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PhD thesis code
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#### PhD Thesis: Change-point detection for a transient change and high-dimensional covering
#### Year: 2021
#### Author: Jack Noonan
#### Email: Noonanj1@cardiff.ac.uk
#############
# This script contains the majority of code used within the thesis.
# Please search (ctrl F) for the chapter and number, e.g. "Chapter 6", to locate the relevant code.
# Note that this script focuses mainly on the Monte Carlo simulations and does not include all code used within the PhD.
# Many of the figures and table in this thesis can be reproduced with the below code. At the beginning of each chunk of code,
# the relevant figure or table should be mentioned.
# Many Monte Carlo simulations will use the R package 'snow' to run in parallel and speed up total simulations.
# Please install snow and specificy how many cores you would like to run simulations on (depends on your processor).
# install.packages('snow').
# Please also install all other packages below.
##############
library('snow')
library('statmod')
library('rARPACK')
library('FrF2')
library('SobolSequence')
library('numDeriv')
number_of_cores = 8 # we will initially run simulations on 8 cores
############################################
#### Code for main results of Chapter 1 ####
############################################
# The below code reproduces Figures 1.1
M=50
A=1
l0=10
l1=20
samp_num=1000
H_seq=seq(3.5,11.5,0.25)*A
GT = function(){
sample_matrix=matrix(0,nrow=samp_num,ncol=1)
track = 0
for (k in 1:samp_num){
sample = rnorm(M+l1,0,1)
max = NULL
for (i in 1:(length(sample)-1)){
for (j in (i+1):length(sample)){
if ((j-i+1)>=l0 & (j-i+1)<=l1 ){
max=max(max, A*(sum(sample[i:j])-0.5*A*(j-i+1)) )
print(i)
}
}
}
sample_matrix[k,]=max
}
sample_matrix
}
cl <- makeCluster(rep('localhost',number_of_cores),type='SOCK')
clusterExport(cl,c('GT','samp_num','n','l0','l1','A'))
data1 = clusterEvalQ(cl,GT())
stopCluster(cl)
Data = unlist(lapply(data1,function(x){x}))
prob_track=NULL
for (H in H_seq){
prob_track=c(prob_track,sum(Data>H)/length(Data))
}
par(mar=c(5.1,6.2,4.1,2.1))
plot(H_seq,prob_track,type='l',ylab='',xlab='H',lty=2,xaxt="n",yaxt='n',lwd=7,xaxs="i",yaxs="i",cex.axis=2.4,cex.lab=2.5)
axis(1, at=seq(3.5,11.5,0.5)*A, cex.axis=2.5)
axis(side = 2, tck = -.015, labels = NA)
axis(side = 2, lwd = 0, line = 0.7, las = 1, cex.axis=2.5)
samp_num=10000
GT = function(){
sample_matrix=matrix(0,nrow=samp_num,ncol=1)
track = 0
for (k in 1:samp_num){
sample = rnorm(3*l1,0,1)
max = NULL
for (i in 1:(length(sample)-1)){
for (j in (i+1):length(sample)){
