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# This library is free software; you can redistribute it and/or
# modify it under the terms of the GNU Library General Public
# License as published by the Free Software Foundation; either
# version 2 of the License, or (at your option) any later version.
#
# This library is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Library General Public License for more details.
#
# You should have received a copy of the GNU Library General
# Public License along with this library; if not, write to the
# Free Foundation, Inc., 59 Temple Place, Suite 330, Boston,
# MA 02111-1307 USA
# Copyrights (C)
# for this R-port:
# 1999 - 2007, Diethelm Wuertz, GPL
# Diethelm Wuertz <wuertz@itp.phys.ethz.ch>
# info@rmetrics.org
# www.rmetrics.org
# for the code accessed (or partly included) from other R-ports:
# see R's copyright and license files
# for the code accessed (or partly included) from contributed R-ports
# and other sources
# see Rmetrics's copyright file
################################################################################
# TAIL DEPENDENCE ESTIMATOR
# derived from the "Mixed Gumbel-SurvivalGumbel-Normal Copula" approach
# supported Marginals: Normal, NIG and Generalized Hyperbolic Student-t
################################################################################
# FUNCTION: GUMBEL COPULA:
# .rgumbelCopula Generates fast Gumbel copula random variates
# .dgumbelCopula Computes Gumbel copula probability
# .pgumbelCopula Computes Gumbel copula probability
# FUNCTION: MIXED GUMBEL-SURVIVALGUMBEL-NORMAL COPULA:
# .rgsgnormCopula Generates G-SG-NORM copula random variates
# .dgsgnormCopula Computes G-SG-NORM copula probability
# .gsgnormCopulaFit Computes G-SG-NORM copula probability
# FUNCTION: NON-PARAMETRIC TAIL DEPENDECY ESTIMATOR:
# .cfgTDE Estimates non-parametrically tail dependence
# FUNCTION: COPULA FIT WITH NIG MARGINALS:
# .normDependencyFit Estimates tail dependence with normal marginals
# .nigDependencyFit Estimates tail dependence with NIG marginals
# .ghtDependencyFit Estimates tail dependence with GHT marginals
################################################################################
################################################################################
# Gumbel Copula
.rgumbelCopula =
function(n, alpha = 2)
{ # A function implemented by Diethelm Wuertz
# Description:
# Generates fast Gumbel copula random variates
# Arguments:
# FUNCTION:
# RVs:
dim = 2
theta <- runif(n, 0, pi)
w <- rexp(n)
b = 1/alpha
a <- sin((1-b)*theta)*(sin(b*theta))^(b/(1-b))/(sin(theta))^(1/(1-b))
fr = (a/w)^((1-b)/b)
fr <- matrix(fr, nrow = n, ncol = dim)
val <- matrix(runif(dim * n), nrow = n)
s = -log(val)/fr
ans = exp(-s^(1/alpha))
control = list(alpha = alpha, copula = "archm", type = "gumbel")
attr(ans, "control") <- unlist(control)
# Return Value:
ans
}
# ------------------------------------------------------------------------------
.dgumbelCopula =
function(u = 0.5, v = u, alpha = 2, output = c("vector", "list"))
{ # A function implemented by Diethelm Wuertz
# Description:
# Computes Bivariate Gumbel Copula Density
# FUNCTION:
# Conveniance Wrapper:
ans = darchmCopula(u, v, alpha, type = "4", output = output)
# Return Value:
ans
}
# ------------------------------------------------------------------------------
.pgumbelCopula =
function(u = 0.5, v = u, alpha = 2, output = c("vector", "list"))
{ # A function implemented by Diethelm Wuertz
