Nothing
R/dC-dc.R
In copula: Multivariate Dependence with Copulas
Defines functions
dlogcdthetaEllipCopula
dlogcdthetaExplicit
dlogcdthetaNumer
dlogcduTCopula
dlogcduNormalCopula
dlogcduExplicit
dlogcduNumer
dCdthetaEllipCopula
dCdthetaEvCopula
dCdthetaExplicit
dCdthetaNumer
plackettFormulaTCopula
plackettFormulaNormalCopula
plackettFormulaDim2TCopula
plackettFormulaDim2NormalCopula
dCduEllipCopula
dCduIndepCopula
dCduExplicit
dCduArchmCopula
dCduNumer
sides
gradControl
## Copyright (C) 2012 Marius Hofert, Ivan Kojadinovic, Martin Maechler, and Jun Yan
##
## This program is free software; you can redistribute it and/or modify it under
## the terms of the GNU General Public License as published by the Free Software
## Foundation; either version 3 of the License, or (at your option) any later
## version.
##
## This program 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 General Public License for more
## details.
##
## You should have received a copy of the GNU General Public License along with
## this program; if not, see <http://www.gnu.org/licenses/>.
##################################################################################
### Partial derivatives of the CDF wrt arguments
##################################################################################
##' @title List of control arguments for grad() / jacobian() in package numDeriv
##' @param eps see ?grad
##' @param d see ?grad
##' @param zero.tol see ?grad
##' @param r see ?grad
##' @param v see ?grad
##' @param show.details see ?grad
##' @return a list of the above
##' @author Ivan Kojadinovic
gradControl <- function(eps = 1e-4, d = 1e-4,
zero.tol = sqrt(.Machine$double.eps / 7e-7),
r = 6, v = 2, show.details = FALSE) {
list(eps = eps, d = d, zero.tol = zero.tol, r = r, v = v,
show.details = show.details)
}
##' @title Returns 'side' vector for grad() / jacobian() when computing
##' partial derivatives at x in (lb, ub) or [lb, ub] or ...
##' @param x point where partial derivatives will be computed
##' @param dimx length of x
##' @param d parameter from grad(); see ?grad
##' @param lb lower bound for x values
##' @param ub upper bound for x values
##' @return the 'side' vector
##' @author Ivan Kojadinovic
sides <- function(x, dimx, d, lb, ub) {
res <- rep(NA, dimx) # two-sided derivatives by default
res[x - lb < d] <- 1 # right derivative
res[ub - x < d] <- -1 # left derivative
res
}
##' @title Basic implementation of dCdu based on numerical differentiation
##' @param copula an object of class "copula"
##' @param u evaluation point in [0,1]^d or matrix of evaluation points
##' @param method.args to be passed to grad()
##' @param ...
##' @return partial derivatives of the copula wrt arguments at u
##' @author Ivan Kojadinovic
dCduNumer <- function(copula, u, method.args = gradControl(d = 0.1), may.warn=TRUE, ...) {
cl <- class(copula)[[1]]
if(may.warn && (is.null(W <- get0("warned.dCdu", .copulaEnv)) || is.na(match(cl, W)))) {
assign("warned.dCdu", if(is.null(W)) cl else c(W, cl), envir = .copulaEnv)
warning(gettextf(
"Function %s() not implemented for %s copulas; numerical differentiation used",
"dCdu", dQuote(cl)), domain=NA)
}
logC <- function(x) log(pCopula(x, copula))
dim <- dim(copula)
res <- t(apply(u, 1, function(u.)
grad(logC, u., side = sides(u., dim, method.args$d, 0, 1),
method.args = method.args))) * pCopula(u, copula)
res[res < 0] <- 0 # because 0 < dCdu < 1 uniformly
res[res > 1] <- 1
res
}
## For archmCopula objects {FIXME not yet used}
dCduArchmCopula <- function(copula, u, ...) {
## TODO: For ACs, the following is better than 'fixed formulas' => use it
## Example:
## require(copula)
## n <- 10
## d <- 4
## family <- "Gumbel"
## th <- 2
## u <- matrix(runif(n*d), ncol=d)
## cop <- onacopulaL(family, nacList=list(th, seq_len(d)))
## iPsi.u <- cop@copula@iPsi(u, theta=th)
stopifnot((d <- copula@dimension) >= 2)
th <- getTheta(copula, freeOnly=FALSE)# need full theta vector
iPsi.u <- iPsi(copula, u)
sumiPsi.u <- rowSums(iPsi.u)
cop <- getAcop(copula)
## j <- ceiling(d/2)
## NEED function absdPsi to make below work for any Archimedean copula in the same way that iPsi exists
sapply(seq_len(d), function(j)
exp(cop@copula@absdPsi(sumiPsi.u , theta=th, degree=1, log=TRUE) -
cop@copula@absdPsi(iPsi.u[,j], theta=th, degree=1, log=TRUE)))
}
## For copulas with explicit cdf
## Warning: This function assumes symmetry in u
dCduExplicit <- function(copula, u, ...) {
d <- dim(copula)
mat <- matrix(NA_real_, nrow(u), d)
algNm <- paste(class(copula)[1], "cdfDerWrtArg.algr", sep=".")
## FIXME: fails if class(copula) *extends* (but is not identical to) say "normalCopula"
if (exists(algNm)) {
alpha <- copula@parameters # typically used in 'eval(der.cdf.u, *)' below
der.cdf.u <- get(algNm)[d]
unames0 <- paste0("u",1:d)
for (j in 1:d) {
unames <- unames0; unames[1] <- unames0[j]; unames[j] <- unames0[1]
colnames(u) <- unames
mat[,j] <- eval(der.cdf.u, data.frame(u))
}
} else if (.hasSlot(copula, "exprdist") && is.language(cdf <- copula@exprdist$cdf)) {
## symbolic derivatives of explicit cdf expressions
params <- getTheta(copula, freeOnly = FALSE, named = TRUE)
colnames(u) <- unames <- paste0("u", 1:d)
u.df <- data.frame(u)
der <- deriv(cdf, unames)
mat <- attr(eval(der, c(u.df, params)), "gradient")
attr(mat, "dimnames") <- NULL
}
else warning("Function dCdu() not implemented for copulas of class '",
class(copula), "'")
mat
}
## This function is used for khoudrajiCopula objects
dCduIndepCopula <- function(copula, u, ...) {
dim <- copula@dimension
mat <- matrix(0, nrow(u), dim)
for (j in 1:dim) {
mat[,j] <- apply(u[, -j, drop=FALSE], 1, prod)
}
mat
}
setMethod("dCdu", signature("parCopula"), dCduNumer)
setMethod("dCdu", signature("joeCopula"), dCduExplicit)
setMethod("dCdu", signature("archmCopula"), dCduExplicit)
setMethod("dCdu", signature("plackettCopula"), dCduExplicit)
setMethod("dCdu", signature("evCopula"), dCduExplicit)
setMethod("dCdu", signature("gumbelCopula"), dCduExplicit)
