In Panxiaoba/StatComp18013: functions and answers for 'Statistical Computing' course
Overview
StatComp18013 is a simple R package developed to present the homework answers for the 'Statistical Computing' course.
The first homework answer in 2018-09-14
The question is :
Write a .Rmd file to implement at least three examples of different types in the above books (texts, numerical results, tables, and figures).
The code is as follows:
#The first example:
x <- rnorm(10)
y <- rnorm(10)
plot(x,y)
#The second example:
data("InsectSprays")
aov.spray <- aov(sqrt(count) ~ spray,data=InsectSprays)
summary(aov.spray)
#The third example:
m1 <- matrix(1,2,2)
m2 <- matrix(2,2,2)
rbind(m1,m2)
cbind(m1,m2)
The second homework answer in 2018-09-21
The question is :
-
A discrete random variable X has probability mass function
x 0 1 2 3 4
p(x) 0.1 0.2 0.2 0.2 0.3
Use the inverse transform method to generate a random sample of size 1000 from the distribution of X. Construct a relative frequency table and compare the empirical with the theoretical probabilities. Repeat using the R sample function.
2 .Write a function to generate a random sample of size n from the Beta(a,b) distribution by the acceptance-rejection method. Generate a random sample of size 1000 from the Beta(3,2) distribution. Graph the histogram of the sample with the theoretical Beta(3,2) density superimposed.
- Simulate a continuous Exponential-Gamma mixture. Suppose that the rate parameter Λ has Gamma(r,β) distribution and Y has Exp(Λ) distribution. That is, $(Y |\Lambda = \lambda) \sim f_{Y}(y|\lambda) = \lambda e^{−\lambda y}$. Generate 1000 random observations from this mixture with r = 4 andβ = 2.
The answer code is as follows:
#The answer of first question:
x <- c(0,1,2,3,4)
p <- c(0.1,0.2,0.2,0.2,0.3)
cp <- cumsum(p)
m <- 1e4
r <- numeric(m)
r <- x[findInterval(runif(m),cp)+1]
ct <- table(r)/m
x <- sample(c(0,1,2,3,4),m,p,replace = TRUE)
y <- table(x)/m
rbind(prob=p,prob1=ct,prob2=y)
#The answer of second question:
beta <- function(n,a,b){
j<-k<-0
y <- numeric(n)
while (k < n) {
u <- runif(1)
j <- j + 1
x <- runif(1)
if (x^(a-1)* (1-x)^(b-1) > u) {
k <- k + 1
y[k] <- x
}
}
return (y)
}
r<-beta(1000,3,2)
hist(r,prob=TRUE,main="the histogram of Beta(3,2) and its density",col="red")
t<-seq(0,1,0.001)
s<-dbeta(t,3,2)
lines(t,s,col="blue",lwd=3)
#The answer of third question:
n <- 1e3; r <- 4; beta <- 2
lambda <- rgamma(n, r, beta)
y <- rexp(n, lambda)
y
The third homework answer in 2018-09-28
The question is :
-
Write a function to compute a Monte Carlo estimate of the Beta(3, 3) cdf,and use the function to estimate F(x) for x = 0.1,0.2,...,0.9. Compare the estimates with the values returned by the pbeta function in R.
-
The Rayleigh density [156, (18.76)] is
$${f(x)}=\frac{x}{\sigma^2}e^{-x^2/{(2\sigma^2)}},\qquad x\geq0,\sigma>0$$
Implement a function to generate samples from a Rayleigh distribution,using antithetic variables. What is the percent reduction in variance of $\frac{x+x'}{2}$ compared with $\frac{x_1+x_2}{2}$ for independent ${x_1+x_2}$.
-
Find two importance functions $f_{1}$ and $f_{2}$ that are supported on $(1,\infty)$ and are close? to $$g(x)=\frac{x^{2}}{\sqrt{2\pi}}e^{\frac{-x^{2}}{2}},\qquad x>1.$$
Which of your two importance functions should produce the smaller variance in estimating$$\int_{1}^{\infty}\frac{x^{2}}{\sqrt{2\pi}}e^{\frac{-x^{2}}{2}}dx$$by importance sampling Explain.
-
Obtain a Monte Carlo estimate of$$\int_{1}^{\infty}\frac{x^{2}}{\sqrt{2\pi}}e^{\frac{-x^{2}}{2}}dx$$
by importance sampling.
The answer code is as follows:
#The answer of first question:
mtbeta <- function(x){
m <- 1e5
t <- runif(m,0,x)
mean(30*t*t*(1-t)*(1-t))*x
}
mbeta <- Vectorize(mtbeta)
s <- seq(0.1,0.9,0.1)
print(round(rbind(x=s,p=pbeta(s,3,3),phat=mbeta(s),se=mbeta(s)-pbeta(s,3,3)),5))
#The answer of second question:
mtrayl <- function(x, R = 10000, antithetic = TRUE) {
u <- runif(R/2)
if (!antithetic) v <- runif(R/2) else
v <- 1 - u
u <- c(u, v)
cdf <- numeric(length(x))
for (i in 1:length(x)) {
g <- x[i]^2*u * exp(-(u * x[i])^2 / 2)
cdf[i] <- mean(g)
}
cdf
}
x <- seq(0, 2.5,0.5)
set.seed(123)
mtr1 <- mtrayl(x, anti = FALSE)
set.seed(123)
mtr2 <- mtrayl(x,anti = TRUE)
print(round(rbind(x, mtr1, mtr2), 5))
m <- 1000
mtr3 <- mtr4 <- numeric(m)
x <- 1
for (i in 1:m) {
mtr3[i] <- mtrayl(x,antithetic = FALSE)
mtr4[i] <- mtrayl(x)
}
print(var(mtr3))
print(var(mtr4))
print((var(mtr3) - var(mtr4))/var(mtr3))
#The answer of third question:
m <- 10000
se <- numeric(2)
g <- function(x) {
exp(-x^2/2)*x^2/(sqrt(2*pi)) * (x > 1)
}
x <- rweibull(m, 2,sqrt(2))
fg1 <- g(x) /(x* exp(-x^2/2))
se[1] <- sd(fg1)
x <- rexp(m, 1)
fg2 <- g(x) / exp(-x)
se[2] <- sd(fg2)
se
#The answer of fourth question:
m <- 10000
g <- function(x) {
exp(-x^2/2)*x^2/(sqrt(2*pi)) * (x > 1)
}
x <- rweibull(m, 2,sqrt(2))
fg1 <- g(x) /(x* exp(-x^2/2))
theta1 <- mean(fg1)
theta1
The fourth homework answer in 2018-10-12
The question is :
-
Let $X$ be a non-negative random variable with $\mu = E[X] < ??$. For a random sample $x_1$ ,...,$x_n$ from the distribution of X, the Gini ratio is defined by $${G}=\frac{1}{2n^2\mu}\sum_{j=1}^n \sum_{i=1}^n |x_i-x_j|$$
The Gini ratio is applied in economics to measure inequality in income distribution (see e.g. [163]). Note that G can be written in terms of the order statistics $x_(i)$ as $${G}=\frac{1}{n^2\mu}\sum_{i=1}^n (2i-n-1)x_{(i)}$$
If the mean is unknown, let $\hat{G}$ be the statistic $G$ with $\mu$ replaced by $\bar{x}$. Estimate by simulation the mean, median and deciles of $\hat{G}$ if $X$ is standard lognormal.Repeat the procedure for the uniform distribution and Bernoulli(0.1). Also construct density histograms of the replicates in each case.
-
Construct an approximate 95% confidence interval for the Gini ratio y = E[G] if X is lognormal with unknown parameters. Assess the coverage rate of the estimation procedure with a Monte Carlo experiment.
-
Tests for association based on Pearson product moment correlation $\rho$, Spear-mans rank correlation coefficient $\rho_s$ , or Kendalls coefficient $\tau$, are implemented in cor.test. Show (empirically) that the nonparametric tests based on $\rho_s$ or $\tau$ are less powerful than the correlation test when the sampled distribution is bivariate normal. Find an example of an alternative (a bivariate distribution $(X,Y)$ such that $X$ and $Y$ are dependent) such that at least one of the nonparametric tests have better empirical power than the correlation test against this alternative.
The answer code is as follows:
#The answer of first question:
m <- 1000
n <- 20
g <- numeric(m)
medians <- means <- numeric(3)
y <- gini1 <- gini2 <- gini3 <- numeric(m)
for (i in 1:m) {
x <- sort(rlnorm(n))
xmean <- mean(x)
for (j in 1:n) {
y[j] <- (2*j-n-1)*x[j]
}
gini1[i] <- 1/n^2/xmean*sum(y[1:n])
}
for (i in 1:m) {
x <- sort(runif(n))
xmean <- mean(x)
for (j in 1:n) {
y[j] <- (2*j-n-1)*x[j]
}
gini2[i] <- 1/n^2/xmean*sum(y[1:n])
}
for (i in 1:m) {
x <- sort(rbinom(n,c(0,1),c(0.1,0.9)))
xmean <- mean(x)
for (j in 1:n) {
y[j] <- (2*j-n-1)*x[j]
}
gini3[i] <- 1/n^2/xmean*sum(y[1:n])
}
par(mfrow=c(3,1))
par(pin=c(2,1))
hist(gini1,prob = TRUE)
lines(density(gini1),col = "red",lwd = 2)
hist(gini2,prob = TRUE)
lines(density(gini2),col = "blue",lwd = 2)
hist(gini3,prob = TRUE)
lines(density(gini3),col = "green",lwd = 2)
medians[1] <- median(gini1)
medians[2] <- median(gini2)
medians[3] <- median(gini3)
medians
quantiles1 <- as.vector(quantile(gini1,seq(0.1,0.9,0.1)))
quantiles1
quantiles2 <- as.vector(quantile(gini2,seq(0.1,0.9,0.1)))
quantiles2
quantiles3 <- as.vector(quantile(gini3,seq(0.1,0.9,0.1)))
quantiles3
means[1] <- mean(gini1)
means[2] <- mean(gini2)
means[3] <- mean(gini3)
means
#The answer of second question:
m <- 1000
n <- 20
gini <- numeric(m)
#Get A series of gini ratios genarating from a lognormal distribution
for (i in 1:m) {
x <- sort(rlnorm(n))
xmean <- mean(x)
for (j in 1:n) {
y[j] <- (2*j-n-1)*x[j]
}
gini[i] <- 1/n^2/xmean*sum(y[1:n])
}
#get the lower confidence interval
LCI<- mean(gini)-sd(gini)*qt(0.975,m-1)
#get the upper confidence interval
UCI <- mean(gini)+sd(gini)*qt(0.975,m-1)
#get the confidence interval
CI <- c(LCI,UCI)
print(CI)
#calculate the converage rte
covrate<-sum(I(gini>CI[1]&gini<CI[2]))/m
print(covrate)
#The answer of third question:
#We need load the MASS package
library(MASS)
mean <- c(0, 0)
sigma <- matrix( c(25,5,
5, 25),nrow=2, ncol=2)
m <- 1000
#Calculate the power using pearson correlation test by setting the parameter method as pearson
pearvalues <- replicate(m, expr = {
mydata1 <- mvrnorm(50, mean, sigma)
x <- mydata1[,1]
y <- mydata1[,2]
peartest <- cor.test(x,y,alternative = "two.sided", method = "pearson")
peartest$p.value
} )
power1 <- mean(pearvalues <= 0.05)
power1
#Calculate the power using spearman correlation test by setting the parameter method as spearman
spearvalues <- replicate(m, expr = {
mydata2 <- mvrnorm(50, mean, sigma)
x <- mydata2[,1]
y <- mydata2[,2]
speartest <- cor.test(x,y,alternative = "two.sided", method = "spearman")
speartest$p.value
} )
power2 <- mean(spearvalues <= 0.05)
power2
#Calculate the power using kendall correlation test by setting the parameter method as kendall
kenvalues <- replicate(m, expr = {
mydata3 <- mvrnorm(50, mean, sigma)
x <- mydata3[,1]
y <- mydata3[,2]
kentest <- cor.test(x,y,alternative = "two.sided", method = "kendall")
kentest$p.value
} )
power3 <- mean(kenvalues <= 0.05)
power3
R <- 1000
m <- 1000
#Calculate the power using pearson correlation test by setting the parameter method as pearson
pearvalues <- replicate(m, expr = {
u <- runif(R/2,0,10)
v <- sin(u)
peartest <- cor.test(u,v,alternative = "two.sided", method = "pearson")
peartest$p.value
} )
power1 <- mean(pearvalues <= 0.05)
power1
#Calculate the power using spearman correlation test by setting the parameter method as spearman
spearvalues <- replicate(m, expr = {
u <- runif(R/2,0,10)
v <- sin(u)
speartest <- cor.test(u,v,alternative = "two.sided", method = "spearman")
speartest$p.value
} )
power2 <- mean(spearvalues <= 0.05)
power2
#Calculate the power using kendall correlation test by setting the parameter method as kendall
kenvalues <- replicate(m, expr = {
u <- runif(R/2,0,10)
v <- sin(u)
kentest <- cor.test(u,v,alternative = "two.sided", method = "kendall")
kentest$p.value
} )
power3 <- mean(kenvalues <= 0.05)
power3
The fifth homework answer in 2018-11-02
The question is :
-
Compute a jackknife estimate of the bias and the standard error of the corre-
lation statistic in Example 7.2.