if ((j-i+1)>=l0 & (j-i+1)<=l1 ){
max=max(max, A*(sum(sample[i:j])-0.5*A*(j-i+1)) )
print(i)
}
}
}
sample_matrix[k,]=max
}
sample_matrix
}
cl <- makeCluster(rep('localhost',number_of_cores),type='SOCK')
clusterExport(cl,c('GT','samp_num','M','l0','l1','A'))
data1 = clusterEvalQ(cl,GT())
stopCluster(cl)
Data = unlist(lapply(data1,function(x){x}))
prob_track2=NULL
for (H in H_seq){
prob_track2=c(prob_track2,sum(Data<H)/length(Data))
}
GT = function(){
sample_matrix=matrix(0,nrow=samp_num,ncol=1)
track = 0
for (k in 1:samp_num){
sample = rnorm(2*l1,0,1)
max = NULL
for (i in 1:(length(sample)-1)){
for (j in (i+1):length(sample)){
if ((j-i+1)>=l0 & (j-i+1)<=l1 ){
max=max(max, A*(sum(sample[i:j])-0.5*A*(j-i+1)) )
print(i)
}
}
}
sample_matrix[k,]=max
}
sample_matrix
}
cl <- makeCluster(rep('localhost',number_of_cores),type='SOCK')
clusterExport(cl,c('GT','samp_num','M','l0','l1','A'))
data1 = clusterEvalQ(cl,GT())
stopCluster(cl)
Data = unlist(lapply(data1,function(x){x}))
prob_track3=NULL
for (H in H_seq){
prob_track3=c(prob_track3,sum(Data<H)/length(Data))
}
lines(H_seq,1-prob_track2*(prob_track2/prob_track3)^(n/l1-2),col='blue',lwd=4)
# The below code reproduces Figures 1.2 and Figure 1.3
## Approximations using results of D. Siegmund (to be run when l_0=1)
M=250
A=1
l0=1
l1=50
samp_num=1000
H_seq=seq(1,9,0.25)*A
GT = function(){
sample_matrix=matrix(0,nrow=samp_num,ncol=1)
track = 0
for (k in 1:samp_num){
sample = rnorm(M+l1,0,1)
max = NULL
for (i in 1:(length(sample)-1)){
for (j in (i+1):length(sample)){
if ((j-i+1)>=l0 & (j-i+1)<=l1 ){
max=max(max, A*(sum(sample[i:j])-0.5*A*(j-i+1)) )
print(i)
}
}
}
sample_matrix[k,]=max
}
sample_matrix
}
cl <- makeCluster(rep('localhost',number_of_cores),type='SOCK')
clusterExport(cl,c('GT','samp_num','M','l0','l1','A'))
data1 = clusterEvalQ(cl,GT())
stopCluster(cl)
Data = unlist(lapply(data1,function(x){x}))
prob_track=NULL
for (H in H_seq){
prob_track=c(prob_track,sum(Data>H)/length(Data))
}
lines(H_seq,prob_track,type='l',lty=2,lwd=7)
par(mar=c(5.1,6.2,4.1,2.1))
plot(H_seq,prob_track,type='l',ylim=c(0,1),ylab='',xlab='H',lty=2,xaxt="n",yaxt='n',lwd=7,xaxs="i",yaxs="i",cex.axis=2.4,cex.lab=2.5)
axis(1, at=H_seq, cex.axis=2.5)
axis(side = 2, tck = -.015, labels = NA)
axis(side = 2, lwd = 0, line = 0.7, las = 1, cex.axis=2.5)
Siegmund_bcp_approx = NULL
for (H in H_seq){
H=H/A
M1=3*l1
f1 = function(b){
(A*(M1*A/2 - b)+3)*exp(-A*b)
}
f1=Vectorize(f1)
p1=min(f1(H+0.583*2),1)
M1=2*l1
f1 = function(b){
(A*(M1*A/2 - b)+3)*exp(-A*b)
}
f1=Vectorize(f1)
p2=min(f1(H+0.583*2),1)
prob= 1-(1-p1)*((1-p1)/(1-p2))^(M/l1-2)
Siegmund_bcp_approx=c(Siegmund_bcp_approx,prob)
