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bayes-copula.R
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bayes-copula.R
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rm(list=ls())
#"Quadratic" function in Owen (2001) (Section 3.14)
Owen <- function(x, thre, gradiente)
{
grad <- t(gradiente)
par <- t(as.matrix(x))
eps <- 1/thre
z <- 1+par%*%grad
ans <- z
lo <- ((z)<eps)
ans[ lo ] <- log(eps) - 1.5 + 2*(z[lo])/eps - 0.5*(z[lo]/eps)^2
ans[ !lo ] <- log( z[!lo] )
-sum(ans)
}
Schennach <- function(x, gradiente)
{
grad <- t(gradiente)
par <- t(as.matrix(x))
z <- exp(par%*%grad)/length(gradiente)
sum(z)
}
#Below the function for the computation of empirical likelihood
##input:
####### 1) SC --> individual contributions of the estimating function.
####### It can be either an n x 1 vector (scalar parameter) or an n x p matrix
####### (vector valued parameter)
##output: a list with elements
######## 1) w0 --> weights associated to each unit
######## 2) conv --> convergence of the algorithm (available only for vector valued parameter)
######## 3) elr --> empirical loglikelihood ratio
######## 4) el --> empirical loglikelihood
EL <- function(SC)
{
n <- NROW(SC)
p <- NCOL(SC)
##find Lagrange multiplier
if(p!=1)
{
OBJ <- optim(par=rep(0,p), fn=Owen, thre=n, gradiente=SC, control=list(maxit=1e4))
molt_lagr_0 <- OBJ$pa
conv <- OBJ$conv
}
else
{
molt_lagr_0 <- optim(par=rep(0,p), fn=Owen, thre=n, gradiente=SC, method="Brent", lower=-1e1, upper=1e1)$pa
}
##weights
w_emp_0 <- as.numeric( 1/(n*(1+molt_lagr_0%*%t(SC))) )
if(p!=1)
{
list(w0=w_emp_0, conv=conv, elr=-2*sum(log(w_emp_0))-2*n*log(n), el=log(w_emp_0))
}
else
{
list(w0=w_emp_0, elr=-2*sum(log(w_emp_0))-2*n*log(n), el=log(w_emp_0) )
}
}
EL.Schennach <- function(SC)
{
n <- NROW(SC)
p <- NCOL(SC)
##find Lagrange multiplier
molt_lagr_0 <- optim(par=rep(0,p), fn=Schennach, gradiente=SC,
method="Brent", lower=-1e1, upper=1e1)$pa
##weights
const=as.numeric(sum(exp(molt_lagr_0%*%t(SC))))
w_emp_0 <- as.numeric( exp(molt_lagr_0%*%t(SC)) / const )
el=log(w_emp_0)
elr=-2*sum(log(w_emp_0))-2*n*log(n)
return( list(w=w_emp_0,el=el,elr=elr ) )
}
### Population mean example
n <- 30
mu <- 1
set.seed(1)
data <- rnorm(n, mu)
est.func <- data-mu
EL(est.func)
EL.Schennach(est.func)
###########################
### ABC for copula models
###########################
install.packages("gmm")
require(gmm)
install.packages("copula")
require(copula)
### Data simulation
theta=1.076 # Spearman =.5
ssize=100
S=1000 #MC sample size
cc<-claytonCopula(theta)
uu<-rCopula(ssize, cc)
### ELCOP function
ELCOP<- function(sample){
n<-dim(sample)[1]
p<-dim(sample)[2]
rr<-rank(sample[,1])
ss<-rank(sample[,2])
vv<-rr*ss
#print(vv)
cor(uu[,1], uu[,2], method="spearman")