-
Refer to Exercise 7.4. Compute 95% bootstrap confidence intervals for the mean time between failures $\frac1\lambda$ by the standard normal, basic, percentile, and BCa methods. Compare the intervals and explain why they may differ.
-
Refer to Exercise 7.7. Obtain the jackknife estimates of bias and standard error of $\hat{\theta}$
-
In Example 7.18, leave-one-out (n-fold) cross validation was used to select the best fitting model. Use leave-two-out cross validation to compare the models.
The answer code is as follows:
#The answer of first question:
LAST <- c(576,635,558,578 ,666 ,580 ,555 ,661 ,651 ,605 ,653 ,575 ,545 ,572 ,594)
GPA <- c(339 ,330 ,281 ,303 ,344 ,307 ,300 ,343 ,336 ,313 ,312 ,274 ,276 ,288 ,296)
n <- length(LAST)
theta.hat <- cor(LAST,GPA) #the Correlation between the LAST and GPA we give above
theta.jack <- numeric(n)
library('bootstrap')
#compute the jackknife replicates, leave-one-out estimates
for (i in 1:n)
theta.jack[i] <- cor(LAST[-i],GPA[-i])
bias <- (n - 1) * (mean(theta.jack) - theta.hat)
#jackknife estimate of bias
print(bias)
se <- sqrt((n-1)* mean((theta.jack - mean(theta.jack))^2))
print(se)
#The answerof second question:
set.seed(1)
air <- c(3,5,7,18,43,85,91,98,100,130,230,487)
library(boot)
lamda.boot <- function(data,ind) {
f <- function (lamda) n/lamda-s
n <- length(data[ind])
s <- sum(data[ind])
#the function uniroot can be used to get the root
root <- uniroot(f,c(0,1))$root
1/root
}
boot.obj <- boot(data=air, statistic = lamda.boot, R = 3000)
print(boot.obj)
print(boot.ci(boot.obj,type = c("basic", "norm", "perc","bca")))
#The answer of third question:
library(bootstrap)
n <- nrow(scor)
#get the eigenvalues
eigenvalues <- eigen(cov(scor))$values
#the estimate computed from the original observed sample
theta.hat <- max(eigenvalues)/sum(eigenvalues)
theta.jack <- numeric(n)
for (i in 1:n)
theta.jack[i] <- max(eigen(cov(scor[-i,]))$values)/sum(eigen(cov(scor[-i,]))$values)
bias <- (n - 1) * (mean(theta.jack) - theta.hat)
print(bias) #jackknife estimate of bias
se <- sqrt((n-1)* mean((theta.jack - mean(theta.jack))^2))
print(se)
#The answer of fourth question:
library(DAAG)
attach(ironslag)
#the length of the chemical
n <- length(chemical)
#A pairwise combination of 1 and n
comb <- t(combn(1:n,2))
e1 <- e2 <- e3 <- e4 <- e5 <- e6 <- e7 <- e8 <- numeric(n)
for (k in 1:n) {
#comb is a matrix with two columns
k1 <- comb[k,1]
k2 <- comb[k,2]
#get the training set
x <- chemical[-k1][-(k2-1)]
y <- magnetic[-k1][-(k2-1)]
#use the training set to get the models
J1 <- lm(y ~ x)
#use the test set to get the prediction error
yhat1 <- J1$coef[1] + J1$coef[2] * chemical[k1]
yhat2 <- J1$coef[1] + J1$coef[2] * chemical[k2]
e1[k] <- magnetic[k1] - yhat1
e2[k] <- magnetic[k2] - yhat2
#the following procedures are the same as the above
J2 <- lm(y ~ x + I(x^2))
yhat3 <- J2$coef[1] + J2$coef[2] * chemical[k1] +
J2$coef[3] * chemical[k1]^2
yhat4 <- J2$coef[1] + J2$coef[2] * chemical[k2] +
J2$coef[3] * chemical[k2]^2
e3[k] <- magnetic[k1] - yhat3
e4[k] <- magnetic[k2] - yhat4
J3 <- lm(log(y) ~ x)
logyhat5 <- J3$coef[1] + J3$coef[2] * chemical[k1]
logyhat6 <- J3$coef[1] + J3$coef[2] * chemical[k2]
yhat5 <- exp(logyhat5)
yhat6 <- exp(logyhat6)
e5[k] <- magnetic[k1] - yhat5
e6[k] <- magnetic[k2] - yhat6
J4 <- lm(log(y) ~ log(x))
logyhat7 <- J4$coef[1] + J4$coef[2] * log(chemical[k1])
logyhat8 <- J4$coef[1] + J4$coef[2] * log(chemical[k2])
yhat7 <- exp(logyhat7)
yhat8 <- exp(logyhat8)
e7[k] <- magnetic[k1] - yhat7
e8[k] <- magnetic[k2] - yhat8
}
# get the prediction error by leave-two-out cross validation.
c(mean(e1^2+e2^2), mean(e3^2+e4^2), mean(e5^2+e6^2), mean(e7^2+e8^2))
The sixth homework answer in 2018-11-16
The question is :
-
Implement the two-sample Cram?? er-von Mises test for equal distributions as a
permutation test. Apply the test to the data in Examples 8.1 and 8.2.
-
Design experiments for evaluating the performance of the NN,energy, and ball methods in various situations??
(1). Unequal variances and equal expectations
(2).Unequal variances and unequal expectations
(3).Non-normal distributions: t distribution with 1 df (heavy-tailed
distribution), bimodel distribution (mixture of two normal distributions)
(4).Unbalanced samples (say, 1 case versus 10 controls)
-
Use the Metropolis-Hastings sampler to generate random variables from a standard Cauchy distribution. Discard the first 1000 of the chain, and compare the deciles of the generated observations with the deciles of the standard Cauchy distribution (see qcauchyor qt with df=1). Recall that a
Cauchy$(\theta,\eta)$ distribution has density function
$${f(x)}=\frac{1}{\theta\pi(1+[(x-\eta)/\theta]^2)},-\infty0$$
-
Rao [220, Sec. 5g] presented an example on genetic linkage of 197 animals in four categories (also discussed in [67, 106, 171, 266]). The group sizes are (125,18,20,34). Assume that the probabilities of the corresponding multinomial distribution are
$$\left(\frac{1}{2}+\frac{\theta}{4},\frac{1-\theta}{4},\frac{1-\theta}{4},\frac{\theta}{4}\right)$$
Estimate the posterior distribution of θ given the observed sample, using one of the methods in this chapter.