setMethod("dCdu", signature("indepCopula"), dCduIndepCopula)
setMethod("dCdu", signature("khoudrajiExplicitCopula"), dCduExplicit)
setMethod("dCdu", signature("mixExplicitCopula"), dCduExplicit)
setMethod("dCdu", signature("rotExplicitCopula"), dCduExplicit)
## For ellipCopula objects
dCduEllipCopula <- function(copula, u, algorithm=NULL,
keepAttr=FALSE, checkCorr=FALSE, ...) {
dim <- copula@dimension
sigma <- getSigma(copula)
## quantile transformation
v <- switch(class(copula),
"normalCopula" = qnorm(u),
"tCopula" = {
df <- getdf(copula) # (needed further)
qt(u, df=df)
},
stop("not implemented for class ", class(copula)))
n <- nrow(u)
mat <- matrix(0,n,dim)
if(dim > 2) lower <- rep(-Inf, dim - 1L)
for (j in 1:dim) {
s <- sigma[-j,-j] - sigma[-j,j,drop=FALSE] %*% sigma[j,-j,drop=FALSE]
switch(class(copula),
"normalCopula" =
{
if (dim == 2) {
rho <- copula@parameters[1] # IK: single parameter only
mat[,j] <- pnorm(v[,-j], rho * v[,j], sqrt(1 - rho^2))
}
else
for (i in 1:n) {
x <- v[i,-j]
if(is.null(algorithm))
algorithm <- pmvnormAlgo(dim, x, checkCorr=checkCorr)
mat[i,j] <- pmvnorm(lower = lower, upper = x,
mean = v[i,j] * sigma[-j,j],
sigma = drop(s)
, algorithm=algorithm , keepAttr=keepAttr
)
}
},
"tCopula" =
{
if (dim == 2) {
rho <- copula@parameters[1] # IK: single parameter only
mat[,j] <- pt(sqrt((df+1)/(df+v[,j]^2)) / sqrt(1 - rho^2)
* (v[,-j] - rho * v[,j]), df=df+1)
}
else {
if(df != as.integer(df))
stop("'df' is not integer; therefore, dCdu() cannot be computed yet")
for (i in 1:n) {
x <- drop(sqrt((df+1)/(df+v[i,j]^2)) * (v[i,-j] - v[i,j] * sigma[-j,j]))
if(is.null(algorithm)) algorithm <- pmvtAlgo(dim, x)
mat[i,j] <- pmvt(lower = lower, upper = x,
sigma = s, df = df + 1
, algorithm=algorithm , keepAttr=keepAttr
)
}
}
})
}
mat
}
setMethod("dCdu", signature("ellipCopula"), dCduEllipCopula)
##################################################################################
### Plackett formula for elliptical copulas
##################################################################################
setGeneric("plackettFormulaDim2", function(copula, x) standardGeneric("plackettFormulaDim2"))
##' @title Derivative of the bivariate standard normal cdf wrt correlation
##' @param copula a bivariate normalCopula object
##' @param x evaluation point
##' @return the derivative at x
##' @author Ivan Kojadinovic
plackettFormulaDim2NormalCopula <- function(copula, x) {
rho <- copula@parameters[1] # JY: single parameter only
ir2 <- 1 - rho^2
as.matrix(exp(-(x[,1]^2 + x[,2]^2 - 2 * rho * x[,1] * x[,2]) /
(2 * ir2)) /
(2 * pi * sqrt(ir2)))
}
setMethod("plackettFormulaDim2", signature("normalCopula"), plackettFormulaDim2NormalCopula)
##' @title Derivative of the bivariate standard t cdf wrt correlation
##' @param copula a bivariate tCopula object
##' @param x evaluation point
##' @return the derivative at x
##' @author Ivan Kojadinovic
plackettFormulaDim2TCopula <- function(copula, x) {
rho <- copula@parameters[1] #JY: single parameter only
ir2 <- 1 - rho^2
df <- getdf(copula)
as.matrix((1 + (x[,1]^2 + x[,2]^2 - 2 * rho * x[,1] * x[,2]) /
(df * ir2))^(-df / 2) / (2 * pi * sqrt(ir2)))
}
setMethod("plackettFormulaDim2", signature("tCopula"), plackettFormulaDim2TCopula)
## JY: is rho a single parameter?
setGeneric("plackettFormula", function(copula, dim, rho, s, m, x, i, j) standardGeneric("plackettFormula"))
## For the multivariate standard normal cdf
plackettFormulaNormalCopula <- function(copula, dim, rho, s, m, x, i, j) {
exp(-(x[i]^2 + x[j]^2 - 2 * rho * x[i] * x[j]) /
(2 * (1 - rho^2))) / (2 * pi * sqrt(1 - rho^2)) *
(if (dim == 3) pnorm(drop((x[-c(i,j)] - m %*% x[c(i,j)])/sqrt(s)))
else pmvnorm(lower = rep(-Inf, dim - 2),
upper = drop(x[-c(i,j)] - m %*% x[c(i,j)]),
sigma = s))
}
setMethod("plackettFormula", signature("normalCopula"), plackettFormulaNormalCopula)
## For the multivariate standard t cdf
plackettFormulaTCopula <- function(copula, dim, rho, s, m, x, i, j) {
stopifnot(dim >= 3)
df <- getdf(copula)
if(df != as.integer(df) && dim > 3)
stop("'df' is not integer; therefore, plackettFormula() cannot be computed yet")
term <- 1 + (x[i]^2 + x[j]^2 - 2 * rho * x[i] * x[j]) / (df * (1 - rho^2))
## return:
term^(-df / 2) / (2 * pi * sqrt(1 - rho^2)) *
if (dim == 3) pt(drop((x[-c(i,j)] - m %*% x[c(i,j)]) / sqrt(term * s)), df= df)
else pmvt(df = df, lower = rep(-Inf, dim - 2),
upper = drop((x[-c(i,j)] - m %*% x[c(i,j)]) / sqrt(term)),
sigma = s)
}
setMethod("plackettFormula", signature("tCopula"), plackettFormulaTCopula)
##################################################################################
### Partial derivatives of the CDF wrt parameters
##################################################################################
##' @title Basic implementation of dCdtheta based on numerical differentiation
##' @param copula an object of class "copula"
##' @param u evaluation point in [0,1]^d or matrix of evaluation points
##' @param method.args to be passed to grad()
##' @param ...
##' @return n by p, partial derivatives of the copula wrt parameters at u
##' @author Ivan Kojadinovic
dCdthetaNumer <- function(copula, u, method.args = gradControl(d = 0.1), may.warn=TRUE, ...) {
cl <- class(copula)[[1]]
if(may.warn && (is.null(W <- get0("warned.dCdtheta", .copulaEnv)) || is.na(match(cl, W)))) {
assign("warned.dCdtheta", if(is.null(W)) cl else c(W, cl), envir = .copulaEnv)
warning(gettextf(
"Function %s() not implemented for %s copulas; numerical differentiation used",
"dCdtheta", dQuote(cl)), domain=NA)
}
logC <- function(theta) {
freeParam(copula) <- theta
log(pCopula(u, copula))
}
theta <- getTheta(copula, attr=TRUE)
p <- length(theta)
lb <- attr(theta, "param.lowbnd")
ub <- attr(theta, "param.upbnd" )
jacobian(logC, theta, side = sides(theta, p, method.args$d, lb, ub),
method.args = method.args) * pCopula(u, copula)
}
## For copulas with explicit cdf
dCdthetaExplicit <- function(copula, u, ...) {
d <- dim(copula)
algNm <- paste(class(copula)[1], "cdfDerWrtPar.algr", sep=".")