S=1000
rh<-runif(S, -1,1)
omega<-rep(0,S)
for (s in 1:S) {
#print(rh[s])
estim= 12 * vv /(n^2-1) - 3*(n+1)/(n-1) - rh[s]
omega[s]<-exp(-EL(estim)$elr)
#print(omega[s])
}
psam<-sample(rh, size=S, rep=T, prob=omega)
sintesi<-c(quantile(psam,.05), quantile(psam,.5), quantile(psam,.95))
return(as.numeric(sintesi))
}
ELCOP.Schennach<- function(sample){
n<-dim(sample)[1]
p<-dim(sample)[2]
rr<-rank(sample[,1])
ss<-rank(sample[,2])
vv<-rr*ss
#print(vv)
cor(uu[,1], uu[,2], method="spearman")
S=1000
rh<-runif(S, -1,1)
omega<-rep(0,S)
for (s in 1:S) {
#print(rh[s])
estim= 12 * vv /(n^2-1) - 3*(n+1)/(n-1) - rh[s]
omega[s]<-exp(-EL.Schennach(estim)$elr)
#print(omega[s])
}
psam<-sample(rh, size=S, rep=T, prob=omega)
sintesi<-c(quantile(psam,.05), quantile(psam,.5), quantile(psam,.95))
return(as.numeric(sintesi))
}
as.numeric(ELCOP(uu))
as.numeric(ELCOP.Schennach(uu))
### Repeat simulations for 100 samples
require(gmm)
require(copula)
sim.cop.clay<-function(theta=1.076, Z=10,ssize=100,
S=1000){
# ssize= Sample Size
# S= MC sample size
cc<-claytonCopula(theta)
sintesi.finale<-matrix(rep(0, 3*Z), nrow=Z)
#
for (i in 1:Z){
uu<-rCopula(ssize, cc)
sintesi.finale[i,]<-ELCOP(uu)
}
boxplot(sintesi.finale, xlab="5th, 50th and 95th percentiles of the posterior of rho")
title("Boxplots: Clayton Copula: true Rho=0.5")
return(sintesi.finale)
}
system.time(prova.clay1<-sim.cop.clay(ssize=50, Z=500, S=1000))
save.image("copula-Clayton.RData")
### Comparison between ABC and classical methods
ELCOP.vs.freq<- function(sample,S=1000){
n<-dim(sample)[1]
p<-dim(sample)[2]
# Ranghi
rr<-rank(sample[,1])
ss<-rank(sample[,2])
vv<-rr*ss
# Frequentist estimate
rhon=12 * sum(vv) /(n*(n+1)*(n-1)) - 3*(n+1)/(n-1)
# Frequentist estimate of the variance
An=sum(rr/(n+1)*ss/(n+1))/n
Bn=sum((rr/(n+1))^2*(ss/(n+1))^2)/n
Cn=0
for(i in 1:length(rr))
{
const=rr[i]/(n+1)*ss[i]/(n+1)
for(j in 1:length(rr))
{
for(k in 1:length(33))
{
Cn=Cn+const*(rr[k]<=rr[i])*(ss[k]<=ss[j])
}
}
}
Cn=Cn/n^3+0.25-An
Dn=0
for(i in 1:length(rr))
{
for(j in 1:length(ss))
{
Dn=Dn+ss[i]*ss[j]/(n+1)^2*max(rr[i]/(n+1),rr[j]/(n+1))
}
}
Dn=Dn/n^2
En=0
for(i in 1:length(rr))
{
for(j in 1:length(ss))
{
En=En+rr[i]*rr[j]/(n+1)^2*max(ss[i]/(n+1),ss[j]/(n+1))
}
}
En=En/n^2
sigman=144*(-9*An^2+Bn+2*Cn+2*Dn+2*En)
ICn=c()
if(sigman>=0)
{
ICn=c(rhon-1.96*sqrt(sigman/n),rhon+1.96*sqrt(sigman/n))
} else {
ICn=rep(NA,2)
}
### ABC
rh<-runif(S, -1,1)
omega<-rep(0,S)
for (s in 1:S) {
estim= 12 * vv /(n^2-1) - 3*(n+1)/(n-1) - rh[s]
omega[s]<-exp(-EL(estim)$elr)
}
psam<-sample(rh, size=S, rep=T, prob=omega)
sintesi<-c(quantile(psam,.025), quantile(psam,.5), quantile(psam,.975))
return(list(rhon=rhon,ICn=ICn,ABC=sintesi))
}
ELCOP.vs.freq.Sch<- function(sample){
n<-dim(sample)[1]
p<-dim(sample)[2]