# Description:
# Computes Bivariate Gumbel Copula Probability
# FUNCTION:
# Conveniance Wrapper:
ans = parchmCopula(u, v, alpha, type = "4", output = output)
# Return Value:
ans
}
################################################################################
# Mixed Gumbel-SurvivalGumbel-Normal Copula
.rgsgnormCopula =
function(n = 1000, alpha = c(2, 2), rho = 0, gamma = c(0.5, 0.5))
{ # A function implemented by Diethelm Wuertz
# Description:
# Computes RVs from a mixed GSG copula
# Example:
# .rgsgnormCopula(20)
# FUNCTION:
# Upper Gumbel = 1 , Lower Gumbel = 2, t = 3:
n1 = round(gamma[1]*n)
n2 = n - n1
n1 = round(gamma[2]*n1)
n2 = round(gamma[2]*n2)
n3 = n - n1 - n2
# Random Variates:
r = rbind(
if (n1 > 0) .rgumbelCopula(n1, alpha[1]),
if (n2 > 0) 1-.rgumbelCopula(n2, alpha[2]),
if (n3 > 0) rellipticalCopula(n3, rho, type = "norm") )
index = sample(1:n)
ans = r[index, ]
N = c(n, n1, n2, n3)
names(N) = c("n", "n1", "n2", "n3")
attr(ans, "control")<-N
# Return Value:
ans
}
# ------------------------------------------------------------------------------
.dgsgnormCopula =
function(u = 0.5, v = u, alpha = c(2, 2), rho = 0, gamma = c(0.5, 0.5))
{ # A function implemented by Diethelm Wuertz
# Description:
# Computes mixed GSG copula density
# Example:
# .dgsgnormCopula(grid2d()$x, grid2d()$y)
# FUNCTION:
# Settings:
if (is.list(u)) {
v = u[[2]]
u = u[[1]]
}
if (is.matrix(u)) {
v = u[, 1]
u = u[, 2]
}
# Mix Gumbel + Survival Gumbel:
dCopula1 = .dgumbelCopula(u, v, alpha[1], output = "list")
dCopula2 = .dgumbelCopula(1-u, 1-v, alpha[2], output = "list")
dCopula12 = dCopula1
dCopula12$z = gamma[1]*dCopula1$z + (1-gamma[1])*dCopula2$z
# Mix Gumbel/SurvivalGumbel + Student-t:
dCopula3 = dellipticalCopula(u, v, rho, type = "norm", output = "list")
dCopula123 = dCopula12
dCopula123$z = gamma[2]*dCopula12$z + (1-gamma[2])*dCopula3$z
ans = dCopula123
# Return Value:
ans
}
# ------------------------------------------------------------------------------
.gsgnormCopulaFit =
function(u, v = NULL, trace = FALSE)
{ # A function implemented by Diethelm Wuertz
# Description:
# Fits parameters for a mixed GSG copula
# FUNCTION:
# Settings:
U <- u
V <- v
if (is.list(u)) {
U = u[[1]]
V = u[[2]]
}
if (is.matrix(u)) {
U = u[, 1]
V = u[, 2]
}
# Estimate Copula:
start = c(1.5, 1.5, 0, 1/3, 1/3)
fun = function(x, U, V, trace)
{
density = .dgsgnormCopula(u = U, v = V,
alpha = x[1:2], rho = x[3], gamma = x[4:5])$z
density = density[!is.na(density)]
f = -mean( log(density) )
if (trace) {
cat("\n Objective Function Value: ", -f)
cat("\n Parameter Estimates: ",
round(c(x[1:3], x[5]*x[4], x[5]*(1-x[4]), 1-x[5]), 4), "\n")
}
f
}
# Fit:
fit = nlminb(start = start, objective = fun,
lower = c( 1, 1, -0.999, 0, 0),
upper = c(Inf, Inf, 0.999, 1, 1), U = U, V = V, trace = trace)
param = fit$par
alpha1 = param[1]
alpha2 = param[2]
gamma1 = param[4]
gamma2 = param[5]
Upper = (gamma2*gamma1*(2 - 2^(1/alpha1)))[[1]]
Lower = (gamma2*(1-gamma1)*(2 - 2^(1/alpha2)))[[1]]
Param = c(param[1:3], param[5]*param[4], param[5]*(1-param[4]), 1-param[5])
names(Param) = c("alpha1", "alpha2", "rho", "gumbel", "survival", "norm")
Lambda = c(lower = Lower, upper = Upper)
# Return Value:
list(param = Param, lambda = Lambda, fitted = fit$par)
}
################################################################################
# Non-Parametric Tail Dependence Estimator
.cfgTDE =
function(x, y)
{ # A function implemented by Diethelm Wuertz