alpha <- getTheta(copula) # typically used in 'eval(*)' below
mat <- matrix(NA_real_, nrow(u), nParam(copula))
if (exists(algNm)) { ## JY: alpha is a scalar parameter
der.cdf.alpha <- get(algNm)[d]
colnames(u) <- paste0("u", 1:d)
mat <- as.matrix(eval(der.cdf.alpha, data.frame(u)))
} else if (.hasSlot(copula, "exprdist") && is.language(cdf <- copula@exprdist$cdf)) {
## symbolic derivatives of explicit cdf expressions
params <- getTheta(copula, freeOnly = FALSE, named = TRUE)
colnames(u) <- paste0("u", 1:d)
u.df <- data.frame(u)
der <- deriv(cdf, names(params)[isFree(copula)])
mat <- attr(eval(der, c(u.df, params)), "gradient")
attr(mat, "dimnames") <- NULL
} else {
warning(gettextf(
"Function %s() not implemented for copulas of class '%s'",
"dCdtheta", class(copula)), domain = NA)
## matrix(NA_real_, nrow(u), length(alpha)) # copula@parameters))
}
## JY: For some rotated explicit copula, dCdtheta returns NA:
## copula:::dCdtheta(khoudrajiCopula(indepCopula(), rotCopula(gumbelCopula(2)), shapes = c(.4, .95)), as.matrix(expand.grid(1:4/5, 1:4/5)))
## In this case, fall back on the numerical version
col.na <- apply(mat, 2, anyNA)
if (any(col.na)) {
warning("dCdtheta() returned NaN in column(s) ", paste(which(col.na), collapse=", "),
" for this explicit copula; falling back to numeric derivative for those columns")
## FIXME: computing for all columns, but need only 'col.na'
mat.num <- dCdthetaNumer(copula, u, may.warn=FALSE, ...)# warned above
mat[,col.na] <- mat.num[,col.na]
}
mat
}
## For evCopula objects
## JY: For single parameter only
dCdthetaEvCopula <- function(copula, u, ...) {
loguv <- log(u[,1], u[,2])
w <- log(u[,2]) / loguv
## return der.cdf.alpha
## JY: single parameter only
as.matrix(pCopula(u, copula) * loguv * dAdtheta(copula, w))
}
setMethod("dCdtheta", signature("parCopula"), dCdthetaNumer)
setMethod("dCdtheta", signature("joeCopula"), dCdthetaExplicit)
setMethod("dCdtheta", signature("tevCopula"), dCdthetaNumer)
setMethod("dCdtheta", signature("archmCopula"), dCdthetaExplicit)
setMethod("dCdtheta", signature("plackettCopula"), dCdthetaExplicit)
setMethod("dCdtheta", signature("evCopula"), dCdthetaExplicit)
setMethod("dCdtheta", signature("gumbelCopula"), dCdthetaExplicit)
setMethod("dCdtheta", signature("khoudrajiExplicitCopula"), dCdthetaExplicit)
setMethod("dCdtheta", signature("mixExplicitCopula"), dCdthetaExplicit)
setMethod("dCdtheta", signature("rotExplicitCopula"), dCdthetaExplicit)
## For ellipCopula objects
dCdthetaEllipCopula <- function(copula, u, ...) {
dim <- copula@dimension
## quantile transformation
v <- switch(class(copula),
"normalCopula" = qnorm(u),
"tCopula" = qt(u, df = getdf(copula)),
stop("not implemented for class ", class(copula)))
val <-
if (dim == 2)
plackettFormulaDim2(copula, v)
else {
n <- nrow(u)
sigma <- getSigma(copula)
if (copula@dispstr %in% c("ex","ar1")) { ## exchangeable or ar1
rho <- copula@parameters[1] # IK: single parameter only
r <- matrix(c(1,-rho,-rho,1),2,2)/(1 - rho^2)
m <- sigma[-c(1,2),c(1,2)] %*% r
s <- sigma[-c(1,2),-c(1,2)] -
sigma[-c(1,2),c(1,2)] %*% r %*% sigma[c(1,2),-c(1,2)]
mat <- matrix(0,n,1)
if (copula@dispstr == "ex") ## exchangeable
for (k in 1:n)
for (j in 1:(dim-1))
for (i in (j+1):dim)
mat[k,1] <- mat[k,1] +
plackettFormula(copula, dim, rho, s, m, v[k,], i, j)
else ## ar1
for (k in 1:n)
for (j in 1:(dim-1))
for (i in (j+1):dim)
mat[k,1] <- mat[k,1] + (i - j) * rho ^ (i - j - 1) *
plackettFormula(copula, dim, rho, s, m, v[k,], i, j)
mat
}
else { # unstructured or toeplitz or ...
mat <- matrix(0,n,dim*(dim-1)/2)
l <- 1
for (j in 1:(dim-1))
for (i in (j+1):dim)
{
rho <- sigma[i,j]
r <- matrix(c(1,-rho,-rho,1),2,2)/(1 - rho^2)
m <- sigma[-c(i,j),c(i,j)] %*% r
s <- sigma[-c(i,j),-c(i,j)] - m %*% sigma[c(i,j),-c(i,j)]
for (k in 1:n)
mat[k,l] <- plackettFormula(copula, dim, rho, s, m, v[k,], i, j)
l <- l + 1
}
if (copula@dispstr == "un") ## unstructured
mat
else if (copula@dispstr == "toep") {
coef <- matrix(0, dim*(dim-1)/2, dim-1)
for (k in 1:(dim-1)) {
m <- row(sigma) == col(sigma) + k
coef[,k] <- P2p(m)
}
mat %*% coef
}
else stop("Not implemented yet for the dispersion structure ",
copula@dispstr)
}
}
free <- isFreeP(copula@parameters)
if (.hasSlot(copula, "df.fixed")) free <- free[-length(free)]
val[, free, drop = FALSE]
}
setMethod("dCdtheta", signature("ellipCopula"), dCdthetaEllipCopula)
##################################################################################
### Partial derivatives of the log PDF wrt arguments
##################################################################################
##' @title Basic implementation of dlogcdu based on numerical differentiation
##' @param copula an object of class "copula"
##' @param u evaluation point in [0,1]^d or matrix of evaluation points
##' @param method.args to be passed to grad()
##' @param ...
##' @return partial derivatives of the log copula density wrt arguments at u
##' @author Ivan Kojadinovic
dlogcduNumer <- function(copula, u, method.args = gradControl(d=1e-5), may.warn = TRUE, ...) {
cl <- class(copula)[[1]]
if(may.warn && (is.null(W <- get0("warned.dlogcdu", .copulaEnv)) || is.na(match(cl, W)))) {
assign("warned.dlogcdu", if(is.null(W)) cl else c(W, cl), envir = .copulaEnv)
warning(gettextf(
"Function %s() not implemented for %s copulas; numerical differentiation used",
"dlogcdu", dQuote(cl)), domain=NA)
}
logc <- function(x) dCopula(x, copula, log = TRUE)
dim <- dim(copula)
t(apply(u, 1, function(u.)
grad(logc, u., side = sides(u., dim, method.args$d, 0, 1),
method.args = method.args)))
}
## For copulas with explicit densities
dlogcduExplicit <- function(copula, u, ...) {
d <- dim(copula)
algNm <- paste(class(copula)[1], "pdfDerWrtArg.algr", sep=".")
mat <- matrix(NA_real_, nrow(u), d)
if(exists(algNm)) {
der.pdf.u <- get(algNm)[d]
alpha <- copula@parameters # typically used in eval() # JY: scalar alpha
unames0 <- paste0("u", 1:d)
for (j in 1:d) {
unames <- unames0; unames[1] <- unames0[j]; unames[j] <- unames0[1]
colnames(u) <- unames
mat[,j] <- eval(der.pdf.u, data.frame(u))
}
} else if (.hasSlot(copula, "exprdist") && is.language(pdf <- copula@exprdist$pdf)) {
## symbolic derivatives of explicit pdf expressions
params <- getTheta(copula, freeOnly = FALSE, named = TRUE)
colnames(u) <- unames <- paste0("u", 1:d)
u.df <- data.frame(u)
der <- deriv(pdf, unames)
mat <- attr(eval(der, c(u.df, params)), "gradient")
attr(mat, "dimnames") <- NULL
} else warning(gettextf(
"Function %s() not implemented for copulas of class '%s'",
"dlogcdu", class(copula)), domain=NA)
mat / dCopula(u, copula)
}
setMethod("dlogcdu", signature("parCopula"), dlogcduNumer)
setMethod("dlogcdu", signature("joeCopula"), dlogcduExplicit)
setMethod("dlogcdu", signature("tevCopula"), dlogcduNumer)
setMethod("dlogcdu", signature("archmCopula"), dlogcduExplicit)
setMethod("dlogcdu", signature("evCopula"), dlogcduExplicit)
setMethod("dlogcdu", signature("plackettCopula"), dlogcduExplicit)
setMethod("dlogcdu", signature("khoudrajiExplicitCopula"), dlogcduExplicit)
setMethod("dlogcdu", signature("mixExplicitCopula"), dlogcduExplicit)
setMethod("dlogcdu", signature("rotExplicitCopula"), dlogcduExplicit)
## For normalCopula objects
dlogcduNormalCopula <- function(copula, u, ...) {
v <- qnorm(u)
(- v %*% solve(getSigma(copula)) + v) / dnorm(v)
}
setMethod("dlogcdu", signature("normalCopula"), dlogcduNormalCopula)
## For tCopula objects
dlogcduTCopula <- function(copula, u, ...) {
df <- getdf(copula)
v <- qt(u,df=df)
w <- dt(v,df=df)
m <- v %*% solve(getSigma(copula))
- (df + copula@dimension) * m / ((df + rowSums(m * v)) * w) +
(df + 1) * v / ((df + v^2) * w)
}
setMethod("dlogcdu", signature("tCopula"), dlogcduTCopula)
##################################################################################
### Partial derivatives of the log PDF wrt parameters
##################################################################################
##' @title Basic implementation of dlogcdu based on numerical differentiation
##' @param copula an object of class "copula"
##' @param u evaluation point in [0,1]^d or matrix of evaluation points
##' @param method.args to be passed to grad()
##' @param ...