# Ranghi
rr<-rank(sample[,1])
ss<-rank(sample[,2])
vv<-rr*ss
# Frequentist estimate
rhon=12 * sum(vv) /(n*(n+1)*(n-1)) - 3*(n+1)/(n-1)
# Frequentist estimate of the variance
An=sum(rr/(n+1)*ss/(n+1))/n
Bn=sum((rr/(n+1))^2*(ss/(n+1))^2)/n
Cn=0
for(i in 1:length(rr))
{
const=rr[i]/(n+1)*ss[i]/(n+1)
for(j in 1:length(rr))
{
for(k in 1:length(33))
{
Cn=Cn+const*(rr[k]<=rr[i])*(ss[k]<=ss[j])
}
}
}
Cn=Cn/n^3+0.25-An
Dn=0
for(i in 1:length(rr))
{
for(j in 1:length(ss))
{
Dn=Dn+ss[i]*ss[j]/(n+1)^2*max(rr[i]/(n+1),rr[j]/(n+1))
}
}
Dn=Dn/n^2
En=0
for(i in 1:length(rr))
{
for(j in 1:length(ss))
{
En=En+rr[i]*rr[j]/(n+1)^2*max(ss[i]/(n+1),ss[j]/(n+1))
}
}
En=En/n^2
sigman=144*(-9*An^2+Bn+2*Cn+2*Dn+2*En)
ICn=c()
if(sigman>=0)
{
ICn=c(rhon-1.96*sqrt(sigman/n),rhon+1.96*sqrt(sigman/n))
} else {
ICn=rep(NA,2)
}
### ABC
S=1000
rh<-runif(S, -1,1)
omega<-rep(0,S)
for (s in 1:S) {
estim= 12 * vv /(n^2-1) - 3*(n+1)/(n-1) - rh[s]
omega[s]<-exp(-EL.Schennach(estim)$elr)
}
psam<-sample(rh, size=S, rep=T, prob=omega)
sintesi<-c(quantile(psam,.025), quantile(psam,.5), quantile(psam,.975))
return(list(rhon=rhon,ICn=ICn,ABC=sintesi))
}
ELCOP.vs.freq(uu)
ELCOP.vs.freq.Sch(uu)
### Repeat simulations for Z=500 samples
require(gmm)
require(copula)
sim.cop.clay.comp<-function(theta=1.076, Z=10,ssize=100,
S=10^4){
cc<-claytonCopula(theta)
sintesi.finale.Sch<-matrix(rep(0, 6*Z), nrow=Z)
sintesi.finale<-matrix(rep(0, 6*Z), nrow=Z)
#
for (i in 1:Z){
uu<-rCopula(ssize, cc)
obj.Sch=ELCOP.vs.freq.Sch(uu)
obj=ELCOP.vs.freq(uu)
sintesi.finale[i,1]=obj[[2]][1]
sintesi.finale[i,2]=obj[[1]]
sintesi.finale[i,3]=obj[[2]][2]
sintesi.finale[i,4:6]=obj[[3]]
sintesi.finale.Sch[i,1]=obj.Sch[[2]][1]
sintesi.finale.Sch[i,2]=obj.Sch[[1]]
sintesi.finale.Sch[i,3]=obj.Sch[[2]][2]
sintesi.finale.Sch[i,4:6]=obj.Sch[[3]]
}
return(list(Schennach=sintesi.finale.Sch,Owen=sintesi.finale))
}
# user system elapsed
#1395.43 1.56 1478.47
prova.comp<-sim.cop.clay.comp(ssize=1000, Z=500, S=10^4)
prova.comp.Sch<-sim.cop.clay.comp(ssize=1000, Z=500, S=10^4)
# user system elapsed
#1553.36 0.78 1677.63
par(mfrow=c(1,2))
plot(prova.comp[,2],type="l",ylim=c(0,1),xlab="Experiment",ylab=expression(rho),
main="Clayton",col="blue",lty=1,lwd=1)
lines(prova.comp[,1],col="blue",lty=3,lwd=0.5)
lines(prova.comp[,3],col="blue",lty=3,lwd=0.5)
lines(prova.comp[,5],col="green",lty=1,lwd=1)
lines(prova.comp[,4],col="green",lty=3,lwd=0.5)
lines(prova.comp[,6],col="green",lty=3,lwd=0.5)
abline(h=0.5,col="red",lwd=1.5)
sum(is.na(prova.comp[,1])) # 0
mean(prova.comp[,3]-prova.comp[,1]) #0.2663587
mean(prova.comp[,6]-prova.comp[,4]) #0.2596814
le.freq=prova.comp[,3]-prova.comp[,1]
le.ABC=prova.comp[,6]-prova.comp[,4]
length(le.ABC)
sum(le.freq>le.ABC)/length(le.ABC)