# Description:
# Estimates non-parametrically tail dependency coefficient
# FUNCTION:
# Upper Tail:
lambda = NULL
n = length(x)
for(i in 1:n){
lambda = c(lambda,
log(sqrt(log(1/x[i])*log(1/y[i]))/log(1/max(x[i],y[i])^2)))
}
upper = 2-2*exp(sum(lambda/n))
# Lower Tail:
x = 1-x
y = 1-y
lambda = NULL
n = length(x)
for(i in 1:n){
lambda = c(lambda,
log(sqrt(log(1/x[i])*log(1/y[i]))/log(1/max(x[i],y[i])^2)))
}
lower = 2-2*exp(sum(lambda/n))
# Return Value:
c(lower = lower, upper = upper)
}
################################################################################
# GSGNORM Parametric Tail Dependence Estimator
.normDependencyFit =
function(x, doplot = TRUE, trace = TRUE)
{ # A function implemented by Diethelm Wuertz
# Description:
# Estimates tail dependency coefficients with Normal marginals
# Arguments:
# x - a multivariate 'timeSeries' object
# FUNCTION:
# Settings:
N = ncol(x)
lowerLambda = upperLambda = 0*diag(N)
assetsNames = colnames(x)
P = NULL
for (i in 1:(N-1)) {
# First asset:
r1 = as.vector(x[, i])
fit1 = nFit(r1)
estim1 = fit1$estimate
p1 = pnorm(r1, estim1[1], estim1[2])
Main1 = assetsNames[i]
P = cbind(P, p1)
for (j in (i+1):N)
{
# Second asset:
r2 = as.vector(x[, j])
fit2 = nFit(r2)
estim2 = fit2$estimate
p2 = pnorm(r2, estim2[1], estim2[2])
Main2 = assetsNames[j]
# Optional Plot:
if (doplot)
{
# Plot Distribution:
MainR = paste("Distribution:", Main1, "-", Main2)
plot(r1, r2, pch = 19, main = MainR)
grid()
# Plot Copula:
MainP = paste("Copula:", Main1, "-", Main2)
plot(p1, p2, pch = 19, main = MainP)
grid()
}
# Fit GSG copula parameters:
fit = .gsgnormCopulaFit(u = p1, v = p2, trace = FALSE)
if (trace)
cat(assetsNames[c(i,j)], round(fit$lambda, 3), "\n")
# Compose lambda Matrix:
lowerLambda[i, j] = lowerLambda[j, i] = fit$lambda[1]
upperLambda[i, j] = upperLambda[j, i] = fit$lambda[2]
}
}
# Result:
colnames(lowerLambda) = rownames(lowerLambda) = assetsNames
colnames(upperLambda) = rownames(upperLambda) = assetsNames
ans = list(lower = lowerLambda, upper = upperLambda)
# Return Value:
ans
}
# ------------------------------------------------------------------------------
.nigDependencyFit =
function(x, doplot = TRUE, trace = TRUE)
{ # A function implemented by Diethelm Wuertz
# Description:
# Estimates tail dependency coefficients with NIG marginals
# Arguments:
# x - a multivariate 'timeSeries' object
# FUNCTION:
# Settings:
N = ncol(x)
lowerLambda = upperLambda = 0*diag(N)
assetsNames = colnames(x)
P = NULL
for (i in 1:(N-1)) {
# First asset:
r1 = as.vector(x[, i])
fit1 = nigFit(r1, doplot = FALSE)
estim1 = fit1@fit$estimate
p1 = pnig(r1, estim1[1], estim1[2], estim1[3], estim1[4])
Main1 = assetsNames[i]
P = cbind(P, p1)
for (j in (i+1):N) {
# Second asset:
r2 = as.vector(x[, j])
fit2 = nigFit(r2, doplot = FALSE)
estim2 = fit2@fit$estimate
p2 = pnig(r2, estim2[1], estim2[2], estim2[3], estim2[4])
Main2 = assetsNames[j]
# Optional Plot:
if (doplot) {
MainR = paste("Distribution:", Main1, "-", Main2)
plot(r1, r2, pch = 19, main = MainR)
grid()
MainP = paste("Copula:", Main1, "-", Main2)
plot(p1, p2, pch = 19, main = MainP)
grid()
}
# Fit GSG copula parameters:
fit = .gsgnormCopulaFit(u = p1, v = p2, trace = FALSE)
if (trace)
cat(assetsNames[c(i,j)], round(fit$lambda, 3), "\n")
# Compose lambda Matrix:
lowerLambda[i, j] = lowerLambda[j, i] = fit$lambda[1]
upperLambda[i, j] = upperLambda[j, i] = fit$lambda[2]
}
}
# Result:
colnames(lowerLambda) = rownames(lowerLambda) = assetsNames
colnames(upperLambda) = rownames(upperLambda) = assetsNames