##' @return partial derivatives of the log copula density wrt parameters at u
##' @author Ivan Kojadinovic
dlogcdthetaNumer <- function(copula, u, method.args = gradControl(d=1e-5), may.warn = TRUE, ...) {
cl <- class(copula)[[1]]
if(may.warn &&
(is.null(W <- get0("warned.dlogcdtheta", .copulaEnv)) || is.na(match(cl, W)))) {
assign("warned.dlogcdtheta", if(is.null(W)) cl else c(W, cl), envir = .copulaEnv)
warning(gettextf(
"Function %s() not implemented for %s copulas; numerical differentiation used",
"dlogcdtheta", dQuote(cl)), domain=NA)
}
logc <- function(theta) {
freeParam(copula) <- theta
dCopula(u, copula, log = TRUE)
}
theta <- getTheta(copula, attr = TRUE)
p <- length(theta)
lb <- attr(theta, "param.lowbnd")
ub <- attr(theta, "param.upbnd" )
jacobian(logc, theta, side = sides(theta, p, method.args$d, lb, ub),
method.args = method.args)
}
## For copulas with explicit densities
dlogcdthetaExplicit <- function(copula, u, ...) {
d <- dim(copula)
algNm <- paste(class(copula)[1], "pdfDerWrtPar.algr", sep=".")
if(exists(algNm) && !is.null((der.pdf.alpha <- get(algNm)[d])[[1]])) {
alpha <- copula@parameters # typically used in val(.)
colnames(u) <- paste0("u", 1:d)
mat <- as.matrix(eval(der.pdf.alpha, data.frame(u))) / dCopula(u, copula)
} else if (.hasSlot(copula, "exprdist") && is.language(pdf <- copula@exprdist$pdf)) {
## symbolic derivatives of explicit pdf expressions
params <- getTheta(copula, freeOnly = FALSE, named = TRUE)
colnames(u) <- paste0("u", 1:d)
u.df <- data.frame(u)
der <- deriv(pdf, names(params)[isFree(copula)])
mat <- attr(eval(der, c(u.df, params)), "gradient")
attr(mat, "dimnames") <- NULL
mat <- mat / dCopula(u, copula)
} else {
warning(gettextf(
"Function %s() not implemented for copulas of class '%s'",
"dlogcdtheta", class(copula)), domain = NA)
mat <- matrix(NA_real_, nrow(u), nParam(copula))
}
## JY: For some rotated explicit copula, dCdtheta returns NA:
## copula:::dlogcdtheta(khoudrajiCopula(indepCopula(), rotCopula(gumbelCopula(2)), shapes = c(.4, .95)), as.matrix(expand.grid(1:4/5, 1:4/5)))
## In this case, fall back on the numerical version
col.na <- apply(mat, 2, anyNA)
if (any(col.na)) {
warning("dlogcdtheta() returned NaN in column(s) ", paste(which(col.na), collapse=", "),
" for this explicit copula; falling back to numeric derivative for those columns")
## FIXME: computing for all columns, but need only 'col.na'
mat.num <- dlogcdthetaNumer(copula, u, may.warn=FALSE, ...)# warned above
mat[,col.na] <- mat.num[,col.na]
}
mat
}
setMethod("dlogcdtheta", signature("parCopula"), dlogcdthetaNumer)
setMethod("dlogcdtheta", signature("joeCopula"), dlogcdthetaExplicit)
setMethod("dlogcdtheta", signature("tevCopula"), dlogcdthetaNumer)
setMethod("dlogcdtheta", signature("archmCopula"), dlogcdthetaExplicit)
setMethod("dlogcdtheta", signature("plackettCopula"), dlogcdthetaExplicit)
setMethod("dlogcdtheta", signature("evCopula"), dlogcdthetaExplicit)
setMethod("dlogcdtheta", signature("khoudrajiExplicitCopula"), dlogcdthetaExplicit)
setMethod("dlogcdtheta", signature("mixExplicitCopula"), dlogcdthetaExplicit)
setMethod("dlogcdtheta", signature("rotExplicitCopula"), dlogcdthetaExplicit)
## For ellipCopula objects
dlogcdthetaEllipCopula <- function(copula, u, ...) {
dim <- copula@dimension
## quantile transformation
v <- switch(clc <- class(copula), ## FIXME: fails if copula *extends* "tCopula" (e.g.)
"normalCopula" = qnorm(u),
"tCopula" = {
df <- getdf(copula)
qt(u, df=df)
},
## else:
stop("class ", sQuote(clc), " not implemented"))
val <-
if (dim == 2) {
rho <- copula@parameters[1] # JY: single rho parameter, free
ir2 <- 1 - rho^2
sv2 <- rowSums(v^2) # == v[,1]^2 + v[,2]^2
as.matrix(switch(clc,
"normalCopula" =
(rho * ir2 - rho * sv2 + (rho^2 + 1) * v[,1] * v[,2])/ir2^2,
"tCopula" =
(1 + df) * rho / -ir2 + (2 + df) * (df * rho + v[,1] * v[,2])
/ (df * ir2 + sv2 - 2 * rho * v[,1] * v[,2])))
} else { ## dim >= 3
n <- nrow(u)
sigma <- getSigma(copula)
detsig <- det(sigma)
invsig <- solve(sigma)
if (copula@dispstr %in% c("ex","ar1")) { ## exchangeable or ar1
rho <- copula@parameters[1] # JY: single rho parameter
dersig <- matrix(1, dim, dim)
if (copula@dispstr == "ex") ## ex
diag(dersig) <- 0
else ## ar1
for (i in 1:dim)
for (j in 1:dim) {
ij <- abs(i - j)
dersig[i,j] <- ij * rho^(ij - 1)
}
## MM: sum(diag(A %*% B)) == sum(A * t(B)) .. and B=dersig is symmetric here
## derdetsig <- detsig * sum(diag(invsig %*% dersig))
derdetsig <- detsig * sum(invsig * dersig)
derinvsig <- - invsig %*% dersig %*% invsig
firstterm <- derdetsig/detsig
mat <-
switch(clc,
"normalCopula" =
- (firstterm + rowSums((v %*% derinvsig) * v))/2,
"tCopula" =
- (firstterm + (df + dim) * rowSums((v %*% derinvsig) * v)
/ (df + rowSums((v %*% invsig) * v)) ) / 2)
as.matrix(mat)
}
else {# unstructured or toeplitz or ...
mat <- matrix(0, n, dim*(dim - 1)/2)
l <- 1
for (j in 1:(dim-1))
for (i in (j+1):dim) {
derdetsig <- 2 * det(sigma[-i,-j,drop=FALSE]) * (-1)^(i+j)
derinvsig <- - invsig[,i] %*% t(invsig[,j]) - invsig[,j] %*% t(invsig[,i])
firstterm <- derdetsig/detsig
mat[,l] <-
switch(clc,
"normalCopula" =
- (firstterm + rowSums((v %*% derinvsig) * v))/2,
"tCopula" =
- (firstterm + (df + dim) * rowSums((v %*% derinvsig) * v)
/ (df + rowSums((v %*% invsig) * v)) ) / 2)
l <- l + 1
}
if (copula@dispstr == "un") ## unstructured
mat
else if (copula@dispstr == "toep") { ## toeplitz: dim-1 parameters
coef <- matrix(0, dim*(dim-1)/2, dim-1)
for (k in 1:(dim-1)) {
m <- row(sigma) == col(sigma) + k
coef[,k] <- P2p(m)
}
mat %*% coef
}
else stop("Not implemented yet for the dispersion structure ", copula@dispstr)
}
} ## dim >= 3
free <- isFreeP(copula@parameters)
if (.hasSlot(copula, "df.fixed")) free <- free[-length(free)]
val[, free, drop = FALSE]
}
setMethod("dlogcdtheta", signature("ellipCopula"), dlogcdthetaEllipCopula)
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Defines functions dlogcdthetaEllipCopula dlogcdthetaExplicit dlogcdthetaNumer dlogcduTCopula dlogcduNormalCopula dlogcduExplicit dlogcduNumer dCdthetaEllipCopula dCdthetaEvCopula dCdthetaExplicit dCdthetaNumer plackettFormulaTCopula plackettFormulaNormalCopula plackettFormulaDim2TCopula plackettFormulaDim2NormalCopula dCduEllipCopula dCduIndepCopula dCduExplicit dCduArchmCopula dCduNumer sides gradControl