# 0.588
# Coverage
rho.comp=rho(claytonCopula(1.076))
sum(prova.comp[,3]>rho.comp&prova.comp[,1]<rho.comp)/500 #99.8%
sum(prova.comp[,6]>rho.comp&prova.comp[,4]<rho.comp)/500 #100%
save.image("copula-Clayton.RData")
#####################
### Multivariate rho
#####################
ELCOP.vs.freq.multi<- function(sam,var.IC,S=10^3){
n<-dim(sam)[1]
d<-dim(sam)[2]
# Ranks
U.hat=matrix(NA,ncol=d,nrow=n)
for(i in 1:d)
{
U.hat[,i]=rank(sam[,i])/n
}
VV1=apply(1-U.hat,1,prod)
VV2=apply(U.hat,1,prod)
# Frequentist estimate
const=(d+1)/(2^d-(d+1))
estim1=const*(2^d/n*sum(VV1)-1)
estim2=const*(2^d/n*sum(VV2)-1)
estim3=mean(estim1,estim2)
# Frequentist estimate of the variance
sigman=var.IC
IC1n=c()
IC2n=c()
if(sigman>=0)
{
IC1n=c(estim1-1.96*sqrt(sigman/n),estim1+1.96*sqrt(sigman/n))
IC2n=c(estim2-1.96*sqrt(sigman/n),estim2+1.96*sqrt(sigman/n))
IC3n=c(estim3-1.96*sqrt(sigman/n),estim3+1.96*sqrt(sigman/n))
} else {
IC1n=rep(NA,2)
IC2n=rep(NA,2)
}
### ABC
lbound=(2^d-factorial(d+1))/(factorial(d)*(2^d-(d+1)))
rho<-runif(S, lbound,1)
omega1<-rep(0,S)
omega2<-rep(0,S)
omega3<-rep(0,S)
for (s in 1:S) {
est3=estim3-rho[s]
est1=const*(2^d/n*sum(VV1)-1) - rho[s]
est2=const*(2^d/n*sum(VV2)-1) - rho[s]
omega1[s]<-exp(-EL(est1)$elr)
omega2[s]<-exp(-EL(est2)$elr)
omega3[s]<-exp(-EL(est3)$elr)
}
psam1<-sample(rho, size=S, rep=T, prob=omega1)
psam2<-sample(rho, size=S, rep=T, prob=omega2)
psam3<-sample(rho, size=S, rep=T, prob=omega3)
return(list(rhon1=estim1,rhon2=estim2,rhon3=estim3,
IC1n=IC1n,IC2n=IC2n,IC3n=IC3n,
ABC1=psam1,ABC2=psam2,ABC3=psam3))
}
uu=rCopula(10^3,cc)
d=2
var.IC=((d+1)^2*(3*(4/3)^d-d-3))/(3*(1+d-2^d)^2)
try1=ELCOP.vs.freq.multi(uu,var.IC)
print("end - Multivariate rho")
# Let's repeat the same procedure 100 times
print("start - Multivariate rho - independence")
multirho.BAYvsFREQ=function(n,cc,var.IC,S)
{
uu=rCopula(n,cc)
try=ELCOP.vs.freq.multi(sam=uu,var.IC=var.IC,S=10^3)
return(try)
}
# Let's try with d=2 and the variance in case of independence
final.multirho.freq <- matrix(NA,nrow=500,ncol=9)
final.multirho.bay1 <- matrix(NA,ncol=500,nrow=10^4)
final.multirho.bay2 <- matrix(NA,ncol=500,nrow=10^4)
final.multirho.bay3 <- matrix(NA,ncol=500,nrow=10^4)
for(i in 1:500)
{
try=multirho.BAYvsFREQ(1000,cc,var.IC,S=10^4)
final.multirho.freq[i,1]=try[[1]]
final.multirho.freq[i,2]=try[[2]]
final.multirho.freq[i,3]=try[[3]]
final.multirho.freq[i,4:5]=try$IC1n
final.multirho.freq[i,6:7]=try$IC2n
final.multirho.freq[i,8:9]=try$IC3n
final.multirho.bay1[,i]=try$ABC1
final.multirho.bay2[,i]=try$ABC2
final.multirho.bay3[,i]=try$ABC3
print(i)
}
new.mat=cbind(final.multirho[,4],final.multirho[,1],final.multirho[,5],
final.multirho[,6],final.multirho[,2],final.multirho[,7],
final.multirho[,8],final.multirho[,3],final.multirho[,9],