}
lines(H_seq,Siegmund_bcp_approx,col='red',lwd=4)
# The below code reproduces Tables 1.4 and Tables 1.5
### Average run length Monte Carlo and simulations
H1 = seq(-5,-2,0.5)
A=1
l0=25
l1=50
samp_num=10000
GT = function(){
sample_matrix=matrix(0,nrow=samp_num,ncol=1)
track = 0
for (k in 1:samp_num){
sample = rnorm(3*l1,0,1)
max = NULL
for (i in 1:(length(sample)-1)){
for (j in (i+1):length(sample)){
if ((j-i+1)>=l0 & (j-i+1)<=l1 ){
max=max(max, A*(sum(sample[i:j])-0.5*A*(j-i+1)) )
print(i)
}
}
}
sample_matrix[k,]=max
}
sample_matrix
}
cl <- makeCluster(rep('localhost',number_of_cores),type='SOCK')
clusterExport(cl,c('GT','samp_num','n','l0','l1','A'))
data1 = clusterEvalQ(cl,GT())
stopCluster(cl)
Data = unlist(lapply(data1,function(x){x}))
GT = function(){
sample_matrix=matrix(0,nrow=samp_num,ncol=1)
track = 0
for (k in 1:samp_num){
sample = rnorm(2*l1,0,1)
max = NULL
for (i in 1:(length(sample)-1)){
for (j in (i+1):length(sample)){
if ((j-i+1)>=l0 & (j-i+1)<=l1 ){
max=max(max, A*(sum(sample[i:j])-0.5*A*(j-i+1)) )
print(i)
}
}
}
sample_matrix[k,]=max
}
sample_matrix
}
cl <- makeCluster(rep('localhost',number_of_cores),type='SOCK')
clusterExport(cl,c('GT','samp_num','n','l0','l1','A'))
data1 = clusterEvalQ(cl,GT())
stopCluster(cl)
Data2 = unlist(lapply(data1,function(x){x}))
F2=NULL
for (H in H1){
F2=c(F2,sum(Data<H)/length(Data))
}
F1=NULL
for (H in H1){
F1=c(F1,sum(Data2<H)/length(Data2))
}
theta=F2/F1
ARL_approx = l1- l1*F2/(theta^2*log(theta))
ARL_sim =NULL
for (H in H1){
GT = function(){
ARL_SIM = NULL
for (k in 1:1000){
track = l1
max=-Inf
n=0
sample = rnorm(n+l1,0,1)
for (i in 1:(length(sample)-1)){
for (j in (i+1):length(sample)){
if ((j-i+1)>=l0 & (j-i+1)<=l1 ){
max=max(max, A*(sum(sample[i:j])-0.5*A*(j-i+1)) )
}
}
}
while (max <H){
sample=c(sample, rnorm(1,0,1))
track=track+1
n = length(sample)
for (i in 1:(length(sample)-1)){
if ((n-i+1)>=l0 & (n-i+1)<=l1 ){
max=max(max, A*(sum(sample[i:n])-0.5*A*(n-i+1)) )
}
}
#print(track)
}
print(k)
ARL_SIM=c(ARL_SIM,track)
}
mean(ARL_SIM)
}
cl <- makeCluster(rep('localhost',8),type='SOCK')
clusterExport(cl,c('GT','samp_num','n','l0','l1','A','ARL_SIM','H'))
data1 = unlist(clusterEvalQ(cl,GT()))
data1=unlist(data1)
stopCluster(cl)
ARL_sim=c(ARL_sim,mean(data1))
print(ARL_sim)
}
ARL_approx
ARL_sim
# The below code reproduces the approximation in Table 1.5
## ARL approx using D. Siegmund approximation
Siegmund_approx = NULL
for (H in H1){
F1 = 1-(A*(A*l1-H/A-2*0.583 )+3)*exp(-A*(H/A+2*0.583))
F2 = 1-(A*(3*A*l1/2-H/A-2*0.583 )+3)*exp(-A*(H/A+2*0.583))
theta = F2/F1
Siegmund_approx=c(Siegmund_approx, l1- l1*F2/(theta^2*log(theta)) )
