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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
################################################################################
# FUNCTION: EMPIRICAL COPULAE PROBABILIY:
# pempiricalCopula Computes empirical copula probability
# FUNCTION: EMPIRICAL COPULAE DENSITY:
# dempiricalCopula Computes empirical copula density
# FUNCTION: DEBYE FUNCTION:
# .Debye Returns the value of the Debye function of order k
# .Debye1 Returns the value of the Debye function of order 1
# FUNCTION:
# .pmoCopula
# .dmoCopula
################################################################################
################################################################################
# FUNCTION: EMPIRICAL COPULAE PROBABILIY:
# pempiricalCopula Computes empirical copula probability
pempiricalCopula <-
function(u, v, N = 10)
{ # A function implemented by Diethelm Wuertz
# Description
# Computes the empirical copula probability
# Source:
# bouye02a.pdf
# FUNCTION:
# Settings:
if (is.list(u)) {
v = u[[2]]
u = u[[1]]
}
if (is.matrix(u)) {
v = u[, 1]
u = u[, 2]
}
# Probability:
p = q = (0:N)/N
h = matrix(rep(0, times = (N+1)^2), N+1)
for ( i in (0:N) ) {
for ( j in (0:N) ) {
z = Heaviside(u, p[i+1]) + Heaviside(v, q[j+1])
h[j+1, i+1] = length(z[z == 0])
}
}
h = h/length(u)
# Return Value:
list(x = p, y = q, z = h)
}
################################################################################
# FUNCTION: EMPIRICAL COPULAE DENSITY:
# dempiricalCopula Computes empirical copula density
dempiricalCopula <-
function(u, v, N = 10)
{
# A function implemented by Diethelm Wuertz
# Description
# Computes the empirical copula probability
# Source:
# bouye02a.pdf
# FUNCTION:
# Settings:
if (is.list(u)) {
v = u[[2]]
u = u[[1]]
}
if (is.matrix(u)) {
v = u[, 1]
u = u[, 2]
}
# Probability:
ans = pempiricalCopula(u, v, N)
X = ans$x
Y = ans$y
C = ans$z
# Density:
M = N+1
x = X[-1] - diff(X)/2
y = Y[-1] - diff(Y)/2
c = C[-1,-1]+C[-M,-M]-C[-1,-M]-C[-M,-1]
# Return Value:
list(x = x, y = y, z = c)
}
################################################################################
# FUNCTION: DEBYE FUNCTION:
# .Debye Returns the value of the Debye function of order k
# .Debye1
.Debye <-
function(x, k = 1)
{
# A function implemented by Diethelm Wuertz
# Description:
# Returns the value of the Debye function of order k
# Arguments:
# x - a numeric value or vector
# k - the order of the Debye function, a positive integer value
# FUNCTION:
# Check:
if (!is.integer(k) | k <= 0)
stop("k must be a positive integer")
# Loop:
D = NULL
error = NULL
for ( i in 1:length(x) ) {
nextD = .Debye1(x[i],k)
D = c(D, nextD[[1]])
error = c(error, nextD[[2]])
}
# Add error attribute:
attr(D, "error") = error
# Return Value:
D
}
# ------------------------------------------------------------------------------
.Debye1 <-
function(x, k = 1)
{
# A function implemented by Diethelm Wuertz
# FUNCTION:
# Function to be integrated:
d = function(x, lambda) { x^lambda / ( exp(x) - 1 ) }
# Integrate:
u = abs(x)
if (x == 0) {
D = 1
error = 0
} else {
ans = integrate(f = d, lower = 0, upper = u, lambda = k)
D = k * ans[[1]] / u^k
error = ans[[2]]
}
if (x < 0) {
D = D + k*u/(k+1)
}
# Return Value:
list(D = D, error = error)
}
################################################################################
# FUNCTION:
# .pmoCopula
# .dmoCopula
.pmoCopula <-
function(u = 0.5, v = u, alpha = NULL)
{
if (is.null(alpha)) alpha = c(0.5, 0.5)
alpha1 = alpha[1]
alpha2 = alpha[2]
U = u^(1-alpha1) * v
V = u * v^(1-alpha2)
UV = cbind(U,V)
apply(UV, 1, max)
}
# ------------------------------------------------------------------------------
.dmoCopula <-
function(u = 0.5, v = u, alpha = NULL)
{
if (is.null(alpha)) alpha = c(0.5, 0.5)
alpha1 = alpha[1]
alpha2 = alpha[2]
U = u^(1-alpha1) * v
V = u * v^(1-alpha2)
UV = cbind(U,V)
apply(UV, 1, max)
}
################################################################################
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