final.multirho[,10:18])
boxplot(new.mat,names=expression(IC1[0.025],rho[1],IC1[0.975],
IC2[0.025],rho[2],IC2[0.975],
IC3[0.025],rho[3],IC3[0.975],
q1[0.025],q1[0.50],q1[0.975],
q2[0.025],q2[0.50],q2[0.975],
q3[0.025],q3[0.50],q3[0.975]),las=2)
ICcov.freq=matrix(NA,500,3)
ICcov.bay=matrix(NA,500,3)
for(i in 1:500)
{
ICcov.freq[i,1]=new.mat[i,1]<0.5 & new.mat[i,3]>0.5
ICcov.freq[i,2]=new.mat[i,4]<0.5 & new.mat[i,6]>0.5
ICcov.freq[i,3]=new.mat[i,7]<0.5 & new.mat[i,9]>0.5
ICcov.bay[i,1]=new.mat[i,10]<0.5 & new.mat[i,12]>0.5
ICcov.bay[i,2]=new.mat[i,13]<0.5 & new.mat[i,15]>0.5
ICcov.bay[i,3]=new.mat[i,16]<0.5 & new.mat[i,18]>0.5
}
apply(ICcov.freq,2,sum)/500
# 0.97, 0.97, 0.97
apply(ICcov.bay,2,sum)/500
# 1, 1, 1
save.image("copula-Clayton.RData")
# By supposing independence the coverage is the one expected
# Let's try with d=6
library(copula)
cc.multi=claytonCopula(1.076,dim=6)
final.multirho6.freq <- matrix(NA,nrow=500,ncol=9)
final.multirho6.bay1 <- matrix(NA,ncol=500,nrow=10^4)
final.multirho6.bay2 <- matrix(NA,ncol=500,nrow=10^4)
final.multirho6.bay3 <- matrix(NA,ncol=500,nrow=10^4)
for(i in 1:500)
{
try6=multirho.BAYvsFREQ(1000,cc.multi,var.IC,S=10^4)
final.multirho6.freq[i,1]=try6[[1]]
final.multirho6.freq[i,2]=try6[[2]]
final.multirho6.freq[i,3]=try6[[3]]
final.multirho6.freq[i,4:5]=try6$IC1n
final.multirho6.freq[i,6:7]=try6$IC2n
final.multirho6.freq[i,8:9]=try6$IC3n
final.multirho6.bay1[,i]=try6$ABC1
final.multirho6.bay2[,i]=try6$ABC2
final.multirho6.bay3[,i]=try6$ABC3
print(i)
}
save.image("copula-Clayton.RData")
new.mat6=cbind(final.multirho6[,4],final.multirho6[,1],final.multirho6[,5],
final.multirho6[,6],final.multirho6[,2],final.multirho6[,7],
final.multirho6[,8],final.multirho6[,3],final.multirho6[,9],
final.multirho6[,10:18])
boxplot(new.mat6,names=expression(IC1[0.025],rho[1],IC1[0.975],
IC2[0.025],rho[2],IC2[0.975],
IC3[0.025],rho[3],IC3[0.975],
q1[0.025],q1[0.50],q1[0.975],
q2[0.025],q2[0.50],q2[0.975],
q3[0.025],q3[0.50],q3[0.975]),las=2)
ICcov6.freq=matrix(NA,500,3)
ICcov6.bay=matrix(NA,500,3)
for(i in 1:500)
{
ICcov6.freq[i,1]=new.mat6[i,1]<0.5 & new.mat6[i,3]>0.5
ICcov6.freq[i,2]=new.mat6[i,4]<0.5 & new.mat6[i,6]>0.5
ICcov6.freq[i,3]=new.mat6[i,7]<0.5 & new.mat6[i,9]>0.5
ICcov6.bay[i,1]=new.mat6[i,10]<0.5 & new.mat6[i,12]>0.5
ICcov6.bay[i,2]=new.mat6[i,13]<0.5 & new.mat6[i,15]>0.5
ICcov6.bay[i,3]=new.mat6[i,16]<0.5 & new.mat6[i,18]>0.5
}
apply(ICcov6.freq,2,sum)/500
# 0.988, 0.000, 0.988
apply(ICcov6.bay,2,sum)/500
# 1, 1, 1
## Not all the intervals contains the 0.5 value for the estimates,
## the intervals for rho2 never contain the value 0.5,
## the covarage of the intervals for rho2 is the one expected
## the covarage of the intervals for rho3 is the one expected
save.image("copula-Clayton.RData")