}
### Comparing the power of three tests.
# The below code reproduces Figure 1.8
## Part 1 - Simulation for MOSUM as a function of lambda
l=20
ARL=500
A=0.5
samp_num=10000
stat2 = NULL
for (L in seq(40,10,-2) ){
stat=NULL
ARL_fun = function(x){
astar =x
a= astar + 0.583*sqrt(2)/sqrt(L)
b=0
p_2 = function(s_2){
f_s = function(s){
M = matrix(0,3,3)
M[1,1] =pnorm(a)
M[2,1] =dnorm(a)
M[3,1] = dnorm(2*a-s)
M[1,2] =pnorm(s)
M[2,2] =dnorm(s)
M[3,2] = dnorm(a)
M[1,3] =pnorm(s+s_2-a)
M[2,3] =dnorm(s+s_2-a)
M[3,3] = dnorm(s_2)
det(M)
}
f_s = Vectorize(f_s)
integrate(f_s,-Inf,a)$value
}
p_2 = Vectorize(p_2)
lambda_num =integrate(p_2,-20,a)$value
ana = function(x){
ana2 = function(k){
M = matrix(0,2,2)
M[1,1] = dnorm(x)
M[1,2] = dnorm(-k-a-b)
M[2,1] = dnorm(a)
M[2,2] = dnorm(-k-x-b)
det(M)*exp(-b^2/2+b*(k+b+x))
}
ana2 = Vectorize(ana2)
integrate(ana2,-x-a-b,10)$value
}
ana = Vectorize(ana)
lambda_denom = integrate(ana,-10,a)$value
lambda = lambda_num/lambda_denom
ARL=-L*lambda_num/(lambda^2*log(lambda))
ARL
}
inverse = function (f, lower, upper) {
function (y) uniroot((function (x) f(x) - y), lower = lower, upper = upper)[1]
}
square_inverse = inverse(ARL_fun, 2,4)
Suitable_ARL_alpha= square_inverse(ARL-L)$root
h = Suitable_ARL_alpha
print(h)
for (u in c(1:samp_num)){
V = c(rnorm(2*L,0,1),rnorm(l,A,1),rnorm(7*L,0,1))
MOSUM = NULL
for (i in c(1:(4*l))){
MOSUM=c(MOSUM,sum(V[(i):(i+L-1)])/sqrt(L))
}
if (all(MOSUM[1:L]<h)){
stat=c(stat,ifelse(any(MOSUM[(L+1):(L+2*l-1)]>h),1,0))
}
#print(u)
}
stat2 = c(stat2,sum(stat)/length(stat))
print(L)
}
stat2
par(mar=c(6.1,5.5,4.1,2.1))
plot(l/seq(40,10,-2),stat2,type='l',ylab='',xlab='',lty=1,lwd=4,xaxt="n",yaxt='n',cex.axis=2.4,cex.lab=2.5)#ylim=c(min(stat2),0.82))
axis(1, at=seq(0,10,0.5), cex.axis=2.5)
axis(side = 2, tck = -.015, labels = NA)
axis(side = 2, lwd = 0, line = 0.35, las = 1, cex.axis=2.5)
mtext(side=1, text=expression(paste(lambda)), line=5,cex=4)
## Part 2 - Generalised MOSUM barrier determination (based on ARL) and power
ARL_sim =NULL
H=3.6
A=0.5
l0=10
l1=40
samp_num=1000
GT = function(){
ARL_SIM = NULL
for (k in 1:samp_num){
track = l1
max=-Inf
n=0
sample = rnorm(n+l1,0,1)
for (i in 1:(length(sample)-1)){
for (j in (i+1):length(sample)){
if ((j-i+1)>=l0 & (j-i+1)<=l1 ){
max=max(max, A*(sum(sample[i:j])-0.5*A*(j-i+1)) )
}
}
}
while (max <H){
sample=c(sample, rnorm(1,0,1))
track=track+1
n = length(sample)
for (i in 1:(length(sample)-1)){
if ((n-i+1)>=l0 & (n-i+1)<=l1 ){
max=max(max, A*(sum(sample[i:n])-0.5*A*(n-i+1)) )