The answer code is as follows:
#The answer of first question:
set.seed(1)
x <- c(158, 171 ,193 ,199 ,230 ,243 ,248 ,248 ,250 ,267 ,271 ,316 ,327 ,329)
y <- c(141 ,148 ,169 ,181 ,203 ,213 ,229 ,244 ,257 ,260 ,271 ,309)
#the function cvm.test is used to calculate the Cramer-von Mises statistic
cvm.test <- function(x,y){
#the empirical distribution function of the x,y
F <- ecdf(x)
G <- ecdf(y)
n <- length(x)
m <- length(y)
s <- numeric(n)
t <- numeric(m)
for (i in 1:n) {
s[i] <- (F(x[i])-G(x[i]))^2
}
s <- sum(s)
for (j in 1:m) {
t[j] <- (F(y[j])-G(y[j]))^2
}
t <- sum(t)
#return the Cramer-von Mises statistic
return (m*n*(s+t)/(m+n)^2)
}
#number of replicates
R <- 999
#pooled sample
z <- c(x, y)
K <- 1:26
#storage for replicates
reps <- numeric(R)
t0 <- cvm.test(x, y)
for (i in 1:R) {
#generate indices k for the first sample
k <- sample(K, size = 14, replace = FALSE)
x1 <- z[k]
y1 <- z[-k] #complement of x1
reps[i] <- cvm.test(x1, y1)
}
p <- mean(c(t0, reps) >= t0)
p
hist(reps, main = "", freq = FALSE, xlab = "T (p = 0.421)",
breaks = "scott")
points(t0, 0, cex = 1, pch = 16)
#The answer of second question:
library(RANN)
library(boot)
library(energy)
library(Ball)
m <- 100; k<-3; p<-2; mu <- 0.2; set.seed(12345)
n1 <- n2 <- 20; R<-50; n <- n1+n2; N = c(n1,n2)
Tn <- function(z, ix, sizes,k) {
n1 <- sizes[1]; n2 <- sizes[2]; n <- n1 + n2
if(is.vector(z)) z <- data.frame(z,0);
z <- z[ix, ];
NN <- nn2(data=z, k=k+1)
block1 <- NN$nn.idx[1:n1,-1]
block2 <- NN$nn.idx[(n1+1):n,-1]
i1 <- sum(block1 < n1 + .5); i2 <- sum(block2 > n1+.5)
(i1 + i2) / (k * n)
}
#the nn method and return the p.values
eqdist.nn <- function(z,sizes,k){
boot.obj <- boot(data=z,statistic=Tn,R=R,
sim = "permutation", sizes = sizes,k=k)
ts <- c(boot.obj$t0,boot.obj$t)
p.value <- mean(ts>=ts[1])
list(statistic=ts[1],p.value=p.value)
}
p.values1 <- matrix(NA,m,3)
p.values2 <- matrix(NA,m,3)
p.values3 <- matrix(NA,m,3)
p.values4 <- matrix(NA,m,3)
for(i in 1:m){
#the sample x is from the standard normal distribution and the sample y is from the normal diisreibution with mean=0,variance is 2
x <- matrix(rnorm(n1*p),ncol=p);
y <- cbind(rnorm(n2,sd=2),rnorm(n2,sd=2));
z <- rbind(x,y)
p.values1[i,1] <- eqdist.nn(z,N,k)$p.value
p.values1[i,2] <- eqdist.etest(z,sizes=N,R=R)$p.value
p.values1[i,3] <- bd.test(x=x,y=y,R=999,seed=i*12345)$p.value
}
alpha <- 0.1;
pow1 <- colMeans(p.values1<alpha)
names(pow1) <- c('NN','energy', 'ball ')
#get the powers by the three method
pow1
#under the situation of unequal variances and unequal expectations
for(i in 1:m){
#the sample x is from the standard normal distribution and the sample y is from the normal diisreibution with mean=0.2 ,variance is 2
x <- matrix(rnorm(n1*p),ncol=p);
y <- cbind(rnorm(n2,mean = mu,sd=2),rnorm(n2,mean=mu,sd=2));
z <- rbind(x,y)
p.values2[i,1] <- eqdist.nn(z,N,k)$p.value
p.values2[i,2] <- eqdist.etest(z,sizes=N,R=R)$p.value
p.values2[i,3] <- bd.test(x=x,y=y,R=999,seed=i*12345)$p.value
}
alpha <- 0.1;
pow2 <- colMeans(p.values2<alpha)
names(pow2) <- c('NN','energy', 'ball ')
#get the powers by the three method
pow2
#under the situation of non-normal distributions
for(i in 1:m){
#the sample x is from t distribution with df=1 and the sample y is from the mixture of two normal distributions
x <- matrix(rt(n1*p,df=1),ncol = p);
y <- cbind(rnorm(n2),rnorm(n2,mean=mu,sd=4));
z <- rbind(x,y)
p.values3[i,1] <- eqdist.nn(z,N,k)$p.value
p.values3[i,2] <- eqdist.etest(z,sizes=N,R=R)$p.value
p.values3[i,3] <- bd.test(x=x,y=y,R=999,seed=i*12345)$p.value
}
alpha <- 0.1;
pow3 <- colMeans(p.values3<alpha)
names(pow3) <- c('NN','energy', 'ball ')
#get the powers by the three method
pow3
#under the situation of Unbalanced samples which the number of x is 200 and y is 20
n1 <- 200
n2 <- 20
n <- n1+n2
N = c(n1,n2)
for(i in 1:m){
x <- matrix(rnorm(n1*p),ncol=p);
y <- cbind(rnorm(n2),rnorm(n2));
z <- rbind(x,y)
p.values4[i,1] <- eqdist.nn(z,N,k)$p.value
p.values4[i,2] <- eqdist.etest(z,sizes=N,R=R)$p.value
p.values4[i,3] <- bd.test(x=x,y=y,R=99,seed=i*12345)$p.value
}
alpha <- 0.18;
pow4 <- colMeans(p.values4<alpha)
names(pow4) <- c('NN','energy', 'ball ')
#get the powers by the three method
pow4
#The answer of third question:
set.seed(1)
#build a standard Cauchy distribution
f <- function(x, x0=0, gamma=1){
out<-1/(pi*gamma*(1+((x-x0)/gamma)^2))
return(out)
}
#the times of simulation
m <- 80000
x <- numeric(m)
#generat the normal proposal distribution with mean=xt ,sd=1
x[1] <- rnorm(1,mean=0,sd=1 )
k <- 0
u <- runif(m)
for (i in 2:m) {
xt <- x[i-1]
y <- rnorm(1, mean = xt,sd=1)
num <- f(y) * dnorm(xt, mean = y,sd=1)
den <- f(xt) * dnorm(y, mean = xt,sd=1)
if (u[i] <= num/den) x[i] <- y else {
x[i] <- xt
k <- k+1 #y is rejected
}
}
#discard the burnin sample
b <- 1001
y <- x[b:m]
a <- ppoints(200)
#quantiles of cauchy distribution
Qcauchy <- qcauchy(a)
#quantiles of sample distribution
Q <- quantile(x, a)
qqplot(Qcauchy, Q,xlim=c(-2,2),ylim=c(-2,2),xlab="Cauchy Quantiles", ylab="Sample Quantiles",main = expression("Q-Q plot for Cauchy distribution"))
hist(y, breaks=50, main="", xlab="", freq=FALSE)
lines(Qcauchy, f(Qcauchy))
#The answer of fourth question:
size <- c(125,18,20,34)
#The following function prob computes the target density
prob <- function(y, size) {
if (y < 0 || y >1)
return (0)
return((1/2+y/4)^size[1] *
((1-y)/4)^size[2] * ((1-y)/4)^size[3] *(y/4)^size[4])
}
#length of the chain
m <- 5000
#width of the uniform support set
w <- .25
#burn-in time
burn <- 1000
#for accept/reject step
u <- runif(m)
#proposal distribution
v <- runif(m, -w, w)
x[1] <- .25
for (i in 2:m) {
y <- x[i-1] + v[i]
if (u[i] <= prob(y, size) / prob(x[i-1], size))
x[i] <- y else
x[i] <- x[i-1]
}
xb <- x[(burn+1):m]
print(mean(xb))
The seventh homework answer in 2018-11-23
The question is :
-
For exercise 9.6, use the Gelman-Rubin method to monitor convergence of the chain, and run the chain until the chain has converged approximately to the target distribution
according to R < 1.2.
-
Find the intersection points A(k) in (0,$\sqrt{A}$)
$$S_{k-1}(a)=P\left(t(k-1)>\sqrt{\frac{a^2(k-1)}{k-a^2}}\right)$$
and
$$S_{k}(a)=P\left(t(k)>\sqrt{\frac{a^2(k)}{k+1-a^2}}\right)$$
for k = 4 : 25,100,500,1000, where t(k) is a Student t random variable with k degrees of freedom. (These intersection points determine the critical values for a t-test for scale-mixture errors proposed by Sz?? ekely [260].)
The answer code is as follows:
#The answer of first question:
set.seed(1)
size <- c(125,18,20,34)
#The following function prob computes the target density
prob <- function(y, size) {
if (y < 0 || y >1)
return (0)
return((1/2+y/4)^size[1] *
((1-y)/4)^size[2] * ((1-y)/4)^size[3] *(y/4)^size[4])
}
Gelman.Rubin <- function(psi) {
# psi[i,j] is the statistic psi(X[i,1:j])
# for chain in i-th row of X
psi <- as.matrix(psi)
n <- ncol(psi)
k <- nrow(psi)
psi.means <- rowMeans(psi) #row means
B <- n * var(psi.means) #between variance est.
psi.w <- apply(psi, 1, "var") #within variances
W <- mean(psi.w) #within est.
v.hat <- W*(n-1)/n + (B/n) #upper variance est.
r.hat <- v.hat / W #G-R statistic
return(r.hat)
}
multinomial.chain <- function(N, X1) {
#generates a Metropolis chain for multino-mial distribution
#with uniform(-w,w) proposal distribution
#and starting value X1
#length of the chain
#width of the uniform support set
w <- 0.15
#burn-in time
burn <- 1000
#for accept/reject step
u <- runif(N)
#proposal distribution
v <- runif(N, -w, w)
x <- rep(0, N)
x[1] <- X1
for (i in 2:N) {
y <- x[i-1] + v[i]
r1 <- prob(y, size)
r2 <- prob(x[i-1], size)
r <- r1/r2
if (u[i] <= r)
x[i] <- y else
x[i] <- x[i-1]
}
return(x)
}
k <- 4 #number of chains to generate
n <- 10000 #length of chains
b <- 2000 #burn-in length
#choose overdispersed initial values
x0 <- c(0.1,0.4,0.6,0.9)
#generate the chains
X <- matrix(0, nrow=k, ncol=n)
for (i in 1:k)
X[i, ] <- multinomial.chain(n, x0[i])
#compute diagnostic statistics
psi <- t(apply(X, 1, cumsum))
for (i in 1:nrow(psi))
psi[i,] <- psi[i,] / (1:ncol(psi))
print(Gelman.Rubin(psi))
#get sequences of the running means ?? for four Metropolis-Hastings chains
par(mfrow=c(2,2))
for (i in 1:k)
plot(psi[i, (b+1):n], type="l",
xlab=i, ylab=bquote(psi))
par(mfrow=c(1,1)) #restore default
rhat <- rep(0, n)
for (j in (b+1):n)
rhat[j] <- Gelman.Rubin(psi[,1:j])
plot(rhat[(b+1):n], type="l", xlab="", ylab="R")
abline(h=1.05, lty=2)
#The answer of second question:
m <- c(4:25,100,500,1000)
root <- numeric(length(m))
f1 <- function(a,k) pt(sqrt(a^2*(k-1)/(k-a^2)),k-1)-pt(sqrt(a^2*(k)/(k+1-a^2)),k)
for (i in 4:25) {
res <- uniroot(f1,c(1e-5,2),k=i)
root[i-3] <- res$root
}
res1 <- uniroot(f1,c(1e-5,2),k=100)
root[23] <- res1$root
res2 <- uniroot(f1,c(1e-5,2),k=500)
root[24] <- res2$root
res3 <- uniroot(f1,c(1e-5,2),k=1000)
root[25] <- res3$root
data <- data.frame(k=m,a=root,row.names=1)
data
The eighth homework answer in 2018-11-30
The question is :
-
Write a function to compute the cdf of the Cauchy distribution, which has density$$\frac{1}{\theta\pi(1+[(x-\eta)/\theta]^{2})}, -\infty
(Also see the source code in pcauchy.c.)
-
- A-B-O blood type problem
- Let the three alleles be A, B, and O.
Genotype AA BB OO AO BO AB AA
Frequency p2 q2 r2 2pr 2qr 2pq 1
Count nAA nBB nOO nAO nBO nAB n
Observed data: $n_{A\cdot}=n_{AA}+n_{AO}=28$ (A-type), $n_{B\cdot}=n_{BB}+n_{BO}=24$ (B-type), $n_{OO}=41$ (O-type), $n_{AB}=70$ (AB-type).
Use EM algorithm to solve MLE of $p$ and $q$ (consider missing data $n_{AA}$ and $n_{BB}$).
Record the maximum likelihood values in M-steps, are they increasing?
