p) for (i in 1:n.sample){ set.seed(i) nu.new=1/rgamma(1,shape=n.star/2,rate=d.star/2) nu.sample[i]=nu.new phi.new=rmvnorm(1,mean=m,sigma=nu.new*C) phi.sample[i,]=phi.new } ``` ```{r} #| label: fig-capstone-histogram-earthquake-parameters-ar(3) #| fig-cap: "Posterior samples of AR(3) parameters for Earthquake Data" par(mfrow=c(2,2)) hist(phi.sample[,1],freq=FALSE,xlab=expression(phi[1]),main="") lines(density(phi.sample[,1]),type='l',col='red') hist(phi.sample[,2],freq=FALSE,xlab=expression(phi[2]),main="") lines(density(phi.sample[,2]),type='l',col='red') hist(phi.sample[,2],freq=FALSE,xlab=expression(phi[3]),main="") lines(density(phi.sample[,2]),type='l',col='red') hist(nu.sample,freq=FALSE,xlab=expression(nu),main="") lines(density(nu.sample),type='l',col='red') ``` ```{r} #| label: fig-capstone-dic-earthquake-ar(3) cal_log_likelihood=function(phi,nu){ mu.y = t(Fmtx)%*%phi log.lik=sapply(1:length(mu.y),function(k){dnorm(Y[k,1],mu.y[k],sqrt(nu),log=TRUE)}) sum(log.lik) } phi.bayes=colMeans(phi.sample) nu.bayes=mean(nu.sample) log.lik.bayes=cal_log_likelihood(phi.bayes,nu.bayes) cat(log.lik.bayes) ``` ```{r} #| label: lst-capstone-posterior-inference-earthquake-ar(3)-check ## get in sample prediction post.pred.y=function(s){ beta.cur=matrix(phi.sample[s,],ncol=1) nu.cur=nu.sample[s] mu.y=t(Fmtx)%*%beta.cur sapply(1:length(mu.y), function(k){rnorm(1,mu.y[k],sqrt(nu.cur))}) } y.post.pred.sample=sapply(1:5000, post.pred.y) ``` ```{r} #| label: fig-capstone-posterior-inference-earthquake-ar(3)-check ## show the result summary.vec95=function(vec){ c(unname(quantile(vec,0.025)),mean(vec),unname(quantile(vec,0.975))) } summary.y=apply(y.post.pred.sample,MARGIN=1,summary.vec95) plot(Y,type='b',xlab='Time',ylab='',ylim=c(-10,35),pch=16) lines(summary.y[2,],type='b',col='grey',lty=2,pch=4) lines(summary.y[1,],type='l',col='purple',lty=3) lines(summary.y[3,],type='l',col='purple',lty=3) legend("topright",legend=c('Truth','Mean','95% C.I.'),lty=1:3,col=c('black','grey','purple'), horiz = T,pch=c(16,4,NA)) ``` ```{r} #| label: fig-capstone-dic-earthquake-DIC-ar(3) post.log.lik=sapply(1:5000, function(k){cal_log_likelihood(phi.sample[k,],nu.sample[k])}) E.post.log.lik=mean(post.log.lik) p_DIC=2*(log.lik.bayes-E.post.log.lik) DIC=-2*log.lik.bayes+2*p_DIC cat(DIC) ``` ## AR(1) 2. For the earthquake data from the previous analysis, you should conclude that the data should be fitted with an AR(1) model. Now fit the data using an AR(1) model, using prior $m_0=(0),C_0=10,n_0=d_0=0.02$. Obtain 5000 posterior samples for all the model parameters and then calculate the DIC using these 5000 posterior sample. What is the DIC value you get? ```{r} #| label: fig-capstone-posterior-inference-earthquake-ar(1) #| fig-cap: "AR(1) parameters for Earthquake Data" ## fit AR($p$) with p=3 and sample from posterior ## read data, you need to make sure the data file is in your current working