}
}
#print(track)
}
print(k)
ARL_SIM=c(ARL_SIM,track)
}
mean(ARL_SIM)
}
cl <- makeCluster(rep('localhost',number_of_cores),type='SOCK')
clusterExport(cl,c('GT','samp_num','l0','l1','A','ARL_SIM','H'))
data1 = unlist(clusterEvalQ(cl,GT()))
data1=unlist(data1)
stopCluster(cl)
ARL_sim=mean(data1)
ARL_sim
H=3.55
G_MOSUM_power=NULL
samp_num=1000
for (u in c(1:samp_num)){
sample = c(rnorm(2*l1,0,1),rnorm(l,A,1),rnorm(5*l1,0,1))
G_MOSUM = NULL
k=l1
while( k <=length(sample)){
max=-Inf
for (i in 1:(l1-1)){
for (j in (i+1):l1){
if ((j-i+1)>=l0 & (j-i+1)<=l1 ){
max=max(max, A*(sum(sample[i:j])-0.5*A*(j-i+1)) )
}
}
}
for (i in 1:(k-1)){
if ((k-i+1)>=l0 & (k-i+1)<=l1 ){
max=max(max, A*(sum(sample[i:k])-0.5*A*(k-i+1)) )
}
}
G_MOSUM=c(G_MOSUM,max)
k=k+1
}
if (all(G_MOSUM[1:l1]<H)){
G_MOSUM_power=c(G_MOSUM_power,ifelse(any(G_MOSUM[(l1+1):(l1+2*l-1)]>H),1,0))
}
print(u)
}
G_MOSUM_power2 = sum(G_MOSUM_power)/length(G_MOSUM_power)
G_MOUSM_power2
abline(h=G_MOSUM_power2,lwd=4,lty=5,col='orange')
## Part 3 - CUSUM barrier determination (ARL) and power
H=37.88 #obtained from paper of moustakides and polunchenko
samp_num = 50000
A=0.5
Change_point=40 ## select large enough ot obtain stationary distribution
CUSUM_statistics = matrix(0,nrow=samp_num,ncol=(Change_point+2*l))
CUSUM_statistics[,1] =rep(1,samp_num)
for (i in c(1:samp_num)){
for (j in c(2:(Change_point+2*l))){
if (j>Change_point & j<=(Change_point+l)){
x = rnorm(1,A,1)
}
else{
x = rnorm(1,0,1)
}
CUSUM_statistics[i,j] = max(1,CUSUM_statistics[i,(j-1)])*exp((-(x-A)^2+x^2)/2)
}
print(i)
}
index_track = NULL
for (i in (1:samp_num)){
if (all(CUSUM_statistics[i,c(1:Change_point)]<H)){
index_track = c(index_track,i)
}
}
CUSUM_statistics2 = CUSUM_statistics[index_track,]
CUSUM_power= sum(apply(CUSUM_statistics2,1,function(x){any((x>=H))}))/dim(CUSUM_statistics2)[1]
CUSUM_power
abline(h=CUSUM_power,lwd=4,col='blue',lty=2)
############################################
#### Code for main results of Chapter 2 ####
############################################
# The below code reproduces Figures 2.1
Cap_T = 0.5
L = 10
M = Cap_T*L
M_star = M+1
samp_number=500000
h = seq(1,3.8,0.2)
MOSUM_function = function(n){
time_series = rnorm(L+n,0,1)
myf = function(i){
sum((time_series[i:(i+L-1)]))
}
normal_to_sum=sapply(c(1:n),myf)
return(normal_to_sum)
}
cl <- makeCluster(rep('localhost',3),type='SOCK')
sample_cores = function(){
sample_matrix_third = matrix(0,nrow=ceiling(samp_number/3),ncol=M_star)
for (i in (1:(ceiling(samp_number/3)))){