## Copyright (C) 2012 Marius Hofert, Ivan Kojadinovic, Martin Maechler, and Jun Yan
##
## This program is free software; you can redistribute it and/or modify it under
## the terms of the GNU General Public License as published by the Free Software
## Foundation; either version 3 of the License, or (at your option) any later
## version.
##
## This program 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 General Public License for more
## details.
##
## You should have received a copy of the GNU General Public License along with
## this program; if not, see <http://www.gnu.org/licenses/>.
##################################################################################
### Partial derivatives of the CDF wrt arguments
##################################################################################
##' @title List of control arguments for grad() / jacobian() in package numDeriv
##' @param eps see ?grad
##' @param d see ?grad
##' @param zero.tol see ?grad
##' @param r see ?grad
##' @param v see ?grad
##' @param show.details see ?grad
##' @return a list of the above
##' @author Ivan Kojadinovic
gradControl <- function(eps = 1e-4, d = 1e-4,
zero.tol = sqrt(.Machine$double.eps / 7e-7),
r = 6, v = 2, show.details = FALSE) {
list(eps = eps, d = d, zero.tol = zero.tol, r = r, v = v,
show.details = show.details)
}
##' @title Returns 'side' vector for grad() / jacobian() when computing
##' partial derivatives at x in (lb, ub) or [lb, ub] or ...
##' @param x point where partial derivatives will be computed
##' @param dimx length of x
##' @param d parameter from grad(); see ?grad
##' @param lb lower bound for x values
##' @param ub upper bound for x values
##' @return the 'side' vector
##' @author Ivan Kojadinovic
sides <- function(x, dimx, d, lb, ub) {
res <- rep(NA, dimx) # two-sided derivatives by default
res[x - lb < d] <- 1 # right derivative
res[ub - x < d] <- -1 # left derivative
res
}
##' @title Basic implementation of dCdu based on numerical differentiation
##' @param copula an object of class "copula"
##' @param u evaluation point in [0,1]^d or matrix of evaluation points
##' @param method.args to be passed to grad()
##' @param ...
##' @return partial derivatives of the copula wrt arguments at u
##' @author Ivan Kojadinovic
dCduNumer <- function(copula, u, method.args = gradControl(d = 0.1), may.warn=TRUE, ...) {
cl <- class(copula)[[1]]
if(may.warn && (is.null(W <- get0("warned.dCdu", .copulaEnv)) || is.na(match(cl, W)))) {
assign("warned.dCdu", if(is.null(W)) cl else c(W, cl), envir = .copulaEnv)
warning(gettextf(
"Function %s() not implemented for %s copulas; numerical differentiation used",
"dCdu", dQuote(cl)), domain=NA)
}
logC <- function(x) log(pCopula(x, copula))
dim <- dim(copula)
res <- t(apply(u, 1, function(u.)
grad(logC, u., side = sides(u., dim, method.args$d, 0, 1),
method.args = method.args))) * pCopula(u, copula)
res[res < 0] <- 0 # because 0 < dCdu < 1 uniformly
res[res > 1] <- 1
res
}
## For archmCopula objects {FIXME not yet used}
dCduArchmCopula <- function(copula, u, ...) {
## TODO: For ACs, the following is better than 'fixed formulas' => use it
## Example:
## require(copula)
## n <- 10
## d <- 4
## family <- "Gumbel"
## th <- 2
## u <- matrix(runif(n*d), ncol=d)
## cop <- onacopulaL(family, nacList=list(th, seq_len(d)))
## iPsi.u <- cop@copula@iPsi(u, theta=th)
stopifnot((d <- copula@dimension) >= 2)
th <- getTheta(copula, freeOnly=FALSE)# need full theta vector
iPsi.u <- iPsi(copula, u)
sumiPsi.u <- rowSums(iPsi.u)
cop <- getAcop(copula)
## j <- ceiling(d/2)
## NEED function absdPsi to make below work for any Archimedean copula in the same way that iPsi exists
sapply(seq_len(d), function(j)
exp(cop@copula@absdPsi(sumiPsi.u , theta=th, degree=1, log=TRUE) -
cop@copula@absdPsi(iPsi.u[,j], theta=th, degree=1, log=TRUE)))
}
## For copulas with explicit cdf
## Warning: This function assumes symmetry in u
dCduExplicit <- function(copula, u, ...) {
d <- dim(copula)
mat <- matrix(NA_real_, nrow(u), d)
algNm <- paste(class(copula)[1], "cdfDerWrtArg.algr", sep=".")
## FIXME: fails if class(copula) *extends* (but is not identical to) say "normalCopula"
if (exists(algNm)) {
alpha <- copula@parameters # typically used in 'eval(der.cdf.u, *)' below
der.cdf.u <- get(algNm)[d]
unames0 <- paste0("u",1:d)
for (j in 1:d) {
unames <- unames0; unames[1] <- unames0[j]; unames[j] <- unames0[1]
colnames(u) <- unames
mat[,j] <- eval(der.cdf.u, data.frame(u))
}
} else if (.hasSlot(copula, "exprdist") && is.language(cdf <- copula@exprdist$cdf)) {
## symbolic derivatives of explicit cdf expressions
params <- getTheta(copula, freeOnly = FALSE, named = TRUE)
colnames(u) <- unames <- paste0("u", 1:d)
u.df <- data.frame(u)
der <- deriv(cdf, unames)
mat <- attr(eval(der, c(u.df, params)), "gradient")
attr(mat, "dimnames") <- NULL
}
else warning("Function dCdu() not implemented for copulas of class '",
class(copula), "'")
mat
}
## This function is used for khoudrajiCopula objects
dCduIndepCopula <- function(copula, u, ...) {
dim <- copula@dimension
mat <- matrix(0, nrow(u), dim)
for (j in 1:dim) {
mat[,j] <- apply(u[, -j, drop=FALSE], 1, prod)
}
mat
}
setMethod("dCdu", signature("parCopula"), dCduNumer)
setMethod("dCdu", signature("joeCopula"), dCduExplicit)
setMethod("dCdu", signature("archmCopula"), dCduExplicit)
setMethod("dCdu", signature("plackettCopula"), dCduExplicit)
setMethod("dCdu", signature("evCopula"), dCduExplicit)
setMethod("dCdu", signature("gumbelCopula"), dCduExplicit)
setMethod("dCdu", signature("indepCopula"), dCduIndepCopula)
setMethod("dCdu", signature("khoudrajiExplicitCopula"), dCduExplicit)
setMethod("dCdu", signature("mixExplicitCopula"), dCduExplicit)
setMethod("dCdu", signature("rotExplicitCopula"), dCduExplicit)