The answer code is as follows:
#The answer of first question:
options(digits=6)
#get the density of cauchy distribution with the parameter theta and eta
f <- function(x, theta, eta) {
1/theta/pi/(1+((x-eta)/theta)^2)
}
#the vector x is used to get the cdf of cauchy distribution when theta is 1,eta is 0
x <- seq(-5,5,0.1)
values <- numeric(length(x))
#the vector x2 is used to get the cdf of cauchy distribution when theta is 1,eta is 10
x2 <- seq(0,10,0.1)
values2 <- numeric(length(x2))
for (i in 1:length(x))
values[i] <- integrate(f, lower=-Inf, upper=x[i],rel.tol=.Machine$double.eps^0.25,theta=1, eta=0)$value
#
data <- data.frame(value=values,truevalues=pcauchy(x),errors=values-pcauchy(x))
data[1:20,]
#compare the cauchy distributions,one of them is used integrate and the other is used pcauchy
plot(x,values,pch=21,xlab="the values of x", ylab="the cauchy distribution function ",main = expression("the compare for Cauchy distribution when \ntheta is 1 and eta is 0 with two methods"))
lines(x,pcauchy(x),col='red',lwd=2)
#the situation when theta is 1 and eta is 10
for (i in 1:length(x2))
values2[i] <- integrate(f, lower=-Inf,upper=x2[i],rel.tol=.Machine$double.eps^0.25,theta=1, eta=10)$value
data2 <- data.frame(value=values2,truevalues=pcauchy(x2,location=10,scale=1),error=values2-pcauchy(x2,location=10,scale=1))
data2[1:20,]
#compare the cauchy distributions,one of them is used integrate and the other is used pcauchy
plot(x2,values2,pch=21,xlab="the values of x", ylab="the cauchy distribution function ",main = expression("the compare for Cauchy distribution when \ntheta is 1 and eta is 10 with two methods"))
lines(x2,pcauchy(x2,location = 10,scale = 1),col='red',lwd=2)
#The answer of second question:
library(nloptr)
# the maximum likehood function
f <- function(x,x1,nA=28,nB=24,nOO=41,nAB=70) {
r1<-1-sum(x1)
nAA<-nA*x1[1]^2/(x1[1]^2+2*x1[1]*r1)
nBB<-nB*x1[2]^2/(x1[2]^2+2*x1[2]*r1)
r<-1-sum(x)
return(-2*nAA*log(x[1])-2*nBB*log(x[2])-2*nOO*log(r)-
(nA-nAA)*log(2*x[1]*r)-(nB-nBB)*log(2*x[2]*r)-nAB*log(2*x[1]*x[2]))
}
# constraint function
g <- function(x,x1,nA=28,nB=24,nOO=41,nAB=70) {
return(sum(x)-0.999999)
}
opts <- list("algorithm"="NLOPT_LN_COBYLA",
"xtol_rel"=1.0e-8)
mle<-NULL
r<-matrix(0,1,2)
r<-rbind(r,c(0.2,0.35))# the beginning value of p0 and q0
j<-2
while (sum(abs(r[j,]-r[j-1,]))>1e-8) {
res <- nloptr( x0=c(0.3,0.25),
eval_f=f,
lb = c(0,0), ub = c(1,1),
eval_g_ineq = g,
opts = opts, x1=r[j,],nA=28,nB=24,nOO=41,nAB=70 )
j<-j+1
r<-rbind(r,res$solution)
mle<-c(mle,f(x=r[j,],x1=r[j-1,]))
}
r #the result of EM algorithm
mle #the max likelihood values
The nineth homework answer in 2018-12-07
The question is :
-
Use both for loops and lapply() to fit linear models to the mtcars using the formulas stored in this list:
formulas <- list(
mpg ~ disp,
mpg ~ I(1 / disp),
mpg ~ disp + wt,
mpg ~ I(1 / disp) + wt
)
-
Fit the model mpg ~ disp to each of the bootstrap replicates of mtcars in the list below by using a for loop and lapply() .Can you do it without an anonymous function?
bootstraps <- lapply(1:10, function(i) {
rows <- sample(1:nrow(mtcars), rep = TRUE)
mtcars[rows, ]
})
-
For each model in the previous two exercises, extract R 2 using the function below.
rsq <- function(mod) summary(mod)$r.squared
-
The following code simulates the performance of a t-test for non-normal data. Use sapply() and an anonymous function to extract the p-value from every trial.
trials <- replicate(
100,
t.test(rpois(10, 10), rpois(7, 10)),
simplify = FALSE
)
Extra challenge: get rid of the anonymous function by using [[ directly.
-
Implement a combination of Map() and vapply() to create an lapply() variant that iterates in parallel over all of its inputs and stores its outputs in a vector (or a matrix). What arguments should the function take?
The answer code is as follows:
#The answer of first question:
formulas <- list(
mpg ~ disp,
mpg ~ I(1 / disp),
mpg ~ disp + wt,
mpg ~ I(1 / disp) + wt
)
lapply(formulas,lm,data=mtcars)
out <- vector("list", length(formulas))
for (i in seq_along(formulas)) {
out[[i]] <- lm(formulas[[i]], data = mtcars)
}
out
#The answer of second question:
set.seed(1)
bootstraps <- lapply(1:10, function(i) {
rows <- sample(1:nrow(mtcars), rep = TRUE)
mtcars[rows, ]
})
getvalue <- function(x){
vars <- c('mpg','disp')
x[vars]
}
data <- lapply(bootstraps, getvalue)
lapply(data,lm)
set.seed(1)
out1 <- vector("list", length(bootstraps))
for (i in seq_along(bootstraps)) {
out1[[i]] <- getvalue(bootstraps[[i]])
}
out2 <- vector("list", length(out1))
for (i in seq_along(out1)) {
out2[[i]] <- lm(out1[[i]])
}
out2
#The answer of third question:
formulas <- list(
mpg ~ disp,
mpg ~ I(1 / disp),
mpg ~ disp + wt,
mpg ~ I(1 / disp) + wt
)
funclist <- lapply(formulas,lm,data=mtcars)
rsq <- function(mod) summary(mod)$r.squared
lapply(funclist, rsq)
getvalue <- function(x){
vars <- c('mpg','disp')
x[vars]
}
data <- lapply(bootstraps, getvalue)
data1 <- lapply(data,lm)
rsq <- function(mod) summary(mod)$r.squared
unlist(lapply(data1, rsq))
#The answer of fourth question:
trials <- replicate(
100,
t.test(rpois(10, 10), rpois(7, 10)),
simplify = FALSE
)
#use an anonymous function
sapply(trials, function(mod) mod$p.value)
#The answer of fifth question:
a <- c(1,2,3,4,5)
b <- c(6,7,8,9,10)
c <- c(2,4,6,8,10)
d <- data.frame(a,b,c)
#the lapply1 function can parallelly get the sum of the input vectors
lapply1 <- function(x,...){
args <- list(x,...)
for (a in args) x <- x+a
x
}
lapply1(a,b,c)
lapply1(d[1],d[2],d[3])
#the lapply2 function can parallelly get the product of the input vectors
lapply2 <- function(x,...){
args <- list(...)
for (a in args) x <- x*a
x
}
lapply2(a,b,c)
The third homework answer in 2018-12-14
The question is :
-
Make a faster version of chisq.test() that only computes the chi-square test statistic when the input is two numeric vectors with no missing values. You can try simplifying chisq.test() or by coding from the mathematical definition (http://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test ).
-
Can you make a faster version of table() for the case of an input of two integer vectors with no missing values? Can you use it to speed up your chi-square test?
The answer code is as follows:
#The answer of first question:
chisq.test1<- function (x, y = NULL, correct = TRUE, p = rep(1/length(x), length(x)),
rescale.p = FALSE, simulate.p.value = FALSE, B = 2000)
{
DNAME <- deparse(substitute(x))
#get the total sum of x
n <- sum(x)
if (is.matrix(x)) {
METHOD <- "changed Pearson's Chi-squared test"
nr <- as.integer(nrow(x))
nc <- as.integer(ncol(x))
if (is.na(nr) || is.na(nc) || is.na(nr * nc))
stop("invalid nrow(x) or ncol(x)", domain = NA)
#get the rowsums of x
sr <- rowSums(x)
#get the colsums of x
sc <- colSums(x)
#The sum of rows and columns that's sr,sc divided by n
E <- outer(sr, sc, "*")/n
v <- function(r, c, n) c * r * (n - r) * (n - c)/n^3
V <- outer(sr, sc, v, n)
dimnames(E) <- dimnames(x)
#the situation of the sum of rows and columns are all not equal 0
if (simulate.p.value && all(sr > 0) && all(sc > 0)) {
setMETH()
tmp <- .Call(C_chisq_sim, sr, sc, B, E)
STATISTIC <- sum(sort((x - E)^2/E, decreasing = TRUE))
PARAMETER <- NA
PVAL <- (1 + sum(tmp >= almost.1 * STATISTIC))/(B +
1)
}
else {
if (simulate.p.value)
warning("cannot compute simulated p-value with zero marginals")
if (correct && nrow(x) == 2L && ncol(x) == 2L) {
YATES <- min(0.5, abs(x - E))
if (YATES > 0)
METHOD <- paste(METHOD, "with Yates' continuity correction")
}
else YATES <- 0
STATISTIC <- sum((abs(x - E) - YATES)^2/E)
PARAMETER <- (nr - 1L) * (nc - 1L)
PVAL <- pchisq(STATISTIC, PARAMETER, lower.tail = FALSE)
}
}
names(STATISTIC) <- "X-squared"
names(PARAMETER) <- "df"
if (any(E < 5) && is.finite(PARAMETER))
warning("Chi-squared approximation may be incorrect")
#output the result
structure(list(statistic = STATISTIC, parameter = PARAMETER,
p.value = PVAL, method = METHOD, data.name = DNAME, observed = x,
expected = E, residuals = (x - E)/sqrt(E), stdres = (x -
E)/sqrt(V)), class = "htest")
}
library(vcd)
Treatment <- Arthritis$Treatment
Improved <- Arthritis$Improved
mytable <- table(Treatment,Improved)
#you can see the changed chisq.test1's output is the same as the original function
chisq.test(mytable)
chisq.test1(mytable)
#compare the calculate speed
set.seed(1)
x <- sample(1e5,1e6,10)
y <- sample(1e5,1e6,10)
system.time(chisq.test(rbind(x,y)))
system.time(chisq.test1(rbind(x,y)))
#The answer of second question:
table2 <- function (... , dnn = list.names(...), deparse.level = 1)
{
list.names <- function(...) {
l <- as.list(substitute(list(...)))[-1L]
nm <- names(l)
fixup <- if (is.null(nm))
seq_along(l)
else nm == ""
dep <- vapply(l[fixup], function(x) switch(deparse.level +
1, "", if (is.symbol(x)) as.character(x) else "",
deparse(x, nlines = 1)[1L]), "")
if (is.null(nm))
dep
else {
nm[fixup] <- dep
nm
}
}
args <- list(...)
if (length(args) == 1L && is.list(args[[1L]])) {
args <- args[[1L]]
if (length(dnn) != length(args))
dnn <- if (!is.null(argn <- names(args)))
argn
else paste(dnn[1L], seq_along(args), sep = ".")
