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## Hierarchical Kendall Copulas as defined in Brechmann, Eike Christian.
## "Hierarchical Kendall copulas: Properties and inference." Canadian Journal
## of Statistics 42.1 (2014): 78-108.
# Slots:
#
# Name: nestingCop clusterCops kenFuns dimension parameters param.names param.lowbnd param.upbnd fullname
# Class: copula list list integer numeric character numeric numeric character
## nestingCop copula
## clusterCop list of list (copula, ind)
# easy constructor
hkCopula <- function(nestingCop, clusterCops, kenFuns=NULL) {
if (is.null(kenFuns)) {
kenFuns <- lapply(clusterCops, function(copInd) getKendallDistr(copInd[[1]]))
}
new("hkCopula",
nestingCop = nestingCop,
clusterCops = clusterCops,
kenFuns = kenFuns,
dimension = as.integer(sum(sapply(clusterCops, function(x) x[[1]]@dimension))+nestingCop@dimension-length(clusterCops)),
parameters = NA_real_,
param.names =NA_character_,
param.lowbnd = NA_real_,
param.upbnd = NA_real_,
fullname = "Hierarchical Kendall Copula")
}
showHkCopula <- function(object) {
cat(object@fullname, "\n")
cat("Dimension: ", object@dimension, "\n")
cat("Nesting copula:\n")
show(object@nestingCop)
cat("Cluster copulas:\n")
for (i in 1:length(object@clusterCops)) {
cmpCop <- object@clusterCops[[i]][[1]]
cat(" ", describeCop(cmpCop, "very short"),
"of dimension", cmpCop@dimension,
"for indices", object@clusterCops[[i]][[2]], "\n")
}
}
setMethod("show", signature("hkCopula"), showHkCopula)
## density
dHkCop <- function(u, copula, log=F, ...) {
stopifnot(ncol(u) == copula@dimension)
lik <- NULL
kenVal <- NULL
for (i in 1:length(copula@clusterCops)) {
cop <- copula@clusterCops[[i]][[1]]
ind <- copula@clusterCops[[i]][[2]]
ken <- copula@kenFuns[[i]]
lik <- cbind(lik, dCopula(u[, ind], cop, log=log))
kenVal <- cbind(kenVal, ken(pCopula(u[, ind], cop)))
}
if (ncol(kenVal) < copula@nestingCop@dimension) {
kenVal <- cbind(kenVal, u[, -sapply(copula@clusterCops, function(x) x[[2]])])
}
lik <- cbind(lik, dCopula(kenVal, copula@nestingCop, log=log))
if (log)
return(apply(lik, 1, sum))
return(apply(lik, 1, prod))
}
setMethod("dCopula", signature("matrix", "hkCopula"), dHkCop)
setMethod("dCopula", signature("numeric", "hkCopula"),
function(u, copula, log, ...) dHkCop(matrix(u, ncol = copula@dimension), copula, log, ...))
rHkCop <- function(n, copula, ...) {
smpl <- matrix(NA, n, copula@dimension)
nestSmpl <- rCopula(n, copula@nestingCop)
for (i in 1:length(copula@clusterCops)) {
cop <- copula@clusterCops[[i]][[1]]
ind <- copula@clusterCops[[i]][[2]]
ken <- copula@kenFuns[[i]]
invKen <- genInvKenFun(ken)
smpl[,ind] <- rCopula_y(invKen(nestSmpl[,i]), cop)
}
if (ncol(nestSmpl) > length(copula@clusterCops)) {
smpl[,-sapply(copula@clusterCops, function(x) x[[2]])] <- nestSmpl[, -c(1:length(copula@clusterCops))]
}
return(smpl)
}
setMethod(rCopula, signature = c("numeric","hkCopula"), rHkCop)
setMethod(pCopula, signature = c("numeric","hkCopula"),
function(u, copula, ...) stop("Please use an empirical representation (i.e. \"genEmpCop\" applied to a sample of this copula)."))
setMethod(pCopula, signature = c("matrix","hkCopula"),
function(u, copula, ...) stop("Please use an empirical representation (i.e. \"genEmpCop\" applied to a sample of this copula)."))
