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ISYE 6420 Homework

Creation date：2024-04-10 16:53:39

Belonging category：计算机computer science

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ISYE 6420 Homework

Problem 1
(a) The original data summarizes the number of breakdowns at each voltage level and we
can convert the outcome to be binary. As shown in our lecture note, we have
yi =
{
1 if zi > 0,
0 if zi ≤ 0,
zi|β ∼ N(β0 + β1xi, 1),
θi = p(yi = 1) = p(zi > 0) = Φ(β0 + β1xi).
Assuming noninformative prior p(β) ∝ 1, the full conditional distributions for zi and β can
be derived as follows:
β|z ∼ N((X ′X)−1X ′z, (X ′X)−1),
p(zi|β, y) =
{
p(zi|β, zi > 0) if yi = 1
p(zi|β, zi ≤ 0) if yi = 0.
The Gibbs sampler is implemented by the following R code:
###################################
library(numDeriv)
library(MASS)
library(coda)
library(mcmc)
library(adaptMCMC)
library(mined)
#Problem#1
voltage=c(1065,1071,1075,1083,1089,1094,1100,1107,1111,1120,1128,1135)
trials=rep(100,12)
breakdown=c(2,3,5,11,10,21,29,48,56,88,98,99)
voltagec=voltage-mean(voltage)#center the voltage
a=glm(cbind(breakdown,trials-breakdown)~voltagec,family=binomial(link="probit"))
summary(a)
m=10000
beta=matrix(0,nrow=m,ncol=2)
z=numeric(100)
1
zbar=matrix(0,nrow=m,ncol=12)
beta[1,]=a$coef
library(MASS)
X=as.matrix(cbind(1,voltagec))
Sinv=solve(t(X)%*%X)
library(truncnorm)
for(i in 2:m)
{
for(j in 1:12)
{
mu=sum(X[j,]*beta[i-1,])
z1=rtruncnorm(breakdown[j],a=0,b=Inf,mean=mu,sd=1)
z0=rtruncnorm(100-breakdown[j],a=-Inf,b=0,mean=mu,sd=1)
z=c(z1,z0)
zbar[i,j]=mean(z)
}
beta[i,]=mvrnorm(1,Sinv%*%t(X)%*%zbar[i,],1/100*Sinv)
}
library(coda)
effectiveSize(beta)
cumuplot(mcmc(beta,start=1,end=m,thin=10))
summary(beta)
#1(a)
par(mfrow=c(1,2))
plot(density(beta[,1]),main=expression(beta[0]),xlab="")
plot(density(beta[,2]),main=expression(beta[1]),xlab="")
par(mfrow=c(1,1))
#1(b)
ld50=-beta[,1]/beta[,2]+mean(voltage)
plot(density(ld50),main="LD50")
###################################
For m = 20, 000, the posterior densities for β0 and β1 are shown in Figure 1.
(b) Since θ = 0.5 ⇐⇒ β0 + β1x = 0 ⇐⇒ x = −β0β1 , the LD50 point corresponds to −
β0
β1
.
The distribution of LD50 is plotted in Figure 2.
Problem 2
2
Figure 1: Posterior densities for β0 and β1.
−0.7 −0.4
0
2
4
6
8
β0
D
en
si
ty
0.055 0.070
0
20
40
60
80
10
0
β1
D
en
si
ty
If we fit a logistic regression model to the above data, the posterior distribution of β is
p(β|y) ∝
12∏
i=1
θyii (1− θi)100−yi
=
12∏
i=1
(
ex
T
i β
1 + ex
T
i β
)yi(
1
1 + ex
T
i β
)100−yi .
Note that yi here is the number of breakdowns instead of 0-1. Both mcmc and adaptMCMC
packages use log of unnormalized posterior as input, and we can obtain the following results
via the R code provided.
###################################################################################
a=glm(cbind(breakdown,trials-breakdown)~voltagec,family=binomial(link="logit"))
summary(a)
logh=function(beta){
val=sum(breakdown*(X%*%beta))-100*sum(log(1+exp(X%*%beta)))
return(val)
}
nlogh=function(theta) -logh(theta)
betahat=optim(a$coef,nlogh)$par
Sigmahat=solve(hessian(nlogh,betahat))
3
Figure 2: Distribution of LD50 using Gibbs sampling.
1102 1103 1104 1105 1106 1107 1108 1109
0.
0
0.
1
0.
2
0.
3
0.
4
0.
5
Density plot for LD50
N = 18001 Bandwidth = 0.09821
D
en
si
ty
#mcmc
eig=eigen(Sigmahat)
sqrtSigma=eig$vec%*%diag(sqrt(eig$val))%*%t(eig$vec)
s=2.4/sqrt(2)*sqrtSigma
m=10000
out=metrop(logh,initial=betahat,nbatch=m,scale=s)
out$accept
beta=out$batch
effectiveSize(beta)
cumuplot(mcmc(beta,start=100,end=m,thin=10))
1-rejectionRate(mcmc(beta))
ld50=-beta[,1]/beta[,2]+mean(voltage)
plot(density(ld50),main="LD50")
#adaptMCMC
s=(2.4/sqrt(2))^2*Sigmahat#specify covariance matrix of the jumping distribution
out=MCMC(logh,n=m,init=betahat,scale=s,adapt=T,acc.rate=.4)
out$accept
beta=out$samples
effectiveSize(beta)
4
cumuplot(mcmc(beta,thin=10))
1-rejectionRate(mcmc(beta))
ld50=-beta[,1]/beta[,2]+mean(voltage)
plot(density(ld50),main="LD50")
#mined
#first find approximate ranges for the parameters
summary(a)
a$coef[1]+3*c(-1,1)*.0945
a$coef[2]+3*c(-1,1)*.0064
l1=-1.2
u1=-.6
l2=.09
u2=.14
X=as.matrix(cbind(1,voltagec))
logh=function(x){
beta=x
beta[1]=l1+(u1-l1)*x[1]
beta[2]=l2+(u2-l2)*x[2]
val=sum(breakdown*(X%*%beta))-100*sum(log(1+exp(X%*%beta)))
return(val)
}
ini=Lattice(109,2)
D=mined(ini,logh)$points
beta=matrix(0,nrow=109,ncol=2)
beta[,1]=l1+(u1-l1)*D[,1]
beta[,2]=l2+(u2-l2)*D[,2]
ld50=-beta[,1]/beta[,2]+mean(voltage)
plot(density(ld50),main="LD50")

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LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 FINC5090 MATH1052 SciComp Elec4622 FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 MAT 237 BISM2201 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SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum

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