}
bin <- 0L
lens <- NULL
dims <- integer()
pd <- 1L
dn <- NULL
for (a in args) {
if (is.null(lens))
lens <- length(a)
else if (length(a) != lens)
stop("all arguments must have the same length")
fact.a <- is.factor(a)
if (!fact.a) {
a0 <- a
a <- factor(a)
}
ll <- levels(a)
a <- as.integer(a)
nl <- length(ll)
dims <- c(dims, nl)
if (prod(dims) > .Machine$integer.max)
stop("attempt to make a table with >= 2^31 elements")
dn <- c(dn, list(ll))
bin <- bin + pd * (a - 1L)
pd <- pd * nl
}
names(dn) <- dnn
bin <- bin[!is.na(bin)]
if (length(bin))
bin <- bin + 1L
y <- array(tabulate(bin, pd), dims, dimnames = dn)
class(y) <- "table"
y
}
set.seed(1)
z <- sample(1:50,1e7,replace = TRUE)
system.time(table(z))
system.time(table2(z))
Panxiaoba/StatComp18013 documentation built on May 5, 2019, 11:06 p.m.
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Overview
StatComp18013 is a simple R package developed to present the homework answers for the 'Statistical Computing' course.
The first homework answer in 2018-09-14
The question is : Write a .Rmd file to implement at least three examples of different types in the above books (texts, numerical results, tables, and figures).
The code is as follows:
#The first example: x <- rnorm(10) y <- rnorm(10) plot(x,y) #The second example: data("InsectSprays") aov.spray <- aov(sqrt(count) ~ spray,data=InsectSprays) summary(aov.spray) #The third example: m1 <- matrix(1,2,2) m2 <- matrix(2,2,2) rbind(m1,m2) cbind(m1,m2)
The second homework answer in 2018-09-21
The question is :
-
A discrete random variable X has probability mass function
x 0 1 2 3 4 p(x) 0.1 0.2 0.2 0.2 0.3
Use the inverse transform method to generate a random sample of size 1000 from the distribution of X. Construct a relative frequency table and compare the empirical with the theoretical probabilities. Repeat using the R sample function.
2 .Write a function to generate a random sample of size n from the Beta(a,b) distribution by the acceptance-rejection method. Generate a random sample of size 1000 from the Beta(3,2) distribution. Graph the histogram of the sample with the theoretical Beta(3,2) density superimposed.
- Simulate a continuous Exponential-Gamma mixture. Suppose that the rate parameter Λ has Gamma(r,β) distribution and Y has Exp(Λ) distribution. That is, $(Y |\Lambda = \lambda) \sim f_{Y}(y|\lambda) = \lambda e^{−\lambda y}$. Generate 1000 random observations from this mixture with r = 4 andβ = 2.
The answer code is as follows:
#The answer of first question: x <- c(0,1,2,3,4) p <- c(0.1,0.2,0.2,0.2,0.3) cp <- cumsum(p) m <- 1e4 r <- numeric(m) r <- x[findInterval(runif(m),cp)+1] ct <- table(r)/m x <- sample(c(0,1,2,3,4),m,p,replace = TRUE) y <- table(x)/m rbind(prob=p,prob1=ct,prob2=y) #The answer of second question: beta <- function(n,a,b){ j<-k<-0 y <- numeric(n) while (k < n) { u <- runif(1) j <- j + 1 x <- runif(1) if (x^(a-1)* (1-x)^(b-1) > u) { k <- k + 1 y[k] <- x } } return (y) } r<-beta(1000,3,2) hist(r,prob=TRUE,main="the histogram of Beta(3,2) and its density",col="red") t<-seq(0,1,0.001) s<-dbeta(t,3,2) lines(t,s,col="blue",lwd=3) #The answer of third question: n <- 1e3; r <- 4; beta <- 2 lambda <- rgamma(n, r, beta) y <- rexp(n, lambda) y
The third homework answer in 2018-09-28
The question is :
-
Write a function to compute a Monte Carlo estimate of the Beta(3, 3) cdf,and use the function to estimate F(x) for x = 0.1,0.2,...,0.9. Compare the estimates with the values returned by the pbeta function in R.
-
The Rayleigh density [156, (18.76)] is $${f(x)}=\frac{x}{\sigma^2}e^{-x^2/{(2\sigma^2)}},\qquad x\geq0,\sigma>0$$ Implement a function to generate samples from a Rayleigh distribution,using antithetic variables. What is the percent reduction in variance of $\frac{x+x'}{2}$ compared with $\frac{x_1+x_2}{2}$ for independent ${x_1+x_2}$.
-
Find two importance functions $f_{1}$ and $f_{2}$ that are supported on $(1,\infty)$ and are close? to $$g(x)=\frac{x^{2}}{\sqrt{2\pi}}e^{\frac{-x^{2}}{2}},\qquad x>1.$$ Which of your two importance functions should produce the smaller variance in estimating$$\int_{1}^{\infty}\frac{x^{2}}{\sqrt{2\pi}}e^{\frac{-x^{2}}{2}}dx$$by importance sampling Explain.
-
Obtain a Monte Carlo estimate of$$\int_{1}^{\infty}\frac{x^{2}}{\sqrt{2\pi}}e^{\frac{-x^{2}}{2}}dx$$ by importance sampling.
The answer code is as follows:
#The answer of first question: mtbeta <- function(x){ m <- 1e5 t <- runif(m,0,x) mean(30*t*t*(1-t)*(1-t))*x } mbeta <- Vectorize(mtbeta) s <- seq(0.1,0.9,0.1) print(round(rbind(x=s,p=pbeta(s,3,3),phat=mbeta(s),se=mbeta(s)-pbeta(s,3,3)),5)) #The answer of second question: mtrayl <- function(x, R = 10000, antithetic = TRUE) { u <- runif(R/2) if (!antithetic) v <- runif(R/2) else v <- 1 - u u <- c(u, v) cdf <- numeric(length(x)) for (i in 1:length(x)) { g <- x[i]^2*u * exp(-(u * x[i])^2 / 2) cdf[i] <- mean(g) } cdf } x <- seq(0, 2.5,0.5) set.seed(123) mtr1 <- mtrayl(x, anti = FALSE) set.seed(123) mtr2 <- mtrayl(x,anti = TRUE) print(round(rbind(x, mtr1, mtr2), 5)) m <- 1000 mtr3 <- mtr4 <- numeric(m) x <- 1 for (i in 1:m) { mtr3[i] <- mtrayl(x,antithetic = FALSE) mtr4[i] <- mtrayl(x) } print(var(mtr3)) print(var(mtr4)) print((var(mtr3) - var(mtr4))/var(mtr3)) #The answer of third question: m <- 10000 se <- numeric(2) g <- function(x) { exp(-x^2/2)*x^2/(sqrt(2*pi)) * (x > 1) } x <- rweibull(m, 2,sqrt(2)) fg1 <- g(x) /(x* exp(-x^2/2)) se[1] <- sd(fg1) x <- rexp(m, 1) fg2 <- g(x) / exp(-x) se[2] <- sd(fg2) se #The answer of fourth question: m <- 10000 g <- function(x) { exp(-x^2/2)*x^2/(sqrt(2*pi)) * (x > 1) } x <- rweibull(m, 2,sqrt(2)) fg1 <- g(x) /(x* exp(-x^2/2)) theta1 <- mean(fg1) theta1
The fourth homework answer in 2018-10-12
The question is :
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Let $X$ be a non-negative random variable with $\mu = E[X] < ??$. For a random sample $x_1$ ,...,$x_n$ from the distribution of X, the Gini ratio is defined by $${G}=\frac{1}{2n^2\mu}\sum_{j=1}^n \sum_{i=1}^n |x_i-x_j|$$ The Gini ratio is applied in economics to measure inequality in income distribution (see e.g. [163]). Note that G can be written in terms of the order statistics $x_(i)$ as $${G}=\frac{1}{n^2\mu}\sum_{i=1}^n (2i-n-1)x_{(i)}$$ If the mean is unknown, let $\hat{G}$ be the statistic $G$ with $\mu$ replaced by $\bar{x}$. Estimate by simulation the mean, median and deciles of $\hat{G}$ if $X$ is standard lognormal.Repeat the procedure for the uniform distribution and Bernoulli(0.1). Also construct density histograms of the replicates in each case.
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Construct an approximate 95% confidence interval for the Gini ratio y = E[G] if X is lognormal with unknown parameters. Assess the coverage rate of the estimation procedure with a Monte Carlo experiment.
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Tests for association based on Pearson product moment correlation $\rho$, Spear-mans rank correlation coefficient $\rho_s$ , or Kendalls coefficient $\tau$, are implemented in cor.test. Show (empirically) that the nonparametric tests based on $\rho_s$ or $\tau$ are less powerful than the correlation test when the sampled distribution is bivariate normal. Find an example of an alternative (a bivariate distribution $(X,Y)$ such that $X$ and $Y$ are dependent) such that at least one of the nonparametric tests have better empirical power than the correlation test against this alternative.