## For ellipCopula objects
dCduEllipCopula <- function(copula, u, algorithm=NULL,
keepAttr=FALSE, checkCorr=FALSE, ...) {
dim <- copula@dimension
sigma <- getSigma(copula)
## quantile transformation
v <- switch(class(copula),
"normalCopula" = qnorm(u),
"tCopula" = {
df <- getdf(copula) # (needed further)
qt(u, df=df)
},
stop("not implemented for class ", class(copula)))
n <- nrow(u)
mat <- matrix(0,n,dim)
if(dim > 2) lower <- rep(-Inf, dim - 1L)
for (j in 1:dim) {
s <- sigma[-j,-j] - sigma[-j,j,drop=FALSE] %*% sigma[j,-j,drop=FALSE]
switch(class(copula),
"normalCopula" =
{
if (dim == 2) {
rho <- copula@parameters[1] # IK: single parameter only
mat[,j] <- pnorm(v[,-j], rho * v[,j], sqrt(1 - rho^2))
}
else
for (i in 1:n) {
x <- v[i,-j]
if(is.null(algorithm))
algorithm <- pmvnormAlgo(dim, x, checkCorr=checkCorr)
mat[i,j] <- pmvnorm(lower = lower, upper = x,
mean = v[i,j] * sigma[-j,j],
sigma = drop(s)
, algorithm=algorithm , keepAttr=keepAttr
)
}
},
"tCopula" =
{
if (dim == 2) {
rho <- copula@parameters[1] # IK: single parameter only
mat[,j] <- pt(sqrt((df+1)/(df+v[,j]^2)) / sqrt(1 - rho^2)
* (v[,-j] - rho * v[,j]), df=df+1)
}
else {
if(df != as.integer(df))
stop("'df' is not integer; therefore, dCdu() cannot be computed yet")
for (i in 1:n) {
x <- drop(sqrt((df+1)/(df+v[i,j]^2)) * (v[i,-j] - v[i,j] * sigma[-j,j]))
if(is.null(algorithm)) algorithm <- pmvtAlgo(dim, x)
mat[i,j] <- pmvt(lower = lower, upper = x,
sigma = s, df = df + 1
, algorithm=algorithm , keepAttr=keepAttr
)
}
}
})
}
mat
}
setMethod("dCdu", signature("ellipCopula"), dCduEllipCopula)
##################################################################################
### Plackett formula for elliptical copulas
##################################################################################
setGeneric("plackettFormulaDim2", function(copula, x) standardGeneric("plackettFormulaDim2"))
##' @title Derivative of the bivariate standard normal cdf wrt correlation
##' @param copula a bivariate normalCopula object
##' @param x evaluation point
##' @return the derivative at x
##' @author Ivan Kojadinovic
plackettFormulaDim2NormalCopula <- function(copula, x) {
rho <- copula@parameters[1] # JY: single parameter only
ir2 <- 1 - rho^2
as.matrix(exp(-(x[,1]^2 + x[,2]^2 - 2 * rho * x[,1] * x[,2]) /
(2 * ir2)) /
(2 * pi * sqrt(ir2)))
}
setMethod("plackettFormulaDim2", signature("normalCopula"), plackettFormulaDim2NormalCopula)
##' @title Derivative of the bivariate standard t cdf wrt correlation
##' @param copula a bivariate tCopula object
##' @param x evaluation point
##' @return the derivative at x
##' @author Ivan Kojadinovic
plackettFormulaDim2TCopula <- function(copula, x) {
rho <- copula@parameters[1] #JY: single parameter only
ir2 <- 1 - rho^2
df <- getdf(copula)
as.matrix((1 + (x[,1]^2 + x[,2]^2 - 2 * rho * x[,1] * x[,2]) /
(df * ir2))^(-df / 2) / (2 * pi * sqrt(ir2)))
}
setMethod("plackettFormulaDim2", signature("tCopula"), plackettFormulaDim2TCopula)
## JY: is rho a single parameter?
setGeneric("plackettFormula", function(copula, dim, rho, s, m, x, i, j) standardGeneric("plackettFormula"))
## For the multivariate standard normal cdf
plackettFormulaNormalCopula <- function(copula, dim, rho, s, m, x, i, j) {
exp(-(x[i]^2 + x[j]^2 - 2 * rho * x[i] * x[j]) /
(2 * (1 - rho^2))) / (2 * pi * sqrt(1 - rho^2)) *
(if (dim == 3) pnorm(drop((x[-c(i,j)] - m %*% x[c(i,j)])/sqrt(s)))
else pmvnorm(lower = rep(-Inf, dim - 2),
upper = drop(x[-c(i,j)] - m %*% x[c(i,j)]),
sigma = s))
}
setMethod("plackettFormula", signature("normalCopula"), plackettFormulaNormalCopula)
## For the multivariate standard t cdf
plackettFormulaTCopula <- function(copula, dim, rho, s, m, x, i, j) {
stopifnot(dim >= 3)
df <- getdf(copula)
if(df != as.integer(df) && dim > 3)
stop("'df' is not integer; therefore, plackettFormula() cannot be computed yet")
term <- 1 + (x[i]^2 + x[j]^2 - 2 * rho * x[i] * x[j]) / (df * (1 - rho^2))
## return:
term^(-df / 2) / (2 * pi * sqrt(1 - rho^2)) *
if (dim == 3) pt(drop((x[-c(i,j)] - m %*% x[c(i,j)]) / sqrt(term * s)), df= df)
else pmvt(df = df, lower = rep(-Inf, dim - 2),
upper = drop((x[-c(i,j)] - m %*% x[c(i,j)]) / sqrt(term)),
sigma = s)
}
setMethod("plackettFormula", signature("tCopula"), plackettFormulaTCopula)
##################################################################################
### Partial derivatives of the CDF wrt parameters
##################################################################################
##' @title Basic implementation of dCdtheta based on numerical differentiation
##' @param copula an object of class "copula"
##' @param u evaluation point in [0,1]^d or matrix of evaluation points
##' @param method.args to be passed to grad()
##' @param ...
##' @return n by p, partial derivatives of the copula wrt parameters at u
##' @author Ivan Kojadinovic
dCdthetaNumer <- function(copula, u, method.args = gradControl(d = 0.1), may.warn=TRUE, ...) {
cl <- class(copula)[[1]]
if(may.warn && (is.null(W <- get0("warned.dCdtheta", .copulaEnv)) || is.na(match(cl, W)))) {
assign("warned.dCdtheta", if(is.null(W)) cl else c(W, cl), envir = .copulaEnv)
warning(gettextf(
"Function %s() not implemented for %s copulas; numerical differentiation used",
"dCdtheta", dQuote(cl)), domain=NA)
}
logC <- function(theta) {
freeParam(copula) <- theta
log(pCopula(u, copula))
}
theta <- getTheta(copula, attr=TRUE)
p <- length(theta)
lb <- attr(theta, "param.lowbnd")
ub <- attr(theta, "param.upbnd" )
jacobian(logC, theta, side = sides(theta, p, method.args$d, lb, ub),
method.args = method.args) * pCopula(u, copula)
}
## For copulas with explicit cdf
dCdthetaExplicit <- function(copula, u, ...) {
d <- dim(copula)
algNm <- paste(class(copula)[1], "cdfDerWrtPar.algr", sep=".")