temporary=MOSUM_function(M_star)
for (j in (1:M_star)){
sample_matrix_third[i,j]=temporary[j]
}
}
sample_matrix_third
}
clusterExport(cl,c('sample_cores','M_star','L','samp_number','MOSUM_function'))
data1 = clusterEvalQ(cl,sample_cores())
stopCluster(cl)
sample_matrix = rbind(data1[[1]],data1[[2]],data1[[3]])
Desired_values = rep(0,length(h))
for (i in (1:length(h))){
Desired_values[i] = sum(apply(sample_matrix,1,function(x){any((x/(sqrt(L) )>=h[i]))}))/samp_number
}
Z=Cap_T/(2-Cap_T)
Diffusion_approximation = rep(0,length(h))
for (i in (1:length(h))){
t1= (1/((2*pi)**0.5))*integrate(function(x){pnorm((h[i]*(Z+1)-x*(-Z+1))/(2*(Z**0.5)))*exp(-(x**2)/2)},-Inf,h[i])$value;
t2=((2/pi)**0.5)*((h[i]*Z*exp(-((h[i])**2)/2))/(Z+1))*pnorm(h[i]*((Z)**0.5))
t3=(((Z)**0.5)*exp((-((h[i])**2)/2)*(Z+1))/(pi*(Z+1)))
Diffusion_approximation[i]= 1 - t1 + t2 +t3
}
CDA= NULL
for (h1 in (h)) {
rho = 0.583/sqrt(M*(1/Z))
Mprime=Z
integrand = function(x){
a = function(t){
(h1-t)/2 +rho
}
b = function(t){
(h1+t)/2
}
a = a(x)
b = b(x)
(1 - pnorm((b*Mprime+a)/sqrt(Mprime)) + exp(-2*a*b)*pnorm((b*Mprime-a)/sqrt(Mprime)))*dnorm(x)
}
CDA = c(CDA,1+integrate(integrand,-37,h1)$value-pnorm(h1))
}
Durbin_approx = ((h*Cap_T)/(((2*pi)**0.5)*Q))*exp(-(h**2)/2)
lambda_b = h*dnorm(h)
PCH_approx = 1-exp(-lambda_b*Cap_T)
par(mar=c(5.1,5.1,4.1,2.1))
plot(h,Desired_values,type='l',lwd=6,lty=2,ylab='',xlab='h',xaxt="n",yaxt='n',xaxs="i",yaxs="i",cex.axis=2.4,cex.lab=2.5)
axis(1, at=h, cex.axis=2.5)
axis(side = 2, tck = -.015, labels = NA)
axis(side = 2, lwd = 0, line = 0.7, las = 1, cex.axis=2.5)
lines(h,Diffusion_approximation,col='green')
lines(h,CDA,col='blue',lwd=2)
lines(h,Durbin_approx,col='red',lwd=2)
lines(h,PCH_approx,col='purple',lwd=2)
# The below code reproduces Figures 2.3
Cap_T = 1
L = 20
M = Cap_T*L
M_star = M+1
samp_number=100000
h = seq(1,3.8,0.2)
MOSUM_function = function(n){
time_series = rnorm(L+n,0,1)
myf = function(i){
sum((time_series[i:(i+L-1)]))
}
normal_to_sum=sapply(c(1:n),myf)
return(normal_to_sum)
}
cl <- makeCluster(rep('localhost',3),type='SOCK')
sample_cores = function(){
sample_matrix_third = matrix(0,nrow=ceiling(samp_number/3),ncol=M_star)
for (i in (1:(ceiling(samp_number/3)))){
temporary=MOSUM_function(M_star)
for (j in (1:M_star)){
sample_matrix_third[i,j]=temporary[j]
}
}
sample_matrix_third
}
clusterExport(cl,c('sample_cores','M_star','L','samp_number','MOSUM_function'))
data1 = clusterEvalQ(cl,sample_cores())
stopCluster(cl)
