## Copyright (C) 2011 Marius Hofert and Martin Maechler ## ## 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 . ### Demo of the two-parameter outer power Clayton copula ####################### ### setup ###################################################################### library(nacopula) library(bbmle) library(lattice) library(grid) ## specify parameters n <- 100 # sample size d <- 10 # dimension thetabase <- 1 # fix thetabase tau <- 0.5 # => psi(t) = psi_thetabase(t^(1/theta)) with (thetabase,theta) = (1,4/3) (see below) ## adjustment for initial value h <- c(0.4,0) # h_-, h_+ ### functions ################################################################## ##' Initial interval for opC ##' ##' @title Initial interval for opC ##' @param U (n x d)-matrix of simulated data ##' @param h non-negative auxiliary parameter for computing initial intervals ##' @param method "etau" via sample version of Kendall's tau (may be slow) ##' "dmle.G" via DMLE of Gumbel (may be inaccurate) ##' @return (2 x 2)-matrix containing the initial interval [1st row: lower, ##' 2nd row: upper; 2 parameters => 2 cols] ##' @author Marius Hofert ii.opC <- function(U, h, method=c("etau","dmle.G")){ stopifnot(h >= 0, length(h) >= 2) I <- matrix(, nrow=2, ncol=2, dimnames=list(c("lower", "upper"), c("thetabase", "theta"))) ## estimate Kendall's tau method <- match.arg(method) tau.hat <- switch(method, "etau" = { # uses sample version of tau, more accurate but slower tau.hat.mat <- cor(U, method="kendall") mean(tau.hat.mat[upper.tri(tau.hat.mat)]) }, "dmle.G" = { # uses DMLE for Gumbel to estimate tau Z <- apply(U, 1, max) theta.hat.G <- log(ncol(U))/(log(length(Z))-log(sum(-log(Z)))) copGumbel@tau(theta.hat.G) }, stop("wrong method:", method)) ## compute largest value of theta (for lower right endpoint of the inital interval) stopifnot(tau.hat > 0) tau.hat.hp <- min(tau.hat+h[2], 0.995) th.max <- 2*tau.hat.hp/(1-tau.hat.hp) I[2,1] <- th.max # largest value for theta (= thetabase) I[1,2] <- 1 # smallest value for beta (= theta) ## compute smallest theta (for lower left endpoint of the inital interval) tau.hat.hm <- max(tau.hat-h[1], 0.005) # tau=0.005 <=> theta=0.01 th.min <- 2*tau.hat.hm/(1-tau.hat.hm) I[1,1] <- th.min ## compute largest beta (for upper left endpoint of the inital interval) b.max <- 2/((2+th.min)*(1-tau.hat.hp)) I[2,2] <- b.max ## result I } ##' -log-likelihood ##' ##' @title -log-likelihood ##' @param thetabase parameter of the base (=Clayton) generator ##' @param theta outer power parameter ##' @param u (n x d)-matrix of simulated data ##' @return -sum(log(density)) ##' @author Marius Hofert nlogl.opC <- function(thetabase, theta, u){ if(!is.matrix(u)) u <- rbind(u) if((d <- ncol(u)) < 2) stop("u should be at least bivariate") # check that d >= 2 cop <- opower(copClayton, thetabase) -sum(cop@dacopula(u, theta=theta, log=TRUE)) } ## vectorized version nlogl.opC. <- function(theta, u) nlogl.opC(theta[1], theta=theta[2], u=u) ### estimation ################################################################# ## determine theta such that tau is matched (for given thetabase) opC <- opower(copClayton, thetabase) # outer power Clayton copula theta <- opC@tauInv(tau) # choose theta such that Kendall's tau equals tau ## define the outer power Clayton copula to be sampled and estimated cop <- onacopulaL(opC, list(theta, 1:d)) ## sample set.seed(1000) U <- rnacopula(n, cop) ## plot splom2(U, cex=0.4, pscales=0, main=paste("Sample of size",n, "from an outer power Clayton copula")) ## initial interval and value I <- ii.opC(U, h) start <- colMeans(I) ## without profiling: optim with method="L-BFGS-B" system.time(optim(par=start, method="L-BFGS-B", fn=function(x) nlogl.opC(x[1], theta=x[2], u=U), lower=c(I[1,1], I[1,2]), upper=c(I[2,1], I[2,2]))) ## with profiling: via mle (uses optimizer="optim" with method="L-BFGS-B") nLL <- function(thetabase, theta) nlogl.opC(thetabase, theta, u=U) system.time(ml <- mle(nLL, method="L-BFGS-B", start=list(thetabase=mean(I[,1]), theta=mean(I[,2])), lower=c(thetabase=I[1,1], theta=I[1,2]), upper=c(thetabase=I[2,1], theta=I[2,2]))) summary(ml) str(ml@details) ## with