rCop_y <- function(y, copula, n=1, n.disc = 1e2) {
stopifnot(copula@dimension == 2)
n.y <- length(y)
stopifnot(n.y == 1 | n == 1)
smpl <- matrix(NA, n.y*n, 2)
for (i in 1:n.y) { # i <- 1
condVals <- seq(y[i], 1-(1-y[i])/n.disc, length.out = n.disc)
uv <- qCopula_v(copula, rep(y[i], n.disc), condVals)
uv <- rbind(uv, qCopula_u(copula, rep(y[i], n.disc), condVals))
uv <- uv[order(uv[,1]),]
dSeq <- cumsum(c(0, apply((uv[-nrow(uv),]-uv[-1,])^2, 1, function (x) sqrt(sum(x)))))
probs <- dCopula(uv, copula)
apFun <- approxfun(dSeq, probs, rule = 2)
probCor <- integrate(apFun, 0, max(dSeq))$value
rContour <- runif(n, 0, probCor)
funAppConPoint <- function(rCont) {
invCDFContour <- function(x) {
abs(integrate(apFun, 0, x)$value - rCont)
}
lContour <- optimise(invCDFContour, c(0, max(dSeq)))$minimum
dSeqInt <- findInterval(lContour, dSeq)
lSeq <- sqrt(sum((uv[dSeqInt,]-uv[dSeqInt+1,])^2))
uv[dSeqInt,] + (lContour - dSeq[dSeqInt])/lSeq * (uv[dSeqInt+1,]-uv[dSeqInt,])
}
if (n == 1) {
appConPoint <- funAppConPoint(rContour)
if (appConPoint[1] > appConPoint[2]) {
smpl[i,] <- qCopula_u(copula, y[i], appConPoint[1])
} else {
smpl[i,] <- qCopula_v(copula, y[i], appConPoint[2])
}
} else {
appConPoint <- t(sapply(rContour, funAppConPoint))
boolLower <- appConPoint[,1] > appConPoint[,2]
smpl[boolLower,] <- qCopula_u(copula, rep(y, sum(boolLower)), appConPoint[boolLower, 1])
smpl[!boolLower,] <- qCopula_v(copula, rep(y, sum(!boolLower)), appConPoint[!boolLower, 2])
}
}
return(smpl)
}
setGeneric("rCopula_y", function(y, copula, n=1, n.disc=1e2) NULL)
setMethod("rCopula_y", signature("numeric", "copula"), rCop_y)
# ## attic
# hkCop <- new("hkCopula", nestingCop=normalCopula(0.6),
# clusterCops=list(list(cop=frankCopula(3), ind=c(1,2))),
# kenFuns = list(getKendallDistr(frankCopula(3))),
# dimension = 3L,
# parameters = NA_real_,
# param.names ="",
# param.lowbnd = NA_real_,
# param.upbnd = NA_real_,
# fullname = "Hierarchical Kendall Copula")
#
# hkCop4D <- new("hkCopula", nestingCop=normalCopula(0.6),
# clusterCops=list(list(cop=frankCopula(3), ind=c(1,2)),
# list(cop=gumbelCopula(5), ind=c(3,4))),
# kenFuns = list(getKendallDistr(frankCopula(3)),
# getKendallDistr(gumbelCopula(5))),
# dimension = 4L,
# parameters = c(0),
# param.names ="",
# param.lowbnd = c(0),
# param.upbnd = 0,
# fullname = "Hierarchical Kendall Copula")
#
# rHkCop(10, hkCop)
#
# smplRHkCop3D <- rCopula(100, hkCop)
#
# library(rgl)
# plot3d(smplRHkCop3D)
# kenHKcop <- genEmpKenFun(hkCop, sample = smplRHkCop3D)
#
# curve(kenHKcop)
#
# showMethods("pCopula")
#
# plot(rCopula_y(0.9, gumbelCopula(5), 100))
# plot(rCopula_y(0.9, tawn3pCopula(c(0.75,.25,5)), 100))
# plot(rCopula_y(0.9, normalCopula(0.4), 100), asp=1)
# points(rCopula_y(0.9, normalCopula(0.8), 100), asp=1, pch=2)
# points(rCopula_y(0.9, normalCopula(-0.8), 100), asp=1, pch=3)
#
# points(rCopula_y(0.4, normalCopula(-0.3), 100), asp=1, pch=4)
# abline(1.9,-1, col="red")
#
# contour(normalCopula(-0.3), pCopula, asp=1)
#
# sum(dHkCop(rHkCop(10, hkCop), hkCop))/10
# sum(dHkCop(rHkCop(10, hkCop4D), hkCop4D))/10
#
# sum(dHkCop(matrix(runif(3000), 1000), hkCop))/1000
# sum(dHkCop(matrix(runif(4000), 1000), hkCop4D))/1000
#
# par(mfrow=c(2,1))
# hist(dHkCop(matrix(runif(4*1e5),ncol = 4), hkCop4D), n=4000, xlim=c(0,10))
# hist(dHkCop(matrix(runif(3*1e5),ncol = 3), hkCop), n=400, xlim=c(0,10))
#
# sum(dHkCop(matrix(runif(4*1e5),ncol = 4), hkCop4D))/1e5
# sum(dHkCop(matrix(runif(3*1e5),ncol = 3), hkCop))/1e5
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