sample_matrix = rbind(data1[[1]],data1[[2]],data1[[3]])
Desired_values = rep(0,length(h))
for (i in (1:length(h))){
Desired_values[i] = sum(apply(sample_matrix,1,function(x){any((x/(sqrt(L) )>=h[i]))}))/samp_number
}
plot(h,Desired_values,type='l',xlim=c(1,3.8),ylab='',xlab='h',lty=2,xaxt="n",yaxt='n',lwd=7,xaxs="i",yaxs="i",cex.axis=2.4,cex.lab=2.5)
axis(1, at=seq(1,3.8,0.4), cex.axis=2.5)
axis(side = 2, tck = -.015, labels = NA)
axis(side = 2, lwd = 0, line = 0.7, las = 1, cex.axis=2.5)
### Corrected shepp to be used if T=1
CSA_1 = NULL
for (h in seq(1,4,0.1)){
h1 = h+0.583*sqrt(2)/sqrt(L)
CSA_1=c(CSA_1,1-pnorm(h)*pnorm(h1) +dnorm(h1)*(h*pnorm(h)+dnorm(h)))
}
lines(seq(1,4,0.1),CSA_1,col=2,lwd=3,lty=1)
### Corrected shepp to be used if T=2
CSA_2 = NULL
for (h in seq(1,4,0.1)){
h1 = h+0.583*sqrt(2)/sqrt(L)
f_test = function(x){
f_testy = function(y2){
M = matrix(0,3,3)
M[1,1] = dnorm(x)
M[1,2] = dnorm(h1-y2)
M[1,3] = pnorm(h1-y2)
M[2,1] = dnorm(h1)
M[2,2] = dnorm(2*h1-x-y2)
M[2,3] = pnorm(2*h1-x-y2)
M[3,1] = dnorm(y2+x)
M[3,2] = dnorm(h1)
M[3,3] = pnorm(h1)
det(M)
}
f_testy = Vectorize(f_testy)
integrate(f_testy,h1-x,Inf)$value
}
f_test = Vectorize(f_test)
F2_c = integrate(f_test,-Inf,h)$value
CSA_2=c(CSA_2,1-F2_c)
}
lines(seq(1,4,0.1),CSA_2,col=2,lwd=3,lty=1)
# The below code reproduces Figures 2.6
Cap_T = 1
L = 20
M = Cap_T*L
M_star = M+1
samp_number=100000
h = seq(1,3.8,0.2)
MOSUM_function = function(n){
time_series = rnorm(L+n,0,1)
myf = function(i){
sum((time_series[i:(i+L-1)]))
}
normal_to_sum=sapply(c(1:n),myf)
return(normal_to_sum)
}
cl <- makeCluster(rep('localhost',3),type='SOCK')
sample_cores = function(){
sample_matrix_third = matrix(0,nrow=ceiling(samp_number/3),ncol=M_star)
for (i in (1:(ceiling(samp_number/3)))){
temporary=MOSUM_function(M_star)
for (j in (1:M_star)){
sample_matrix_third[i,j]=temporary[j]
}
}
sample_matrix_third
}
clusterExport(cl,c('sample_cores','M_star','L','samp_number','MOSUM_function'))
data1 = clusterEvalQ(cl,sample_cores())
stopCluster(cl)
sample_matrix = rbind(data1[[1]],data1[[2]],data1[[3]])
Desired_values = rep(0,length(h))
for (i in (1:length(h))){
Desired_values[i] = sum(apply(sample_matrix,1,function(x){any((x/(sqrt(L) )>=h[i]))}))/samp_number
}
plot(h,Desired_values,type='l',xlim=c(1,3.8),ylab='',xlab='h',lty=2,xaxt="n",yaxt='n',lwd=7,xaxs="i",yaxs="i",cex.axis=2.4,cex.lab=2.5)
axis(1, at=seq(1,3.8,0.4), cex.axis=2.5)
axis(side = 2, tck = -.015, labels = NA)
axis(side = 2, lwd = 0, line = 0.7, las = 1, cex.axis=2.5)