The answer code is as follows:
#The answer of first question: m <- 1000 n <- 20 g <- numeric(m) medians <- means <- numeric(3) y <- gini1 <- gini2 <- gini3 <- numeric(m) for (i in 1:m) { x <- sort(rlnorm(n)) xmean <- mean(x) for (j in 1:n) { y[j] <- (2*j-n-1)*x[j] } gini1[i] <- 1/n^2/xmean*sum(y[1:n]) } for (i in 1:m) { x <- sort(runif(n)) xmean <- mean(x) for (j in 1:n) { y[j] <- (2*j-n-1)*x[j] } gini2[i] <- 1/n^2/xmean*sum(y[1:n]) } for (i in 1:m) { x <- sort(rbinom(n,c(0,1),c(0.1,0.9))) xmean <- mean(x) for (j in 1:n) { y[j] <- (2*j-n-1)*x[j] } gini3[i] <- 1/n^2/xmean*sum(y[1:n]) } par(mfrow=c(3,1)) par(pin=c(2,1)) hist(gini1,prob = TRUE) lines(density(gini1),col = "red",lwd = 2) hist(gini2,prob = TRUE) lines(density(gini2),col = "blue",lwd = 2) hist(gini3,prob = TRUE) lines(density(gini3),col = "green",lwd = 2) medians[1] <- median(gini1) medians[2] <- median(gini2) medians[3] <- median(gini3) medians quantiles1 <- as.vector(quantile(gini1,seq(0.1,0.9,0.1))) quantiles1 quantiles2 <- as.vector(quantile(gini2,seq(0.1,0.9,0.1))) quantiles2 quantiles3 <- as.vector(quantile(gini3,seq(0.1,0.9,0.1))) quantiles3 means[1] <- mean(gini1) means[2] <- mean(gini2) means[3] <- mean(gini3) means #The answer of second question: m <- 1000 n <- 20 gini <- numeric(m) #Get A series of gini ratios genarating from a lognormal distribution for (i in 1:m) { x <- sort(rlnorm(n)) xmean <- mean(x) for (j in 1:n) { y[j] <- (2*j-n-1)*x[j] } gini[i] <- 1/n^2/xmean*sum(y[1:n]) } #get the lower confidence interval LCI<- mean(gini)-sd(gini)*qt(0.975,m-1) #get the upper confidence interval UCI <- mean(gini)+sd(gini)*qt(0.975,m-1) #get the confidence interval CI <- c(LCI,UCI) print(CI) #calculate the converage rte covrate<-sum(I(gini>CI[1]&gini<CI[2]))/m print(covrate) #The answer of third question: #We need load the MASS package library(MASS) mean <- c(0, 0) sigma <- matrix( c(25,5, 5, 25),nrow=2, ncol=2) m <- 1000 #Calculate the power using pearson correlation test by setting the parameter method as pearson pearvalues <- replicate(m, expr = { mydata1 <- mvrnorm(50, mean, sigma) x <- mydata1[,1] y <- mydata1[,2] peartest <- cor.test(x,y,alternative = "two.sided", method = "pearson") peartest$p.value } ) power1 <- mean(pearvalues <= 0.05) power1 #Calculate the power using spearman correlation test by setting the parameter method as spearman spearvalues <- replicate(m, expr = { mydata2 <- mvrnorm(50, mean, sigma) x <- mydata2[,1] y <- mydata2[,2] speartest <- cor.test(x,y,alternative = "two.sided", method = "spearman") speartest$p.value } ) power2 <- mean(spearvalues <= 0.05) power2 #Calculate the power using kendall correlation test by setting the parameter method as kendall kenvalues <- replicate(m, expr = { mydata3 <- mvrnorm(50, mean, sigma) x <- mydata3[,1] y <- mydata3[,2] kentest <- cor.test(x,y,alternative = "two.sided", method = "kendall") kentest$p.value } ) power3 <- mean(kenvalues <= 0.05) power3 R <- 1000 m <- 1000 #Calculate the power using pearson correlation test by setting the parameter method as pearson pearvalues <- replicate(m, expr = { u <- runif(R/2,0,10) v <- sin(u) peartest <- cor.test(u,v,alternative = "two.sided", method = "pearson") peartest$p.value } ) power1 <- mean(pearvalues <= 0.05) power1 #Calculate the power using spearman correlation test by setting the parameter method as spearman spearvalues <- replicate(m, expr = { u <- runif(R/2,0,10) v <- sin(u) speartest <- cor.test(u,v,alternative = "two.sided", method = "spearman") speartest$p.value } ) power2 <- mean(spearvalues <= 0.05) power2 #Calculate the power using kendall correlation test by setting the parameter method as kendall kenvalues <- replicate(m, expr = { u <- runif(R/2,0,10) v <- sin(u) kentest <- cor.test(u,v,alternative = "two.sided", method = "kendall") kentest$p.value } ) power3 <- mean(kenvalues <= 0.05) power3
The fifth homework answer in 2018-11-02
The question is :
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Compute a jackknife estimate of the bias and the standard error of the corre- lation statistic in Example 7.2.
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Refer to Exercise 7.4. Compute 95% bootstrap confidence intervals for the mean time between failures $\frac1\lambda$ by the standard normal, basic, percentile, and BCa methods. Compare the intervals and explain why they may differ.
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Refer to Exercise 7.7. Obtain the jackknife estimates of bias and standard error of $\hat{\theta}$
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In Example 7.18, leave-one-out (n-fold) cross validation was used to select the best fitting model. Use leave-two-out cross validation to compare the models.
The answer code is as follows:
#The answer of first question: LAST <- c(576,635,558,578 ,666 ,580 ,555 ,661 ,651 ,605 ,653 ,575 ,545 ,572 ,594) GPA <- c(339 ,330 ,281 ,303 ,344 ,307 ,300 ,343 ,336 ,313 ,312 ,274 ,276 ,288 ,296) n <- length(LAST) theta.hat <- cor(LAST,GPA) #the Correlation between the LAST and GPA we give above theta.jack <- numeric(n) library('bootstrap') #compute the jackknife replicates, leave-one-out estimates for (i in 1:n) theta.jack[i] <- cor(LAST[-i],GPA[-i]) bias <- (n - 1) * (mean(theta.jack) - theta.hat) #jackknife estimate of bias print(bias) se <- sqrt((n-1)* mean((theta.jack - mean(theta.jack))^2)) print(se) #The answerof second question: set.seed(1) air <- c(3,5,7,18,43,85,91,98,100,130,230,487) library(boot) lamda.boot <- function(data,ind) { f <- function (lamda) n/lamda-s n <- length(data[ind]) s <- sum(data[ind]) #the function uniroot can be used to get the root root <- uniroot(f,c(0,1))$root 1/root } boot.obj <- boot(data=air, statistic = lamda.boot, R = 3000) print(boot.obj) print(boot.ci(boot.obj,type = c("basic", "norm", "perc","bca"))) #The answer of third question: library(bootstrap) n <- nrow(scor) #get the eigenvalues eigenvalues <- eigen(cov(scor))$values #the estimate computed from the original observed sample theta.hat <- max(eigenvalues)/sum(eigenvalues) theta.jack <- numeric(n) for (i in 1:n) theta.jack[i] <- max(eigen(cov(scor[-i,]))$values)/sum(eigen(cov(scor[-i,]))$values) bias <- (n - 1) * (mean(theta.jack) - theta.hat) print(bias) #jackknife estimate of bias se <- sqrt((n-1)* mean((theta.jack - mean(theta.jack))^2)) print(se) #The answer of fourth question: library(DAAG) attach(ironslag) #the length of the chemical n <- length(chemical) #A pairwise combination of 1 and n comb <- t(combn(1:n,2)) e1 <- e2 <- e3 <- e4 <- e5 <- e6 <- e7 <- e8 <- numeric(n) for (k in 1:n) { #comb is a matrix with two columns k1 <- comb[k,1] k2 <- comb[k,2] #get the training set x <- chemical[-k1][-(k2-1)] y <- magnetic[-k1][-(k2-1)] #use the training set to get the models J1 <- lm(y ~ x) #use the test set to get the prediction error yhat1 <- J1$coef[1] + J1$coef[2] * chemical[k1] yhat2 <- J1$coef[1] + J1$coef[2] * chemical[k2] e1[k] <- magnetic[k1] - yhat1 e2[k] <- magnetic[k2] - yhat2 #the following procedures are the same as the above J2 <- lm(y ~ x + I(x^2)) yhat3 <- J2$coef[1] + J2$coef[2] * chemical[k1] + J2$coef[3] * chemical[k1]^2 yhat4 <- J2$coef[1] + J2$coef[2] * chemical[k2] + J2$coef[3] * chemical[k2]^2 e3[k] <- magnetic[k1] - yhat3 e4[k] <- magnetic[k2] - yhat4 J3 <- lm(log(y) ~ x) logyhat5 <- J3$coef[1] + J3$coef[2] * chemical[k1] logyhat6 <- J3$coef[1] + J3$coef[2] * chemical[k2] yhat5 <- exp(logyhat5) yhat6 <- exp(logyhat6) e5[k] <- magnetic[k1] - yhat5 e6[k] <- magnetic[k2] - yhat6 J4 <- lm(log(y) ~ log(x)) logyhat7 <- J4$coef[1] + J4$coef[2] * log(chemical[k1]) logyhat8 <- J4$coef[1] + J4$coef[2] * log(chemical[k2]) yhat7 <- exp(logyhat7) yhat8 <- exp(logyhat8) e7[k] <- magnetic[k1] - yhat7 e8[k] <- magnetic[k2] - yhat8 } # get the prediction error by leave-two-out cross validation. c(mean(e1^2+e2^2), mean(e3^2+e4^2), mean(e5^2+e6^2), mean(e7^2+e8^2))
The sixth homework answer in 2018-11-16
The question is :
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Implement the two-sample Cram?? er-von Mises test for equal distributions as a permutation test. Apply the test to the data in Examples 8.1 and 8.2.
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Design experiments for evaluating the performance of the NN,energy, and ball methods in various situations?? (1). Unequal variances and equal expectations (2).Unequal variances and unequal expectations (3).Non-normal distributions: t distribution with 1 df (heavy-tailed distribution), bimodel distribution (mixture of two normal distributions) (4).Unbalanced samples (say, 1 case versus 10 controls)
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Use the Metropolis-Hastings sampler to generate random variables from a standard Cauchy distribution. Discard the first 1000 of the chain, and compare the deciles of the generated observations with the deciles of the standard Cauchy distribution (see qcauchyor qt with df=1). Recall that a Cauchy$(\theta,\eta)$ distribution has density function $${f(x)}=\frac{1}{\theta\pi(1+[(x-\eta)/\theta]^2)},-\infty
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Rao [220, Sec. 5g] presented an example on genetic linkage of 197 animals in four categories (also discussed in [67, 106, 171, 266]). The group sizes are (125,18,20,34). Assume that the probabilities of the corresponding multinomial distribution are $$\left(\frac{1}{2}+\frac{\theta}{4},\frac{1-\theta}{4},\frac{1-\theta}{4},\frac{\theta}{4}\right)$$ Estimate the posterior distribution of θ given the observed sample, using one of the methods in this chapter.