alpha <- getTheta(copula) # typically used in 'eval(*)' below
mat <- matrix(NA_real_, nrow(u), nParam(copula))
if (exists(algNm)) { ## JY: alpha is a scalar parameter
der.cdf.alpha <- get(algNm)[d]
colnames(u) <- paste0("u", 1:d)
mat <- as.matrix(eval(der.cdf.alpha, data.frame(u)))
} else if (.hasSlot(copula, "exprdist") && is.language(cdf <- copula@exprdist$cdf)) {
## symbolic derivatives of explicit cdf expressions
params <- getTheta(copula, freeOnly = FALSE, named = TRUE)
colnames(u) <- paste0("u", 1:d)
u.df <- data.frame(u)
der <- deriv(cdf, names(params)[isFree(copula)])
mat <- attr(eval(der, c(u.df, params)), "gradient")
attr(mat, "dimnames") <- NULL
} else {
warning(gettextf(
"Function %s() not implemented for copulas of class '%s'",
"dCdtheta", class(copula)), domain = NA)
## matrix(NA_real_, nrow(u), length(alpha)) # copula@parameters))
}
## JY: For some rotated explicit copula, dCdtheta returns NA:
## copula:::dCdtheta(khoudrajiCopula(indepCopula(), rotCopula(gumbelCopula(2)), shapes = c(.4, .95)), as.matrix(expand.grid(1:4/5, 1:4/5)))
## In this case, fall back on the numerical version
col.na <- apply(mat, 2, anyNA)
if (any(col.na)) {
warning("dCdtheta() returned NaN in column(s) ", paste(which(col.na), collapse=", "),
" for this explicit copula; falling back to numeric derivative for those columns")
## FIXME: computing for all columns, but need only 'col.na'
mat.num <- dCdthetaNumer(copula, u, may.warn=FALSE, ...)# warned above
mat[,col.na] <- mat.num[,col.na]
}
mat
}
## For evCopula objects
## JY: For single parameter only
dCdthetaEvCopula <- function(copula, u, ...) {
loguv <- log(u[,1], u[,2])
w <- log(u[,2]) / loguv
## return der.cdf.alpha
## JY: single parameter only
as.matrix(pCopula(u, copula) * loguv * dAdtheta(copula, w))
}
setMethod("dCdtheta", signature("parCopula"), dCdthetaNumer)
setMethod("dCdtheta", signature("joeCopula"), dCdthetaExplicit)
setMethod("dCdtheta", signature("tevCopula"), dCdthetaNumer)
setMethod("dCdtheta", signature("archmCopula"), dCdthetaExplicit)
setMethod("dCdtheta", signature("plackettCopula"), dCdthetaExplicit)
setMethod("dCdtheta", signature("evCopula"), dCdthetaExplicit)
setMethod("dCdtheta", signature("gumbelCopula"), dCdthetaExplicit)
setMethod("dCdtheta", signature("khoudrajiExplicitCopula"), dCdthetaExplicit)
setMethod("dCdtheta", signature("mixExplicitCopula"), dCdthetaExplicit)
setMethod("dCdtheta", signature("rotExplicitCopula"), dCdthetaExplicit)
## For ellipCopula objects
dCdthetaEllipCopula <- function(copula, u, ...) {
dim <- copula@dimension
## quantile transformation
v <- switch(class(copula),
"normalCopula" = qnorm(u),
"tCopula" = qt(u, df = getdf(copula)),
stop("not implemented for class ", class(copula)))
val <-
if (dim == 2)
plackettFormulaDim2(copula, v)
else {
n <- nrow(u)
sigma <- getSigma(copula)
if (copula@dispstr %in% c("ex","ar1")) { ## exchangeable or ar1
rho <- copula@parameters[1] # IK: single parameter only
r <- matrix(c(1,-rho,-rho,1),2,2)/(1 - rho^2)
m <- sigma[-c(1,2),c(1,2)] %*% r
s <- sigma[-c(1,2),-c(1,2)] -
sigma[-c(1,2),c(1,2)] %*% r %*% sigma[c(1,2),-c(1,2)]
mat <- matrix(0,n,1)
if (copula@dispstr == "ex") ## exchangeable
for (k in 1:n)
for (j in 1:(dim-1))
for (i in (j+1):dim)
mat[k,1] <- mat[k,1] +
plackettFormula(copula, dim, rho, s, m, v[k,], i, j)
else ## ar1
for (k in 1:n)
for (j in 1:(dim-1))
for (i in (j+1):dim)
mat[k,1] <- mat[k,1] + (i - j) * rho ^ (i - j - 1) *
plackettFormula(copula, dim, rho, s, m, v[k,], i, j)
mat
}
else { # unstructured or toeplitz or ...
mat <- matrix(0,n,dim*(dim-1)/2)
l <- 1
for (j in 1:(dim-1))
for (i in (j+1):dim)
{
rho <- sigma[i,j]
r <- matrix(c(1,-rho,-rho,1),2,2)/(1 - rho^2)
m <- sigma[-c(i,j),c(i,j)] %*% r
s <- sigma[-c(i,j),-c(i,j)] - m %*% sigma[c(i,j),-c(i,j)]
for (k in 1:n)
mat[k,l] <- plackettFormula(copula, dim, rho, s, m, v[k,], i, j)
l <- l + 1
}
if (copula@dispstr == "un") ## unstructured
mat
else if (copula@dispstr == "toep") {
coef <- matrix(0, dim*(dim-1)/2, dim-1)
for (k in 1:(dim-1)) {
m <- row(sigma) == col(sigma) + k
coef[,k] <- P2p(m)
}
mat %*% coef
}
else stop("Not implemented yet for the dispersion structure ",
copula@dispstr)
}
}
free <- isFreeP(copula@parameters)
if (.hasSlot(copula, "df.fixed")) free <- free[-length(free)]
val[, free, drop = FALSE]
}
setMethod("dCdtheta", signature("ellipCopula"), dCdthetaEllipCopula)
##################################################################################
### Partial derivatives of the log PDF wrt arguments
##################################################################################
##' @title Basic implementation of dlogcdu based on numerical differentiation
##' @param copula an object of class "copula"
##' @param u evaluation point in [0,1]^d or matrix of evaluation points
##' @param method.args to be passed to grad()
##' @param ...
##' @return partial derivatives of the log copula density wrt arguments at u
##' @author Ivan Kojadinovic
dlogcduNumer <- function(copula, u, method.args = gradControl(d=1e-5), may.warn = TRUE, ...) {
cl <- class(copula)[[1]]
if(may.warn && (is.null(W <- get0("warned.dlogcdu", .copulaEnv)) || is.na(match(cl, W)))) {
assign("warned.dlogcdu", if(is.null(W)) cl else c(W, cl), envir = .copulaEnv)
warning(gettextf(
"Function %s() not implemented for %s copulas; numerical differentiation used",
"dlogcdu", dQuote(cl)), domain=NA)
}
logc <- function(x) dCopula(x, copula, log = TRUE)
dim <- dim(copula)
t(apply(u, 1, function(u.)
grad(logc, u., side = sides(u., dim, method.args$d, 0, 1),
method.args = method.args)))
}
## For copulas with explicit densities
dlogcduExplicit <- function(copula, u, ...) {
d <- dim(copula)
algNm <- paste(class(copula)[1], "pdfDerWrtArg.algr", sep=".")
mat <- matrix(NA_real_, nrow(u), d)
if(exists(algNm)) {
der.pdf.u <- get(algNm)[d]
alpha <- copula@parameters # typically used in eval() # JY: scalar alpha
unames0 <- paste0("u", 1:d)
for (j in 1:d) {
unames <- unames0; unames[1] <- unames0[j]; unames[j] <- unames0[1]
colnames(u) <- unames
mat[,j] <- eval(der.pdf.u, data.frame(u))
}
} else if (.hasSlot(copula, "exprdist") && is.language(pdf <- copula@exprdist$pdf)) {
## symbolic derivatives of explicit pdf expressions
params <- getTheta(copula, freeOnly = FALSE, named = TRUE)
colnames(u) <- unames <- paste0("u", 1:d)
u.df <- data.frame(u)
der <- deriv(pdf, unames)
mat <- attr(eval(der, c(u.df, params)), "gradient")
attr(mat, "dimnames") <- NULL
} else warning(gettextf(
"Function %s() not implemented for copulas of class '%s'",
"dlogcdu", class(copula)), domain=NA)
mat / dCopula(u, copula)
}
setMethod("dlogcdu", signature("parCopula"), dlogcduNumer)
setMethod("dlogcdu", signature("joeCopula"), dlogcduExplicit)
setMethod("dlogcdu", signature("tevCopula"), dlogcduNumer)
setMethod("dlogcdu", signature("archmCopula"), dlogcduExplicit)
setMethod("dlogcdu", signature("evCopula"), dlogcduExplicit)
setMethod("dlogcdu", signature("plackettCopula"), dlogcduExplicit)
setMethod("dlogcdu", signature("khoudrajiExplicitCopula"), dlogcduExplicit)
setMethod("dlogcdu", signature("mixExplicitCopula"), dlogcduExplicit)
setMethod("dlogcdu", signature("rotExplicitCopula"), dlogcduExplicit)
## For normalCopula objects
dlogcduNormalCopula <- function(copula, u, ...) {
v <- qnorm(u)
(- v %*% solve(getSigma(copula)) + v) / dnorm(v)
}
setMethod("dlogcdu", signature("normalCopula"), dlogcduNormalCopula)
## For tCopula objects
dlogcduTCopula <- function(copula, u, ...) {
df <- getdf(copula)
v <- qt(u,df=df)
w <- dt(v,df=df)
m <- v %*% solve(getSigma(copula))
- (df + copula@dimension) * m / ((df + rowSums(m * v)) * w) +
(df + 1) * v / ((df + v^2) * w)
}
setMethod("dlogcdu", signature("tCopula"), dlogcduTCopula)
##################################################################################
### Partial derivatives of the log PDF wrt parameters
##################################################################################
##' @title Basic implementation of dlogcdu based on numerical differentiation
##' @param copula an object of class "copula"
##' @param u evaluation point in [0,1]^d or matrix of evaluation points
##' @param method.args to be passed to grad()
##' @param ...