#######################################
### Bootstrap estimate of the variance
#######################################
require(gmm)
require(copula)
bootvar.multirho<- function(sam,B=10000){
n<-dim(sam)[1]
d<-dim(sam)[2]
const=(d+1)/(2^d-(d+1))
estim1=c()
estim2=c()
estim3=c()
for(j in 1:B)
{
U.hat=matrix(NA,ncol=d,nrow=n)
n.sam=sample(1:n,size=n,rep=T)
sam.boot=sam[n.sam,]
for(i in 1:d)
{
U.hat[,i]=rank(sam.boot[,i])/n
}
VV1=apply(1-U.hat,1,prod)
VV2=apply(U.hat,1,prod)
# Frequentist estimate
estim1[j]=const*(2^d/n*sum(VV1)-1)
estim2[j]=const*(2^d/n*sum(VV2)-1)
estim3[j]=mean(estim1[j],estim2[j])
}
return(cbind(estim1,estim2,estim3))
}
ssize=1000
cc.multi=claytonCopula(1.076,dim=6)
uu=rCopula(ssize,cc.multi)
try6.boot=bootvar.multirho(uu)
var6.boot=apply(try6.boot,2,var)
var6.boot
ssize=100
cc=claytonCopula(1.076,dim=2)
uu=rCopula(ssize,cc)
try2.boot=bootvar.multirho(uu)
var2.boot=apply(try2.boot,2,var)
var2.boot
# Monte Carlo estimate of the variance
MCvar.multirho<- function(cc,n,B=10000){
d<-dim(cc)
const=(d+1)/(2^d-(d+1))
estim1=c()
estim2=c()
estim3=c()
for(j in 1:B)
{
U.hat=matrix(NA,ncol=d,nrow=n)
uu=rCopula(n,cc)
for(i in 1:d)
{
U.hat[,i]=rank(uu[,i])/n
}
VV1=apply(1-U.hat,1,prod)
VV2=apply(U.hat,1,prod)
# Frequentist estimate
estim1[j]=const*(2^d/n*sum(VV1)-1)
estim2[j]=const*(2^d/n*sum(VV2)-1)
}
return(cbind(estim1,estim2))
}
ssize=1000
cc.multi=claytonCopula(1.076,dim=6)
try6.MC=MCvar.multirho(cc.multi,n=ssize)
var6.MC=apply(try6.MC,2,var)
var6.MC
# Let's repeat the same procedure 500 times
finalboot.multirho=matrix(NA,nrow=500,ncol=18)
for(i in 1:nrow(finalboot.multirho))
{
try.boot=multirho.BAYvsFREQ(1000,cc,max(var2.boot),S=10^4)
finalboot.multirho[i,1]=try.boot[[1]]
finalboot.multirho[i,2]=try.boot[[2]]
finalboot.multirho[i,3]=try.boot[[3]]
finalboot.multirho[i,4:5]=try.boot$IC1n
finalboot.multirho[i,6:7]=try.boot$IC2n
finalboot.multirho[i,8:9]=try.boot$IC3n
finalboot.multirho[i,10:12]=try.boot$ABC1
finalboot.multirho[i,13:15]=try.boot$ABC2
finalboot.multirho[i,16:18]=try.boot$ABC3
}
new.matboot=cbind(finalboot.multirho[,4],finalboot.multirho[,1],finalboot.multirho[,5],
finalboot.multirho[,6],finalboot.multirho[,2],finalboot.multirho[,7],
finalboot.multirho[,8],finalboot.multirho[,3],finalboot.multirho[,9],
finalboot.multirho[,10:18])
boxplot(new.matboot,names=expression(IC1[0.025],rho[1],IC1[0.975],
IC2[0.025],rho[2],IC2[0.975],
IC3[0.025],rho[3],IC3[0.975],
q1[0.025],q1[0.50],q1[0.975],
q2[0.025],q2[0.50],q2[0.975],
q3[0.025],q3[0.50],q3[0.975]),las=2)
ICcovboot.freq=matrix(NA,500,3)
ICcovboot.bay=matrix(NA,500,3)
for(i in 1:500)
{
ICcovboot.freq[i,1]=new.matboot[i,1]<0.5 & new.matboot[i,3]>0.5
ICcovboot.freq[i,2]=new.matboot[i,4]<0.5 & new.matboot[i,6]>0.5
