Modelling dependence with copulas. Full article at: http://datascienceplus.com/modelling-dependence-with-copulas/

copulas_example.R

#Load library mass and set seed
library(MASS)
set.seed(100)

# We are going to use 3 random variables
m <- 3

# Number of samples to be drawn
n <- 2000

# Covariance matrix
sigma <- matrix(c(1, 0.4, 0.2,
                  0.4, 1, -0.8,
                  0.2, -0.8, 1), 
                nrow=3)
# Random samples
z <- mvrnorm(n,mu=rep(0, m),Sigma=sigma,empirical=T)

# Pairplot library
library(psych)

# Correlation calculation
cor(z,method='spearman')

# Pairplot
pairs.panels(z)

# Transform samples into uniform data within [0,1]
u <- pnorm(z)
pairs.panels(u)

# 3D plot library
library(rgl)

# 3D plot
plot3d(u[,1],u[,2],u[,3],pch=20,col='navyblue')

# Apply the inverse of the selected marginals
x1 <- qgamma(u[,1],shape=2,scale=1)
x2 <- qbeta(u[,2],2,2)
x3 <- qt(u[,3],df=5)

#3D plot
plot3d(x1,x2,x3,pch=20,col='blue')

# Check correlation
df <- cbind(x1,x2,x3)
pairs.panels(df)
cor(df,meth='spearman')

# Shorten the steps using the copula package
library(copula)
set.seed(100)
myCop <- normalCopula(param=c(0.4,0.2,-0.8), dim = 3, dispstr = "un")
myMvd <- mvdc(copula=myCop, margins=c("gamma", "beta", "t"),
              paramMargins=list(list(shape=2, scale=1),
                                list(shape1=2, shape2=2), 
                                list(df=5)) )
                                
# Generate random samples
Z2 <- rmvdc(myMvd, 2000)
colnames(Z2) <- c("x1", "x2", "x3")
pairs.panels(Z2)

#------------------------------------------------------------------------------------------------------------------------------
# Example with real data

# Load data
cree <- read.csv('cree_r.csv',header=F)$V2
yahoo <- read.csv('yahoo_r.csv',header=F)$V2

# Plot and correlation checks
plot(cree,yahoo,pch='.')
abline(lm(yahoo~cree),col='red',lwd=1)
cor(cree,yahoo,method='spearman')

# Select the copula to be used
library(VineCopula)
u <- pobs(as.matrix(cbind(cree,yahoo)))[,1]
v <- pobs(as.matrix(cbind(cree,yahoo)))[,2]
selectedCopula <- BiCopSelect(u,v,familyset=NA)
selectedCopula

# Fit a t-copula
t.cop <- tCopula(dim=2)
set.seed(500)
m <- pobs(as.matrix(cbind(cree,yahoo)))
fit <- fitCopula(t.cop,m,method='ml')
coef(fit)

# Set the parameters
rho <- coef(fit)[1]
df <- coef(fit)[2]

# Plot the density
persp(tCopula(dim=2,rho,df=df),dCopula)

# Sample random observation from the copula
u <- rCopula(3965,tCopula(dim=2,rho,df=df))
plot(u[,1],u[,2],pch='.',col='blue')
cor(u,method='spearman')

# Estimate marginals parameters
cree_mu <- mean(cree)
cree_sd <- sd(cree)
yahoo_mu <- mean(yahoo)
yahoo_sd <- sd(yahoo)

# Plot the histograms of the marginals and the fitted lines
hist(cree,breaks=80,main='Cree returns',freq=F,density=30,col='cyan',ylim=c(0,20),xlim=c(-0.2,0.3))
lines(seq(-0.5,0.5,0.01),dnorm(seq(-0.5,0.5,0.01),cree_mu,cree_sd),col='red',lwd=2)
legend('topright',c('Fitted normal'),col=c('red'),lwd=2)
hist(yahoo,breaks=80,main='Yahoo returns',density=30,col='cyan',freq=F,ylim=c(0,20),xlim=c(-0.2,0.2))
lines(seq(-0.5,0.5,0.01),dnorm(seq(-0.5,0.5,0.01),yahoo_mu,yahoo_sd),col='red',lwd=2)
legend('topright',c('Fitted normal'),col=c('red'),lwd=2)

# Build the distribution from the copula
copula_dist <- mvdc(copula=tCopula(rho,dim=2,df=df), margins=c("norm","norm"),
                    paramMargins=list(list(mean=cree_mu, sd=cree_sd),
                                      list(mean=yahoo_mu, sd=yahoo_sd)))
# Sample from the distribution
sim <- rmvdc(copula_dist, 3965)

# Compare observed and simulated values
plot(cree,yahoo,main='Returns')
points(sim[,1],sim[,2],col='red')
legend('bottomright',c('Observed','Simulated'),col=c('black','red'),pch=21)

# Take a look at what would have happened for df=1 and df=8
set.seed(4258)
copula_dist <- mvdc(copula=tCopula(rho,dim=2,df=1), margins=c("norm","norm"),
                    paramMargins=list(list(mean=cree_mu, sd=cree_sd),
                                      list(mean=yahoo_mu, sd=yahoo_sd)))
sim <- rmvdc(copula_dist, 3965)
plot(cree,yahoo,main='Returns')
points(sim[,1],sim[,2],col='red')
legend('bottomright',c('Observed','Simulated df=1'),col=c('black','red'),pch=21)
cor(sim[,1],sim[,2],method='spearman')

copula_dist <- mvdc(copula=tCopula(rho,dim=2,df=8), margins=c("norm","norm"),
                    paramMargins=list(list(mean=cree_mu, sd=cree_sd),
                                      list(mean=yahoo_mu, sd=yahoo_sd)))
sim <- rmvdc(copula_dist, 3965)
plot(cree,yahoo,main='Returns')
points(sim[,1],sim[,2],col='red')
legend('bottomright',c('Observed','Simulated df=8'),col=c('black','red'),pch=21)
cor(sim[,1],sim[,2],method='spearman')

Could you please help me to write the algorithm for how to generate data from a copula model, for example Clayton copula

Could you please help me to write the algorithm for how to generate data from a copula model, for example, Clayton copula
I think you can refer to the code's line 118,131,140, which is the generate date from the copula model, I think all you need to do is substitute the t-copula into Clayton copula

Could you please help me to do the code of the dependence structure of copula vary with time (dynamic copula)?

code is slighlty outdated as of 2023 with copula 1.1.2:

Z2 <- rmvdc(myMvd, 2000)
Error: 'rmvdc' is defunct.
Use 'rMvdc' instead.

New code should apparently read:

Z2 <- rMvdc(2000, myMvd)

thanks!