ans = list(lower = lowerLambda, upper = upperLambda)
# Return Value:
ans
}
# ------------------------------------------------------------------------------
.ghtDependencyFit =
function(x, doplot = TRUE, trace = TRUE)
{ # A function implemented by Diethelm Wuertz
# Description:
# Estimates tail dependency coefficients with GH Student-t marginals
# Arguments:
# x - a multivariate 'timeSeries' object
# FUNCTION:
# Settings:
N = ncol(x)
lowerLambda = upperLambda = 0*diag(N)
assetsNames = colnames(x)
P = NULL
for (i in 1:(N-1)) {
# First asset:
r1 = as.vector(x[, i])
fit1 = nigFit(r1, doplot = FALSE)
estim1 = fit1@fit$estimate
p1 = pnig(r1, estim1[1], estim1[2], estim1[3], estim1[4])
Main1 = assetsNames[i]
P = cbind(P, p1)
for (j in (i+1):N) {
# Second asset:
r2 = as.vector(x[, j])
fit2 = nigFit(r2, doplot = FALSE)
estim2 = fit2@fit$estimate
p2 = pnig(r2, estim2[1], estim2[2], estim2[3], estim2[4])
Main2 = assetsNames[j]
# Optional Plot:
if (doplot) {
MainR = paste("Distribution:", Main1, "-", Main2)
plot(r1, r2, pch = 19, main = MainR)
grid()
MainP = paste("Copula:", Main1, "-", Main2)
plot(p1, p2, pch = 19, main = MainP)
grid()
}
# Fit GSG copula parameters:
fit = .gsgnormCopulaFit(u = p1, v = p2, trace = FALSE)
if (trace)
cat(assetsNames[c(i,j)], round(fit$lambda, 3), "\n")
# Compose lambda Matrix:
lowerLambda[i, j] = lowerLambda[j, i] = fit$lambda[1]
upperLambda[i, j] = upperLambda[j, i] = fit$lambda[2]
}
}
# Result:
colnames(lowerLambda) = rownames(lowerLambda) = assetsNames
colnames(upperLambda) = rownames(upperLambda) = assetsNames
ans = list(lower = lowerLambda, upper = upperLambda)
# Return Value:
ans
}
################################################################################
# Examples
if (FALSE)
{
require(fCopulae)
# Simulated data:
x = .rnormCopula(1000, rho = 0.7)
.gsgnormCopulaFit(x, trace = TRUE)
# LPP Portfolio:
require(fPortfolio)
data(LPP2005REC)
x = 100 * as.timeSeries(LPP2005REC)[, 1:6]
head(x)
# Tail Dependency Estimation:
par(mfrow = c(2,2), cex = 0.7)
ans = .nigDependencyFit(x)
ans
# Lower Upper
# SBI SPI 0 0
# SBI SII 0.055 0
# SBI LMI 0.064 0.069
# SBI MPI 0 0
# SBI ALT 0 0
# SPI SII 0 0.064
# SPI LMI 0 0.072
# SPI MPI 0.352 0.214
# SPI ALT 0.273 0.048
# SII LMI 0.075 0
# SII MPI 0 0.164
# SII ALT 0 0.152
# LMI MPI 0 0
# LMI ALT 0 0
# MPI ALT 0.124 0.012
#
# $lower
# SBI SPI SII LMI MPI ALT
# SBI 0.00000000 0.0000000 0.05524575 0.06369211 0.0000000 0.0000000
# SPI 0.00000000 0.0000000 0.00000000 0.00000000 0.3517273 0.2728653
# SII 0.05524575 0.0000000 0.00000000 0.07541669 0.0000000 0.0000000
# LMI 0.06369211 0.0000000 0.07541669 0.00000000 0.0000000 0.0000000
# MPI 0.00000000 0.3517273 0.00000000 0.00000000 0.0000000 0.1236074
# ALT 0.00000000 0.2728653 0.00000000 0.00000000 0.1236074 0.0000000
#
# $upper
# SBI SPI SII LMI MPI ALT
# SBI 0.00000000 0.00000000 0.0000000 0.06935723 0.00000000 0.00000000
# SPI 0.00000000 0.00000000 0.0638653 0.07169038 0.21421052 0.04785965
# SII 0.00000000 0.06386530 0.0000000 0.00000000 0.16401986 0.15228270
# LMI 0.06935723 0.07169038 0.0000000 0.00000000 0.00000000 0.00000000
# MPI 0.00000000 0.21421052 0.1640199 0.00000000 0.00000000 0.01209013
# ALT 0.00000000 0.04785965 0.1522827 0.00000000 0.01209013 0.00000000
par(mfrow = c(1,1))
.assetsStarPlot(ans$lower, main = "Lower Tail Relations")
.assetsStarPlot(ans$upper, main = "Lower Tail Relations")
}
################################################################################
|