ICcovboot.freq[i,3]=new.matboot[i,7]<0.5 & new.matboot[i,9]>0.5
ICcovboot.bay[i,1]=new.matboot[i,10]<0.5 & new.matboot[i,12]>0.5
ICcovboot.bay[i,2]=new.matboot[i,13]<0.5 & new.matboot[i,15]>0.5
ICcovboot.bay[i,3]=new.matboot[i,16]<0.5 & new.matboot[i,18]>0.5
}
apply(ICcovboot.freq,2,sum)/500
# 0.202, 0.208, 0.202
# rho1 rho2 rho3
# Coverage under 0.95 for all the statistics
apply(ICcovboot.bay,2,sum)/500
# 1.00 1.00 1.00
# rho1 rho2 rho3
save.image("copula-Clayton.RData")
#### Let's try with dimension 6
library(copula)
cc.multi=claytonCopula(1.076,dim=6)
finalboot6.multirho.freq <- matrix(NA,nrow=500,ncol=9)
finalboot6.multirho.bay1 <- matrix(NA,ncol=500,nrow=10^4)
finalboot6.multirho.bay2 <- matrix(NA,ncol=500,nrow=10^4)
finalboot6.multirho.bay3 <- matrix(NA,ncol=500,nrow=10^4)
for(i in 1:500)
{
try6.boot=multirho.BAYvsFREQ(1000,cc.multi,max(var6.boot),S=10^4)
finalboot6.multirho.freq[i,1]=try6.boot[[1]]
finalboot6.multirho.freq[i,2]=try6.boot[[2]]
finalboot6.multirho.freq[i,3]=try6.boot[[3]]
finalboot6.multirho.freq[i,4:5]=try6.boot$IC1n
finalboot6.multirho.freq[i,6:7]=try6.boot$IC2n
finalboot6.multirho.freq[i,8:9]=try6.boot$IC3n
finalboot6.multirho.bay1[,i]=try6.boot$ABC1
finalboot6.multirho.bay2[,i]=try6.boot$ABC2
finalboot6.multirho.bay3[,i]=try6.boot$ABC3
print(i)
}
save.image("copula-Clayton.RData")
new.matboot6=cbind(finalboot6.multirho[,4],finalboot6.multirho[,1],finalboot6.multirho[,5],
finalboot6.multirho[,6],finalboot6.multirho[,2],finalboot6.multirho[,7],
finalboot6.multirho[,8],finalboot6.multirho[,3],finalboot6.multirho[,9],
finalboot6.multirho[,10:18])
### What is the true rho which we are simulating from?
install.packages("cubature")
library(cubature)
# Rho1
f.clayton=function(x,theta)
{
d=length(x)
temp=(sum(x^(0-theta))-d+1)^(-1/theta)
return(temp)
}
int.clayton=adaptIntegrate(f.clayton,theta=1.076,lowerLimit=rep(0,6),upperLimit=rep(1,6))
d=6
rho.clayton1=((d+1)/(2^d-d-1)) * (2^d*int.clayton$integral-1)
rho.clayton1
# 0.5144599
# Rho2
f.rho2=function(x,cop)
{
d=length(x)
ind.cop=sum(log(x))
cop.dens=dCopula(x,cop,log=TRUE)
temp=ind.cop+cop.dens
return(exp(temp))
}
int2.clayton=adaptIntegrate(f2.clayton,cop=cc.multi,
lowerLimit=rep(0,6),upperLimit=rep(1,6))
d=6
rho.clayton2=((d+1)/(2^d-d-1)) * (2^d*int2.clayton$integral-1)
rho.clayton2
# 0.346185
ICcovboot6.freq=matrix(NA,500,2)
ICcovboot6.bay=matrix(NA,500,2)
for(i in 1:500)
{
ICcovboot6.freq[i,1]=new.matboot6[i,1]<rho.clayton1 & new.matboot6[i,3]>rho.clayton1
ICcovboot6.freq[i,2]=new.matboot6[i,4]<rho.clayton2 & new.matboot6[i,6]>rho.clayton2
ICcovboot6.bay[i,1]=new.matboot6[i,10]<rho.clayton1 & new.matboot6[i,12]>rho.clayton1
ICcovboot6.bay[i,2]=new.matboot6[i,13]<rho.clayton2 & new.matboot6[i,15]>rho.clayton2