profiling: via mle2 (uses optimizer="optim" with method="L-BFGS-B") system.time(ml2 <- mle2(nlogl.opC, data=list(u=U), method="L-BFGS-B", start=list(thetabase=mean(I[,1]), theta=mean(I[,2])), lower=c(thetabase=I[1,1], theta=I[1,2]), upper=c(thetabase=I[2,1], theta=I[2,2]))) summary(ml2) str(ml2@details) ### plots ###################################################################### ### profile likelihood plots ################################################### prof <- profile(ml) if(FALSE) { ## FIXME (?) ## maybe this helps: https://stat.ethz.ch/pipermail/r-help/2005-July/076003.html ci <- confint(prof) ci plot(prof) } prof2 <- profile(ml2) (ci <- confint(prof2)) plot(prof2) ### -log-likelihood plots ###################################################### ## for the plots, we use the standard mathematical notation (theta, beta) ## instead of (thetabase, theta) ## build grid m <- 20 # number of grid points = number of intervals + 1 th <- seq(I[1,1], I[2,1], length.out=m) # grid points for thetabase beta <- seq(I[1,2], I[2,2], length.out=m) # grid points for theta grid <- expand.grid(theta=th, beta=beta) # grid val.grid <- apply(grid, 1, nlogl.opC., u=U) # value of the -log-likelihood on the grid ### wireframe ## plot settings true.theta <- c(thetabase, theta) true.val <- c(true.theta, nlogl.opC.(true.theta, u=U)) # theoretical optimum opt <- ml@coef # optimizer-optimum opt.val <- c(opt, nlogl.opC.(opt, u=U)) # optimizer-optimum and its value pts <- rbind(true.val, opt.val) # points to add to wireframe plot title <- "-log-likelihood of an outer power Clayton copula" # title sub <- substitute(italic(n) == N ~~~ italic(d)== D ~~~ tau == TAU ~~~ "#{eval}:" ~ NIT, list(N=n, D=d, TAU= tau, NIT= ml@details$counts[[1]])) ## lattice bug: sub <- as.expression(sub) ## wireframe wireframe(val.grid~grid[,1]*grid[,2], screen=list(z=70, x=-55), zoom=0.95, xlab=expression(italic(theta)), ylab=expression(italic(beta)), zlab = list(as.expression(-log~L * group("(",list(theta,beta),")")), rot=90), main=title, sub=sub, pts=pts, scales=list(col=1, arrows=FALSE), par.settings=list(axis.line=list(col="transparent"), clip=list(panel="off")), zlim=c(min(val.grid, pts[,3]), max(val.grid, pts[,3])), aspect=1, panel.3d.wireframe = function(x,y,z,xlim,ylim,zlim,xlim.scaled, ylim.scaled,zlim.scaled,pts,...){ panel.3dwire(x=x, y=y, z=z, xlim=xlim, ylim=ylim, zlim=zlim, xlim.scaled=xlim.scaled, ylim.scaled=ylim.scaled, zlim.scaled=zlim.scaled, alpha.regions=0.8, ...) panel.3dscatter(x=pts[,1], y=pts[,2], z=pts[,3], xlim=xlim, ylim=ylim, zlim=zlim, xlim.scaled=xlim.scaled, ylim.scaled=ylim.scaled, zlim.scaled=zlim.scaled, type="p", col=c("red","blue"), pch=c(3,4), lex=2, cex=1.4, .scale=TRUE, ...) }, key=list(x=0.64, y=1.01, points=list(pch=c(3,4), col=c("red","blue"), lwd=2, cex=1.4), text=list(c("True value", "Optimum of optimizer")), padding.text=3, cex=1, align=TRUE, transparent=TRUE)) ### levelplot ## plot settings xlim. <- c(min(grid[,1]),max(grid[,1])) ylim. <- c(min(grid[,2]),max(grid[,2])) xeps <- (xlim.[2] - xlim.[1]) * 0.04 yeps <- (ylim.[2] - ylim.[1]) * 0.04 cols <- adjustcolor(colorRampPalette(c("darkgreen", "green", "orange", "yellow"), space="Lab")(100), 0.8) ## levelplot levelplot(val.grid~grid[,1]*grid[,2], par.settings=list(layout.heights=list(main=3, sub=2), regions=list(col=cols)), xlim=c(xlim.[1]-xeps, xlim.[2]+xeps), ylim=c(ylim.[1]-yeps, ylim.[2]+yeps), xlab=expression(italic(theta)), ylab=expression(italic(beta)), main=title, sub=sub, pts=pts, aspect=1, scales=list(alternating=c(1,1), tck=c(1,0)), contour=TRUE, panel=function(x, y, z, pts, ...){ panel.levelplot(x=x, y=y, z=z, ...) grid.points(x=pts[1,1], y=pts[1,2], pch=3, gp=gpar(lwd=2, col="red")) # + true value grid.points(x=pts[2,1], y=pts[2,2], pch=4, gp=gpar(lwd=2, col="blue")) # x optimum }, key=list(x=0.18, y=1.08, points=list(pch=c(3,4), col=c("red","blue"), lwd=2, cex=1.4), columns=2, text=list(c("True value", "Optimum of optimizer")), align=TRUE, transparent=TRUE))