##' @return partial derivatives of the log copula density wrt parameters at u
##' @author Ivan Kojadinovic
dlogcdthetaNumer <- function(copula, u, method.args = gradControl(d=1e-5), may.warn = TRUE, ...) {
cl <- class(copula)[[1]]
if(may.warn &&
(is.null(W <- get0("warned.dlogcdtheta", .copulaEnv)) || is.na(match(cl, W)))) {
assign("warned.dlogcdtheta", if(is.null(W)) cl else c(W, cl), envir = .copulaEnv)
warning(gettextf(
"Function %s() not implemented for %s copulas; numerical differentiation used",
"dlogcdtheta", dQuote(cl)), domain=NA)
}
logc <- function(theta) {
freeParam(copula) <- theta
dCopula(u, copula, log = TRUE)
}
theta <- getTheta(copula, attr = TRUE)
p <- length(theta)
lb <- attr(theta, "param.lowbnd")
ub <- attr(theta, "param.upbnd" )
jacobian(logc, theta, side = sides(theta, p, method.args$d, lb, ub),
method.args = method.args)
}
## For copulas with explicit densities
dlogcdthetaExplicit <- function(copula, u, ...) {
d <- dim(copula)
algNm <- paste(class(copula)[1], "pdfDerWrtPar.algr", sep=".")
if(exists(algNm) && !is.null((der.pdf.alpha <- get(algNm)[d])[[1]])) {
alpha <- copula@parameters # typically used in val(.)
colnames(u) <- paste0("u", 1:d)
mat <- as.matrix(eval(der.pdf.alpha, data.frame(u))) / dCopula(u, copula)
} else if (.hasSlot(copula, "exprdist") && is.language(pdf <- copula@exprdist$pdf)) {
## symbolic derivatives of explicit pdf expressions
params <- getTheta(copula, freeOnly = FALSE, named = TRUE)
colnames(u) <- paste0("u", 1:d)
u.df <- data.frame(u)
der <- deriv(pdf, names(params)[isFree(copula)])
mat <- attr(eval(der, c(u.df, params)), "gradient")
attr(mat, "dimnames") <- NULL
mat <- mat / dCopula(u, copula)
} else {
warning(gettextf(
"Function %s() not implemented for copulas of class '%s'",
"dlogcdtheta", class(copula)), domain = NA)
mat <- matrix(NA_real_, nrow(u), nParam(copula))
}
## JY: For some rotated explicit copula, dCdtheta returns NA:
## copula:::dlogcdtheta(khoudrajiCopula(indepCopula(), rotCopula(gumbelCopula(2)), shapes = c(.4, .95)), as.matrix(expand.grid(1:4/5, 1:4/5)))
## In this case, fall back on the numerical version
col.na <- apply(mat, 2, anyNA)
if (any(col.na)) {
warning("dlogcdtheta() returned NaN in column(s) ", paste(which(col.na), collapse=", "),
" for this explicit copula; falling back to numeric derivative for those columns")
## FIXME: computing for all columns, but need only 'col.na'
mat.num <- dlogcdthetaNumer(copula, u, may.warn=FALSE, ...)# warned above
mat[,col.na] <- mat.num[,col.na]
}
mat
}
setMethod("dlogcdtheta", signature("parCopula"), dlogcdthetaNumer)
setMethod("dlogcdtheta", signature("joeCopula"), dlogcdthetaExplicit)
setMethod("dlogcdtheta", signature("tevCopula"), dlogcdthetaNumer)
setMethod("dlogcdtheta", signature("archmCopula"), dlogcdthetaExplicit)
setMethod("dlogcdtheta", signature("plackettCopula"), dlogcdthetaExplicit)
setMethod("dlogcdtheta", signature("evCopula"), dlogcdthetaExplicit)
setMethod("dlogcdtheta", signature("khoudrajiExplicitCopula"), dlogcdthetaExplicit)
setMethod("dlogcdtheta", signature("mixExplicitCopula"), dlogcdthetaExplicit)
setMethod("dlogcdtheta", signature("rotExplicitCopula"), dlogcdthetaExplicit)
## For ellipCopula objects
dlogcdthetaEllipCopula <- function(copula, u, ...) {
dim <- copula@dimension
## quantile transformation
v <- switch(clc <- class(copula), ## FIXME: fails if copula *extends* "tCopula" (e.g.)
"normalCopula" = qnorm(u),
"tCopula" = {
df <- getdf(copula)
qt(u, df=df)
},
## else:
stop("class ", sQuote(clc), " not implemented"))
val <-
if (dim == 2) {
rho <- copula@parameters[1] # JY: single rho parameter, free
ir2 <- 1 - rho^2
sv2 <- rowSums(v^2) # == v[,1]^2 + v[,2]^2
as.matrix(switch(clc,
"normalCopula" =
(rho * ir2 - rho * sv2 + (rho^2 + 1) * v[,1] * v[,2])/ir2^2,
"tCopula" =
(1 + df) * rho / -ir2 + (2 + df) * (df * rho + v[,1] * v[,2])
/ (df * ir2 + sv2 - 2 * rho * v[,1] * v[,2])))
} else { ## dim >= 3
n <- nrow(u)
sigma <- getSigma(copula)
detsig <- det(sigma)
invsig <- solve(sigma)
if (copula@dispstr %in% c("ex","ar1")) { ## exchangeable or ar1
rho <- copula@parameters[1] # JY: single rho parameter
dersig <- matrix(1, dim, dim)
if (copula@dispstr == "ex") ## ex
diag(dersig) <- 0
else ## ar1
for (i in 1:dim)
for (j in 1:dim) {
ij <- abs(i - j)
dersig[i,j] <- ij * rho^(ij - 1)
}
## MM: sum(diag(A %*% B)) == sum(A * t(B)) .. and B=dersig is symmetric here
## derdetsig <- detsig * sum(diag(invsig %*% dersig))
derdetsig <- detsig * sum(invsig * dersig)
derinvsig <- - invsig %*% dersig %*% invsig
firstterm <- derdetsig/detsig
mat <-
switch(clc,
"normalCopula" =
- (firstterm + rowSums((v %*% derinvsig) * v))/2,
"tCopula" =
- (firstterm + (df + dim) * rowSums((v %*% derinvsig) * v)
/ (df + rowSums((v %*% invsig) * v)) ) / 2)
as.matrix(mat)
}
else {# unstructured or toeplitz or ...
mat <- matrix(0, n, dim*(dim - 1)/2)
l <- 1
for (j in 1:(dim-1))
for (i in (j+1):dim) {
derdetsig <- 2 * det(sigma[-i,-j,drop=FALSE]) * (-1)^(i+j)
derinvsig <- - invsig[,i] %*% t(invsig[,j]) - invsig[,j] %*% t(invsig[,i])
firstterm <- derdetsig/detsig
mat[,l] <-
switch(clc,
"normalCopula" =
- (firstterm + rowSums((v %*% derinvsig) * v))/2,
"tCopula" =
- (firstterm + (df + dim) * rowSums((v %*% derinvsig) * v)
/ (df + rowSums((v %*% invsig) * v)) ) / 2)
l <- l + 1
}
if (copula@dispstr == "un") ## unstructured
mat
else if (copula@dispstr == "toep") { ## toeplitz: dim-1 parameters
coef <- matrix(0, dim*(dim-1)/2, dim-1)
for (k in 1:(dim-1)) {
m <- row(sigma) == col(sigma) + k
coef[,k] <- P2p(m)
}
mat %*% coef
}
else stop("Not implemented yet for the dispersion structure ", copula@dispstr)
}
} ## dim >= 3
free <- isFreeP(copula@parameters)
if (.hasSlot(copula, "df.fixed")) free <- free[-length(free)]
val[, free, drop = FALSE]
}
setMethod("dlogcdtheta", signature("ellipCopula"), dlogcdthetaEllipCopula)
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