directory earthquakes.dat <- read.delim("data/earthquakes.txt") earthquakes.dat$Quakes=as.numeric(earthquakes.dat$Quakes) y.dat=earthquakes.dat$Quakes[1:100] ## this is the training data y.new=earthquakes.dat$Quakes[101:103] ## this is the test data y.sample=y.dat ## 1. setup library(mvtnorm) n.all=length(y.sample) # Total number of observations p=1 # AR order m0=0 # Prior mean of AR coefficients C0=10 # Prior covariance (diffuse) n0=0.02; d0=0.02 # Prior parameters for innovation variance (Inverse-Gamma) ## 2. prepare data matrices Y = matrix(y.sample[(p+1):n.all], ncol=1) #Fmtx = matrix(c(y.sample[1:(n.all - 1)]),nrow=p,byrow=TRUE) Fmtx = matrix(y.sample[1:(n.all - 1)], ncol = 1) #Fmtx = matrix(y.sample[1:(n.all - 1)], ncol = 1) #Fmtx = matrix(c(y.sample[2:(n.all-1)],y.sample[1:(n.all-p)]),nrow=p,byrow=TRUE) #Fmtx = t(embed(y.sample, p+1)[, 2:(p+1)]) n=length(Y) ## 3. Posterior update #e = Y - t(Fmtx) %*% m0 # residuals under prior mean e = Y - Fmtx %*% m0 #Q = t(Fmtx) %*% C0 %*% Fmtx + diag(n) # precision-like matrix #Q = C0 * Fmtx %*% t(Fmtx) + diag(n) Q = as.numeric(C0) * sum(Fmtx^2) + 1 cat(Q) Q.inv = 1 / Q A = C0 %*% t(Fmtx) * Q.inv # (1×1) * (1×n) = (1×n) m = m0 + A %*% (Y - Fmtx %*% m0) # m0: (1×1), A: (1×n), rest: (n×1) C = C0 - A %*% t(A) * Q # (1×1) - (1×n)(n×1) * (scalar) n.star = n + n0 # post parameters for innovation variance #d.star = t(Y - t(Fmtx) %*% m) %*% Q.inv %*% (Y - t(Fmtx) %*% m0) + d0 # post parameters for variance #d.star = t(Y - t(Fmtx) %*% m) %*% (Y - t(Fmtx) %*% m) + d0 mu.m = Fmtx %*% m # fitted values under posterior mean mu.0 = Fmtx %*% m0 # fitted values under prior mean res1 = Y - mu.m res2 = Y - mu.0 d.star = as.numeric(t(Y - mu.m) %*% (Y - mu.m)) + d0 n.sample=5000 nu.sample=rep(0,n.sample) phi.sample=matrix(0,nrow=n.sample,ncol=p) for (i in 1:n.sample){ set.seed(i) nu.new=1/rgamma(1,shape=n.star/2,rate=d.star/2) nu.sample[i]=nu.new phi.new=rmvnorm(1,mean=m,sigma=nu.new*C) phi.sample[i,]=phi.new } ``` ```{r} #| label: fig-capstone-histogram-earthquake-parameters-ar(1) par(mfrow=c(1,2)) hist(phi.sample[,1],freq=FALSE,xlab=expression(phi[1]),main="") lines(density(phi.sample[,1]),type='l',col='red') hist(nu.sample,freq=FALSE,xlab=expression(nu),main="") lines(density(nu.sample),type='l',col='red') ``` ### Model Checking by In-sample Point and Interval Estimation To check whether the model fits well, we plot the posterior point and interval estimate for each point. ```{r} #| label: lst-capstone-posterior-inference-earthquake-ar(1)-check ## get in sample prediction post.pred.y=function(s){ beta.cur=matrix(phi.sample[s,],ncol=1) nu.cur=nu.sample[s] #mu.y=t(Fmtx)%*%beta.cur mu.y = Fmtx %*% beta.cur sapply(1:length(mu.y), function(k){rnorm(1,mu.y[k],sqrt(nu.cur))}) } y.post.pred.sample=sapply(1:5000, post.pred.y) ``` ```{r} #| label: fig-capstone-posterior-inference-earthquake-ar(1)-check ## show the result summary.vec95=function(vec){ c(unname(quantile(vec,0.025)),mean(vec),unname(quantile(vec,0.975))) } summary.y=apply(y.post.pred.sample,MARGIN=1,summary.vec95) plot(Y,type='b',xlab='Time',ylab='',ylim=c(-10,35),pch=16) lines(summary.y[2,],type='b',col='grey',lty=2,pch=4) lines(summary.y[1,],type='l',col='purple',lty=3) lines(summary.y[3,],type='l',col='purple',lty=3) legend("topright",legend=c('Truth','Mean','95% C.I.'),lty=1:3,col=c('black','grey','purple'), horiz = T,pch=c(16,4,NA)) ``` ```{r} #| label: fig-capstone-dic-earthquake-ar(1) cal_log_likelihood=function(phi,nu){ #mu.y = t(Fmtx)%*%phi mu.y = Fmtx %*% phi log.lik=sapply(1:length(mu.y),function(k){dnorm(Y[k,1],mu.y[k],sqrt(nu),log=TRUE)}) sum(log.lik) } phi.bayes=colMeans(phi.sample) nu.bayes=mean(nu.sample) log.lik.bayes=cal_log_likelihood(phi.bayes,nu.bayes) cat(log.lik.bayes) ``` ```{r} #| label: fig-capstone-dic-earthquake-DIC-ar(1) post.log.lik=sapply(1:5000, function(k){cal_log_likelihood(phi.sample[k,],nu.sample[k])}) E.post.log.lik=mean(post.log.lik) cat("E.post.log.lik=",E.post.log.lik) p_DIC=2*(log.lik.bayes-E.post.log.lik) cat(" p_DIC=",p_DIC) DIC=-2*log.lik.bayes+2*p_DIC cat(" DIC=",DIC) ``` ```{r} covid.dat <- read.delim("data/GoogleSearchIndex.txt") covid.dat$Week=as.Date(as.character(covid.dat$Week),format = "%Y-%m-%d") y.dat=covid.dat$covid[1:57] ## this is the training data y.new=covid.dat$covid[58:60] ## this is the test data y=y.dat n.all=length(y) p <- 1 #m0 <- matrix(0, ncol=1) m0=0 C0 <- 10 n0 <- d0 <- 0.02 #Y <- matrix(y[2:100], ncol=1) Y <- matrix(y[(p+1):length(y)], ncol = 1) #Fmtx <- matrix(y[1:99], ncol=1) Fmtx <- matrix(y[1:(length(y)-p)], ncol = 1) n <- length(Y) #Q <- as.numeric(t(Fmtx) %*% C0 %*% Fmtx + 1) Q <- as.numeric(C0) * sum(Fmtx^2) + 1 Qinv <- 1 / Q #A <- C0 %*% t(Fmtx) * Qinv A <- (C0 * Qinv) * t(Fmtx) # A is (1 × n) m <- m0 + A %*% (Y - Fmtx %*% m0) #C <- C0 - A %*% t(A) * Q C <- C0 - C0^2 * sum(Fmtx^2) * Qinv res <- Y - Fmtx %*% m d.star <- as.numeric(t(res) %*% res) + d0 n.star <- n + n0 #library(mvtnorm) n.sample <- 5000 phi.sample <- numeric(n.sample) nu.sample <- numeric(n.sample) set.seed(1) for (i in 1:n.sample) { nu <- 1 / rgamma(1, shape = n.star / 2, rate = d.star / 2) #phi <- rmvnorm(1, mean = m, sigma = nu * C) phi <- rnorm(1, mean = m, sd = sqrt(nu * C)) phi.sample[i] <- phi nu.sample[i] <- nu } loglik <- function(phi, nu) { mu <- Fmtx %*% phi sum(dnorm(Y, mean = mu, sd = sqrt(nu), log = TRUE)) } phi.bayes <- mean(phi.sample) nu.bayes <- mean(nu.sample) loglik.bayes <- loglik(phi.bayes, nu.bayes) post.loglik <- sapply(1:n.sample, function(i) loglik(phi.sample[i], nu.sample[i])) E.loglik <- mean(post.loglik) pDIC <- 2 * (loglik.bayes - E.loglik) DIC <- -2 * loglik.bayes + 2 * pDIC #DIC <- -2 * -203.777 + 2 * 1.983 cat("E.post.log.lik:", E.loglik, "\n", "p_DIC:", pDIC, "\n", "DIC:", DIC, "\n") Goal_DIC <- -2 * -203.777 + 2 * 1.983 cat("Goal DIC:", Goal_DIC, "\n") ``` :::::