The answer code is as follows:
#The answer of first question: set.seed(1) x <- c(158, 171 ,193 ,199 ,230 ,243 ,248 ,248 ,250 ,267 ,271 ,316 ,327 ,329) y <- c(141 ,148 ,169 ,181 ,203 ,213 ,229 ,244 ,257 ,260 ,271 ,309) #the function cvm.test is used to calculate the Cramer-von Mises statistic cvm.test <- function(x,y){ #the empirical distribution function of the x,y F <- ecdf(x) G <- ecdf(y) n <- length(x) m <- length(y) s <- numeric(n) t <- numeric(m) for (i in 1:n) { s[i] <- (F(x[i])-G(x[i]))^2 } s <- sum(s) for (j in 1:m) { t[j] <- (F(y[j])-G(y[j]))^2 } t <- sum(t) #return the Cramer-von Mises statistic return (m*n*(s+t)/(m+n)^2) } #number of replicates R <- 999 #pooled sample z <- c(x, y) K <- 1:26 #storage for replicates reps <- numeric(R) t0 <- cvm.test(x, y) for (i in 1:R) { #generate indices k for the first sample k <- sample(K, size = 14, replace = FALSE) x1 <- z[k] y1 <- z[-k] #complement of x1 reps[i] <- cvm.test(x1, y1) } p <- mean(c(t0, reps) >= t0) p hist(reps, main = "", freq = FALSE, xlab = "T (p = 0.421)", breaks = "scott") points(t0, 0, cex = 1, pch = 16) #The answer of second question: library(RANN) library(boot) library(energy) library(Ball) m <- 100; k<-3; p<-2; mu <- 0.2; set.seed(12345) n1 <- n2 <- 20; R<-50; n <- n1+n2; N = c(n1,n2) Tn <- function(z, ix, sizes,k) { n1 <- sizes[1]; n2 <- sizes[2]; n <- n1 + n2 if(is.vector(z)) z <- data.frame(z,0); z <- z[ix, ]; NN <- nn2(data=z, k=k+1) block1 <- NN$nn.idx[1:n1,-1] block2 <- NN$nn.idx[(n1+1):n,-1] i1 <- sum(block1 < n1 + .5); i2 <- sum(block2 > n1+.5) (i1 + i2) / (k * n) } #the nn method and return the p.values eqdist.nn <- function(z,sizes,k){ boot.obj <- boot(data=z,statistic=Tn,R=R, sim = "permutation", sizes = sizes,k=k) ts <- c(boot.obj$t0,boot.obj$t) p.value <- mean(ts>=ts[1]) list(statistic=ts[1],p.value=p.value) } p.values1 <- matrix(NA,m,3) p.values2 <- matrix(NA,m,3) p.values3 <- matrix(NA,m,3) p.values4 <- matrix(NA,m,3) for(i in 1:m){ #the sample x is from the standard normal distribution and the sample y is from the normal diisreibution with mean=0,variance is 2 x <- matrix(rnorm(n1*p),ncol=p); y <- cbind(rnorm(n2,sd=2),rnorm(n2,sd=2)); z <- rbind(x,y) p.values1[i,1] <- eqdist.nn(z,N,k)$p.value p.values1[i,2] <- eqdist.etest(z,sizes=N,R=R)$p.value p.values1[i,3] <- bd.test(x=x,y=y,R=999,seed=i*12345)$p.value } alpha <- 0.1; pow1 <- colMeans(p.values1<alpha) names(pow1) <- c('NN','energy', 'ball ') #get the powers by the three method pow1 #under the situation of unequal variances and unequal expectations for(i in 1:m){ #the sample x is from the standard normal distribution and the sample y is from the normal diisreibution with mean=0.2 ,variance is 2 x <- matrix(rnorm(n1*p),ncol=p); y <- cbind(rnorm(n2,mean = mu,sd=2),rnorm(n2,mean=mu,sd=2)); z <- rbind(x,y) p.values2[i,1] <- eqdist.nn(z,N,k)$p.value p.values2[i,2] <- eqdist.etest(z,sizes=N,R=R)$p.value p.values2[i,3] <- bd.test(x=x,y=y,R=999,seed=i*12345)$p.value } alpha <- 0.1; pow2 <- colMeans(p.values2<alpha) names(pow2) <- c('NN','energy', 'ball ') #get the powers by the three method pow2 #under the situation of non-normal distributions for(i in 1:m){ #the sample x is from t distribution with df=1 and the sample y is from the mixture of two normal distributions x <- matrix(rt(n1*p,df=1),ncol = p); y <- cbind(rnorm(n2),rnorm(n2,mean=mu,sd=4)); z <- rbind(x,y) p.values3[i,1] <- eqdist.nn(z,N,k)$p.value p.values3[i,2] <- eqdist.etest(z,sizes=N,R=R)$p.value p.values3[i,3] <- bd.test(x=x,y=y,R=999,seed=i*12345)$p.value } alpha <- 0.1; pow3 <- colMeans(p.values3<alpha) names(pow3) <- c('NN','energy', 'ball ') #get the powers by the three method pow3 #under the situation of Unbalanced samples which the number of x is 200 and y is 20 n1 <- 200 n2 <- 20 n <- n1+n2 N = c(n1,n2) for(i in 1:m){ x <- matrix(rnorm(n1*p),ncol=p); y <- cbind(rnorm(n2),rnorm(n2)); z <- rbind(x,y) p.values4[i,1] <- eqdist.nn(z,N,k)$p.value p.values4[i,2] <- eqdist.etest(z,sizes=N,R=R)$p.value p.values4[i,3] <- bd.test(x=x,y=y,R=99,seed=i*12345)$p.value } alpha <- 0.18; pow4 <- colMeans(p.values4<alpha) names(pow4) <- c('NN','energy', 'ball ') #get the powers by the three method pow4 #The answer of third question: set.seed(1) #build a standard Cauchy distribution f <- function(x, x0=0, gamma=1){ out<-1/(pi*gamma*(1+((x-x0)/gamma)^2)) return(out) } #the times of simulation m <- 80000 x <- numeric(m) #generat the normal proposal distribution with mean=xt ,sd=1 x[1] <- rnorm(1,mean=0,sd=1 ) k <- 0 u <- runif(m) for (i in 2:m) { xt <- x[i-1] y <- rnorm(1, mean = xt,sd=1) num <- f(y) * dnorm(xt, mean = y,sd=1) den <- f(xt) * dnorm(y, mean = xt,sd=1) if (u[i] <= num/den) x[i] <- y else { x[i] <- xt k <- k+1 #y is rejected } } #discard the burnin sample b <- 1001 y <- x[b:m] a <- ppoints(200) #quantiles of cauchy distribution Qcauchy <- qcauchy(a) #quantiles of sample distribution Q <- quantile(x, a) qqplot(Qcauchy, Q,xlim=c(-2,2),ylim=c(-2,2),xlab="Cauchy Quantiles", ylab="Sample Quantiles",main = expression("Q-Q plot for Cauchy distribution")) hist(y, breaks=50, main="", xlab="", freq=FALSE) lines(Qcauchy, f(Qcauchy)) #The answer of fourth question: size <- c(125,18,20,34) #The following function prob computes the target density prob <- function(y, size) { if (y < 0 || y >1) return (0) return((1/2+y/4)^size[1] * ((1-y)/4)^size[2] * ((1-y)/4)^size[3] *(y/4)^size[4]) } #length of the chain m <- 5000 #width of the uniform support set w <- .25 #burn-in time burn <- 1000 #for accept/reject step u <- runif(m) #proposal distribution v <- runif(m, -w, w) x[1] <- .25 for (i in 2:m) { y <- x[i-1] + v[i] if (u[i] <= prob(y, size) / prob(x[i-1], size)) x[i] <- y else x[i] <- x[i-1] } xb <- x[(burn+1):m] print(mean(xb))
The seventh homework answer in 2018-11-23
The question is :
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For exercise 9.6, use the Gelman-Rubin method to monitor convergence of the chain, and run the chain until the chain has converged approximately to the target distribution according to R < 1.2.
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Find the intersection points A(k) in (0,$\sqrt{A}$) $$S_{k-1}(a)=P\left(t(k-1)>\sqrt{\frac{a^2(k-1)}{k-a^2}}\right)$$ and $$S_{k}(a)=P\left(t(k)>\sqrt{\frac{a^2(k)}{k+1-a^2}}\right)$$ for k = 4 : 25,100,500,1000, where t(k) is a Student t random variable with k degrees of freedom. (These intersection points determine the critical values for a t-test for scale-mixture errors proposed by Sz?? ekely [260].)
The answer code is as follows:
#The answer of first question: set.seed(1) size <- c(125,18,20,34) #The following function prob computes the target density prob <- function(y, size) { if (y < 0 || y >1) return (0) return((1/2+y/4)^size[1] * ((1-y)/4)^size[2] * ((1-y)/4)^size[3] *(y/4)^size[4]) } Gelman.Rubin <- function(psi) { # psi[i,j] is the statistic psi(X[i,1:j]) # for chain in i-th row of X psi <- as.matrix(psi) n <- ncol(psi) k <- nrow(psi) psi.means <- rowMeans(psi) #row means B <- n * var(psi.means) #between variance est. psi.w <- apply(psi, 1, "var") #within variances W <- mean(psi.w) #within est. v.hat <- W*(n-1)/n + (B/n) #upper variance est. r.hat <- v.hat / W #G-R statistic return(r.hat) } multinomial.chain <- function(N, X1) { #generates a Metropolis chain for multino-mial distribution #with uniform(-w,w) proposal distribution #and starting value X1 #length of the chain #width of the uniform support set w <- 0.15 #burn-in time burn <- 1000 #for accept/reject step u <- runif(N) #proposal distribution v <- runif(N, -w, w) x <- rep(0, N) x[1] <- X1 for (i in 2:N) { y <- x[i-1] + v[i] r1 <- prob(y, size) r2 <- prob(x[i-1], size) r <- r1/r2 if (u[i] <= r) x[i] <- y else x[i] <- x[i-1] } return(x) } k <- 4 #number of chains to generate n <- 10000 #length of chains b <- 2000 #burn-in length #choose overdispersed initial values x0 <- c(0.1,0.4,0.6,0.9) #generate the chains X <- matrix(0, nrow=k, ncol=n) for (i in 1:k) X[i, ] <- multinomial.chain(n, x0[i]) #compute diagnostic statistics psi <- t(apply(X, 1, cumsum)) for (i in 1:nrow(psi)) psi[i,] <- psi[i,] / (1:ncol(psi)) print(Gelman.Rubin(psi)) #get sequences of the running means ?? for four Metropolis-Hastings chains par(mfrow=c(2,2)) for (i in 1:k) plot(psi[i, (b+1):n], type="l", xlab=i, ylab=bquote(psi)) par(mfrow=c(1,1)) #restore default rhat <- rep(0, n) for (j in (b+1):n) rhat[j] <- Gelman.Rubin(psi[,1:j]) plot(rhat[(b+1):n], type="l", xlab="", ylab="R") abline(h=1.05, lty=2) #The answer of second question: m <- c(4:25,100,500,1000) root <- numeric(length(m)) f1 <- function(a,k) pt(sqrt(a^2*(k-1)/(k-a^2)),k-1)-pt(sqrt(a^2*(k)/(k+1-a^2)),k) for (i in 4:25) { res <- uniroot(f1,c(1e-5,2),k=i) root[i-3] <- res$root } res1 <- uniroot(f1,c(1e-5,2),k=100) root[23] <- res1$root res2 <- uniroot(f1,c(1e-5,2),k=500) root[24] <- res2$root res3 <- uniroot(f1,c(1e-5,2),k=1000) root[25] <- res3$root data <- data.frame(k=m,a=root,row.names=1) data
The eighth homework answer in 2018-11-30
The question is :
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Write a function to compute the cdf of the Cauchy distribution, which has density$$\frac{1}{\theta\pi(1+[(x-\eta)/\theta]^{2})}, -\infty
(Also see the source code in pcauchy.c.) -
- A-B-O blood type problem
- Let the three alleles be A, B, and O.
Genotype AA BB OO AO BO AB AA Frequency p2 q2 r2 2pr 2qr 2pq 1 Count nAA nBB nOO nAO nBO nAB n Observed data: $n_{A\cdot}=n_{AA}+n_{AO}=28$ (A-type), $n_{B\cdot}=n_{BB}+n_{BO}=24$ (B-type), $n_{OO}=41$ (O-type), $n_{AB}=70$ (AB-type).
Use EM algorithm to solve MLE of $p$ and $q$ (consider missing data $n_{AA}$ and $n_{BB}$).
Record the maximum likelihood values in M-steps, are they increasing?