}
apply(ICcovboot6.freq,2,sum)/500
# 0.054 0.050
# rho1 rho2
# Coverage under 0.95 for all the statistics
apply(ICcovboot6.bay,2,sum)/500
# 1.00 1.00
# rho1 rho2
save.image("copula-Clayton.RData")
## Rho1, frequentist and Bayesian estimates
par(mfrow=c(1,2))
plot(new.matboot6[,2],type="l",ylim=c(-1,1),xlab="Experiment",ylab=expression(rho[1]),
main="Clayton 1",col="blue")
lines(new.matboot6[,1],col="blue",lty=3)
lines(new.matboot6[,3],col="blue",lty=3)
lines(new.matboot6[,11],ylim=c(-1,1),col="green")
lines(new.matboot6[,10],col="green",lty=3)
lines(new.matboot6[,12],col="green",lty=3)
abline(h=rho.clayton1,col="red",lwd=1.5)
for(i in 1:length(new.matboot6[,2]))
{
segments(x0=i,y0=new.matboot6[i,1],x1=i,y=new.matboot6[i,3],col="blue",lwd=2)
}
points(new.matboot6[,1],col="blue")
points(new.matboot6[,3],col="blue")
abline(h=0.5,col="red",lwd=1.5)
plot(new.matboot6[,11],pch=20,ylim=c(-1,1),xlab="Experiment",ylab=expression(rho[1]),
main="Bayesian")
for(i in 1:length(new.matboot6[,11]))
{
segments(x0=i,y0=new.matboot6[i,10],x1=i,y=new.matboot6[i,12],col="blue",lwd=2)
}
points(new.matboot6[,10],col="blue")
points(new.matboot6[,12],col="blue")
abline(h=rho.clayton1,col="red",lwd=1.5)
# Let's try to see if the length of the confidence intervals varies
# with the dimensionality
d.incr=c(2:10,25,50)
rho.incr.dim=list()
for(i in 1:length(d.incr))
{
mat.rho.incrd=matrix(NA,ncol=12,nrow=50)
cc.incrd=claytonCopula(1.076,dim=d.incr[i])
uu=rCopula(ssize,cc.incrd)
try.boot=bootvar.multirho(uu)
var.boot=apply(try.boot,2,var)
for(j in 1:50)
{
try6.boot=multirho.BAYvsFREQ(1000,cc.incrd,max(var.boot),S=10^4)
mat.rho.incrd[j,3]=try6.boot[[1]]
mat.rho.incrd[j,1:2]=try6.boot$IC1n
mat.rho.incrd[j,6]=try6.boot[[2]]
mat.rho.incrd[j,4:5]=try6.boot$IC2n
mat.rho.incrd[j,7:9]=try6.boot$ABC1
mat.rho.incrd[j,10:12]=try6.boot$ABC2
print(paste(d.incr[i],j,sep="_"))
}
rho.incr.dim[[i]]=mat.rho.incrd
}
le.incrd=matrix(NA,ncol=4,nrow=length(d.incr))
for(i in 1:length(d.incr))
{
le.incrd[i,1]=mean(rho.incr.dim[[i]][,2]-rho.incr.dim[[i]][,1])
le.incrd[i,2]=mean(rho.incr.dim[[i]][,5]-rho.incr.dim[[i]][,4])
le.incrd[i,3]=mean(rho.incr.dim[[i]][,9]-rho.incr.dim[[i]][,7])
le.incrd[i,4]=mean(rho.incr.dim[[i]][,12]-rho.incr.dim[[i]][,10])
}
le.incrd
save.image("copula-Clayton.RData")
# Let's see if the value of rho(1) is fixed by varying the dimension d
# for any value of theta
theta.seq=seq(0.1,10,0.1)
rho1clayton.mat=matrix(NA,nrow=length(theta.seq),ncol=5)
for (i in 1:length(theta.seq))
{
for(j in 1:5)
{
d=d.incr[j]
int.clayton=adaptIntegrate(f.clayton,theta=theta.seq[i],
lowerLimit=rep(0,d),upperLimit=rep(1,d))
rho1clayton.mat[i,j]=((d+1)/(2^d-d-1)) *
(2^d*int.clayton$integral-1)
print(paste(d,i,sep="_"))
}