The answer code is as follows:
#The answer of first question: options(digits=6) #get the density of cauchy distribution with the parameter theta and eta f <- function(x, theta, eta) { 1/theta/pi/(1+((x-eta)/theta)^2) } #the vector x is used to get the cdf of cauchy distribution when theta is 1,eta is 0 x <- seq(-5,5,0.1) values <- numeric(length(x)) #the vector x2 is used to get the cdf of cauchy distribution when theta is 1,eta is 10 x2 <- seq(0,10,0.1) values2 <- numeric(length(x2)) for (i in 1:length(x)) values[i] <- integrate(f, lower=-Inf, upper=x[i],rel.tol=.Machine$double.eps^0.25,theta=1, eta=0)$value # data <- data.frame(value=values,truevalues=pcauchy(x),errors=values-pcauchy(x)) data[1:20,] #compare the cauchy distributions,one of them is used integrate and the other is used pcauchy plot(x,values,pch=21,xlab="the values of x", ylab="the cauchy distribution function ",main = expression("the compare for Cauchy distribution when \ntheta is 1 and eta is 0 with two methods")) lines(x,pcauchy(x),col='red',lwd=2) #the situation when theta is 1 and eta is 10 for (i in 1:length(x2)) values2[i] <- integrate(f, lower=-Inf,upper=x2[i],rel.tol=.Machine$double.eps^0.25,theta=1, eta=10)$value data2 <- data.frame(value=values2,truevalues=pcauchy(x2,location=10,scale=1),error=values2-pcauchy(x2,location=10,scale=1)) data2[1:20,] #compare the cauchy distributions,one of them is used integrate and the other is used pcauchy plot(x2,values2,pch=21,xlab="the values of x", ylab="the cauchy distribution function ",main = expression("the compare for Cauchy distribution when \ntheta is 1 and eta is 10 with two methods")) lines(x2,pcauchy(x2,location = 10,scale = 1),col='red',lwd=2) #The answer of second question: library(nloptr) # the maximum likehood function f <- function(x,x1,nA=28,nB=24,nOO=41,nAB=70) { r1<-1-sum(x1) nAA<-nA*x1[1]^2/(x1[1]^2+2*x1[1]*r1) nBB<-nB*x1[2]^2/(x1[2]^2+2*x1[2]*r1) r<-1-sum(x) return(-2*nAA*log(x[1])-2*nBB*log(x[2])-2*nOO*log(r)- (nA-nAA)*log(2*x[1]*r)-(nB-nBB)*log(2*x[2]*r)-nAB*log(2*x[1]*x[2])) } # constraint function g <- function(x,x1,nA=28,nB=24,nOO=41,nAB=70) { return(sum(x)-0.999999) } opts <- list("algorithm"="NLOPT_LN_COBYLA", "xtol_rel"=1.0e-8) mle<-NULL r<-matrix(0,1,2) r<-rbind(r,c(0.2,0.35))# the beginning value of p0 and q0 j<-2 while (sum(abs(r[j,]-r[j-1,]))>1e-8) { res <- nloptr( x0=c(0.3,0.25), eval_f=f, lb = c(0,0), ub = c(1,1), eval_g_ineq = g, opts = opts, x1=r[j,],nA=28,nB=24,nOO=41,nAB=70 ) j<-j+1 r<-rbind(r,res$solution) mle<-c(mle,f(x=r[j,],x1=r[j-1,])) } r #the result of EM algorithm mle #the max likelihood values
The nineth homework answer in 2018-12-07
The question is :
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Use both for loops and lapply() to fit linear models to the mtcars using the formulas stored in this list:
formulas <- list( mpg ~ disp, mpg ~ I(1 / disp), mpg ~ disp + wt, mpg ~ I(1 / disp) + wt )
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Fit the model mpg ~ disp to each of the bootstrap replicates of mtcars in the list below by using a for loop and lapply() .Can you do it without an anonymous function?
bootstraps <- lapply(1:10, function(i) { rows <- sample(1:nrow(mtcars), rep = TRUE) mtcars[rows, ] })
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For each model in the previous two exercises, extract R 2 using the function below.
rsq <- function(mod) summary(mod)$r.squared
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The following code simulates the performance of a t-test for non-normal data. Use sapply() and an anonymous function to extract the p-value from every trial. trials <- replicate( 100, t.test(rpois(10, 10), rpois(7, 10)), simplify = FALSE ) Extra challenge: get rid of the anonymous function by using [[ directly.
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Implement a combination of Map() and vapply() to create an lapply() variant that iterates in parallel over all of its inputs and stores its outputs in a vector (or a matrix). What arguments should the function take?
The answer code is as follows:
#The answer of first question: formulas <- list( mpg ~ disp, mpg ~ I(1 / disp), mpg ~ disp + wt, mpg ~ I(1 / disp) + wt ) lapply(formulas,lm,data=mtcars) out <- vector("list", length(formulas)) for (i in seq_along(formulas)) { out[[i]] <- lm(formulas[[i]], data = mtcars) } out #The answer of second question: set.seed(1) bootstraps <- lapply(1:10, function(i) { rows <- sample(1:nrow(mtcars), rep = TRUE) mtcars[rows, ] }) getvalue <- function(x){ vars <- c('mpg','disp') x[vars] } data <- lapply(bootstraps, getvalue) lapply(data,lm) set.seed(1) out1 <- vector("list", length(bootstraps)) for (i in seq_along(bootstraps)) { out1[[i]] <- getvalue(bootstraps[[i]]) } out2 <- vector("list", length(out1)) for (i in seq_along(out1)) { out2[[i]] <- lm(out1[[i]]) } out2 #The answer of third question: formulas <- list( mpg ~ disp, mpg ~ I(1 / disp), mpg ~ disp + wt, mpg ~ I(1 / disp) + wt ) funclist <- lapply(formulas,lm,data=mtcars) rsq <- function(mod) summary(mod)$r.squared lapply(funclist, rsq) getvalue <- function(x){ vars <- c('mpg','disp') x[vars] } data <- lapply(bootstraps, getvalue) data1 <- lapply(data,lm) rsq <- function(mod) summary(mod)$r.squared unlist(lapply(data1, rsq)) #The answer of fourth question: trials <- replicate( 100, t.test(rpois(10, 10), rpois(7, 10)), simplify = FALSE ) #use an anonymous function sapply(trials, function(mod) mod$p.value) #The answer of fifth question: a <- c(1,2,3,4,5) b <- c(6,7,8,9,10) c <- c(2,4,6,8,10) d <- data.frame(a,b,c) #the lapply1 function can parallelly get the sum of the input vectors lapply1 <- function(x,...){ args <- list(x,...) for (a in args) x <- x+a x } lapply1(a,b,c) lapply1(d[1],d[2],d[3]) #the lapply2 function can parallelly get the product of the input vectors lapply2 <- function(x,...){ args <- list(...) for (a in args) x <- x*a x } lapply2(a,b,c)
The third homework answer in 2018-12-14
The question is :
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Make a faster version of chisq.test() that only computes the chi-square test statistic when the input is two numeric vectors with no missing values. You can try simplifying chisq.test() or by coding from the mathematical definition (http://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test ).
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Can you make a faster version of table() for the case of an input of two integer vectors with no missing values? Can you use it to speed up your chi-square test?
The answer code is as follows:
#The answer of first question: chisq.test1<- function (x, y = NULL, correct = TRUE, p = rep(1/length(x), length(x)), rescale.p = FALSE, simulate.p.value = FALSE, B = 2000) { DNAME <- deparse(substitute(x)) #get the total sum of x n <- sum(x) if (is.matrix(x)) { METHOD <- "changed Pearson's Chi-squared test" nr <- as.integer(nrow(x)) nc <- as.integer(ncol(x)) if (is.na(nr) || is.na(nc) || is.na(nr * nc)) stop("invalid nrow(x) or ncol(x)", domain = NA) #get the rowsums of x sr <- rowSums(x) #get the colsums of x sc <- colSums(x) #The sum of rows and columns that's sr,sc divided by n E <- outer(sr, sc, "*")/n v <- function(r, c, n) c * r * (n - r) * (n - c)/n^3 V <- outer(sr, sc, v, n) dimnames(E) <- dimnames(x) #the situation of the sum of rows and columns are all not equal 0 if (simulate.p.value && all(sr > 0) && all(sc > 0)) { setMETH() tmp <- .Call(C_chisq_sim, sr, sc, B, E) STATISTIC <- sum(sort((x - E)^2/E, decreasing = TRUE)) PARAMETER <- NA PVAL <- (1 + sum(tmp >= almost.1 * STATISTIC))/(B + 1) } else { if (simulate.p.value) warning("cannot compute simulated p-value with zero marginals") if (correct && nrow(x) == 2L && ncol(x) == 2L) { YATES <- min(0.5, abs(x - E)) if (YATES > 0) METHOD <- paste(METHOD, "with Yates' continuity correction") } else YATES <- 0 STATISTIC <- sum((abs(x - E) - YATES)^2/E) PARAMETER <- (nr - 1L) * (nc - 1L) PVAL <- pchisq(STATISTIC, PARAMETER, lower.tail = FALSE) } } names(STATISTIC) <- "X-squared" names(PARAMETER) <- "df" if (any(E < 5) && is.finite(PARAMETER)) warning("Chi-squared approximation may be incorrect") #output the result structure(list(statistic = STATISTIC, parameter = PARAMETER, p.value = PVAL, method = METHOD, data.name = DNAME, observed = x, expected = E, residuals = (x - E)/sqrt(E), stdres = (x - E)/sqrt(V)), class = "htest") } library(vcd) Treatment <- Arthritis$Treatment Improved <- Arthritis$Improved mytable <- table(Treatment,Improved) #you can see the changed chisq.test1's output is the same as the original function chisq.test(mytable) chisq.test1(mytable) #compare the calculate speed set.seed(1) x <- sample(1e5,1e6,10) y <- sample(1e5,1e6,10) system.time(chisq.test(rbind(x,y))) system.time(chisq.test1(rbind(x,y))) #The answer of second question: table2 <- function (... , dnn = list.names(...), deparse.level = 1) { list.names <- function(...) { l <- as.list(substitute(list(...)))[-1L] nm <- names(l) fixup <- if (is.null(nm)) seq_along(l) else nm == "" dep <- vapply(l[fixup], function(x) switch(deparse.level + 1, "", if (is.symbol(x)) as.character(x) else "", deparse(x, nlines = 1)[1L]), "") if (is.null(nm)) dep else { nm[fixup] <- dep nm } } args <- list(...) if (length(args) == 1L && is.list(args[[1L]])) { args <- args[[1L]] if (length(dnn) != length(args)) dnn <- if (!is.null(argn <- names(args))) argn else paste(dnn[1L], seq_along(args), sep = ".") } bin <- 0L lens <- NULL dims <- integer() pd <- 1L dn <- NULL for (a in args) { if (is.null(lens)) lens <- length(a) else if (length(a) != lens) stop("all arguments must have the same length") fact.a <- is.factor(a) if (!fact.a) { a0 <- a a <- factor(a) } ll <- levels(a) a <- as.integer(a) nl <- length(ll) dims <- c(dims, nl) if (prod(dims) > .Machine$integer.max) stop("attempt to make a table with >= 2^31 elements") dn <- c(dn, list(ll)) bin <- bin + pd * (a - 1L) pd <- pd * nl } names(dn) <- dnn bin <- bin[!is.na(bin)] if (length(bin)) bin <- bin + 1L y <- array(tabulate(bin, pd), dims, dimnames = dn) class(y) <- "table" y } set.seed(1) z <- sample(1:50,1e7,replace = TRUE) system.time(table(z)) system.time(table2(z))
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