copBasic (version 2.0.1)

PlackettPlackettNP: Parameters and L-comoments of a Composition of Two Plackett Copulas with Two Compositing Parameters

Description

EXPERIMENTAL---These data contain a stochastically generated parameter space of a Plackett-Plackett composited copula to support a fast lookup of the relation between four copula parameters and the four L-comoments of L-correlation, L-coskew, and L-cokurtosis---hence, support the copulatic method of L-comoments. The NP in the name indicates that the first Plackett copula was only configured for negative or anticorrelation and the second Plackett copula was only configured for positive correlation. This particular data set is provided in the copBasic package for experimental self-contained parameter estimation.

Specifically, these data contain the stochastically generated parameter space of the $\alpha$ and $\beta$ mixing or compositing parameters of the composited copula in which the two copulas for composition $\mathbf{A}$ and $\mathbf{B}$ are set as Plackett copulas (see PLACKETTcop). Then the composition is made by

$$\mathbf{C}_{\alpha,\beta}(u,v) = \mathbf{A}(u^\alpha, v^\beta) \cdot \mathbf{B}(u^{1-\alpha},v^{1-\beta})\mbox{.}$$ defines a family of copulas $\mathbf{C}_{\alpha,\beta,\kappa,\gamma}$, with compositing parameters $\alpha,\beta \in \mathcal{I}:[0,1]$.

The first Plackett copula named $\mathbf{A}$ (see PLACKETTcop) was randomly generated between the product copulas $\mathbf{\Pi}$ (see P) and the Fréchet{Frechet}-Hoeffding lower bound copula $\mathbf{W}(u,v)$ (see W). The second Plackett copula named $\mathbf{B}$ (PLACKETTcop) was randomly generated between the product copulas $\mathbf{\Pi}$ (see P) and the Fréchet{Frechet}-Hoeffding upper bound copula $\mathbf{M}(u,v)$ (see M). The $\mathbf{A}$ and $\mathbf{B}$ generated in this way means that the first copula controls the negative correlative aspects of a data set and the second copula controls the positive aspects.

To further clarify, $\mathbf{A}_\Theta$ parameter is on the interval $[0,1]$ and the $\mathbf{B}_\Theta$ parameter is on the inteval $[1,\infty]$. In reality, the $\Theta$ parameters were generated uniformly in log-space and then transformed. The log of $\mathbf{A}_\Theta$ was on the interval $[-5,0]$, and the log of $\mathbf{B}_\Theta$ was on the interval $[0,5]$. (See such limits of generation in the Source section.)

Usage

data(PlackettPlackettNP)

encoding

utf8

source

This data set was created using the simcompositeCOP function, which in turn defaults to use of the composite2COP function for compositing of the two Plackett copulas. The usage is shown below. mainpara <- list(cop1=PLACKETTcop, cop2=PLACKETTcop, para1gen=function() { return(10^runif(1, min=-5, max=0)) }, para2gen=function() { return(10^runif(1, min=0, max=5)) })

n <- 10 # really used 20500 for generation of the data set PlackettPlackettNP <- simcompositeCOP(n=1000, nsim=n, parent=mainpara) save(PlackettPlackettNP, file="PlackettPlackettNP.RData", compress="xz") For the generation process, the mainpara list specifies the two copulas to use in the composite, and two generation functions for the parameters of the respective copulas are housed in para1gen and para2gen. For each of 20,500 simulations of size $n=1{,}000$ of the stochastically generated Plackett-Plackett composited copulas, the L-correlation, L-coskew, L-cokurtosis matrices were computed. The opposing diagonals of the L-correlation, L-coskew, L-cokurtosis of the matrices were requested internally by simcompositeCOP and are shown. The L-comoment matrices were computed by the lcomoms2(nmom=4) function of the lmomco package.

References

Salvadori, G., De Michele, C., Kottegoda, N.T., and Rosso, R., 2007, Extremes in Nature---An approach using copulas: Springer, 289 p.

Serfling, R., and Xiao, P., 2007, A contribution to multivariate L-moments---L-comoment matrices: Journal of Multivariate Analysis, v. 98, pp. 1765--1781.

Examples

data(PlackettPlackettNP)
summary(PlackettPlackettNP)

PlackettPlackettNP <- as.data.frame(PlackettPlackettNP)
# Let us now visualize the parameter space
plot.complcom.space <- function(text=NULL) {
   U12 <- PlackettPlackettNP$T2.12; U21 <- PlackettPlackettNP$T2.21
   V12 <- PlackettPlackettNP$T3.12; V21 <- PlackettPlackettNP$T3.21
   U <- c(U12, U21); V <- c(V12, V21)
   plot(U,V, type="n", xlim=c(-1,1), ylim=c(-0.2,0.4),
        xlab="L-CORRELATION", ylab="L-COSKEW")
   # plot transparent blue L-correlation and L-coskew first
   points(U12,V12, col=rgb(   0, 0, 1, 0.12), pch=16)
   # plot transparent purple L-correlation and L-coskew second
   points(U21,V21, col=rgb(0.85, 0, 1, 0.12), pch=16)
   abline(v=0); abline(h=0) # cross lines
   mtext(text)
}
my.text <- "L-COMOMENTS OF PLACKETT(negative)-PLACKETT(postive) COPULA"
plot.complcom.space(text=my.text) # plot the parameter space

# Let us now visualize the parameter space
plot.complcom.space2 <- function(text=NULL) {
   U12 <- PlackettPlackettNP$T3.12; U21 <- PlackettPlackettNP$T3.21
   V12 <- PlackettPlackettNP$T4.12; V21 <- PlackettPlackettNP$T4.21
   U <- c(U12, U21); V <- c(V12, V21)
   plot(U,V, type="n", xlim=c(-0.2,0.4), ylim=c(-0.2,0.2),
        xlab="L-COSKEW", ylab="L-COKURTOSIS")
   # plot transparent blue L-correlation and L-coskew first
   points(U12,V12, col=rgb(   0, 0, 1, 0.12), pch=16)
   # plot transparent purple L-correlation and L-coskew second
   points(U21,V21, col=rgb(0.85, 0, 1, 0.12), pch=16)
   abline(v=0); abline(h=0) # cross lines
   mtext(text)
}
my.text <- "L-COMOMENTS OF PLACKETT(negative)-PLACKETT(postive) COPULA"
plot.complcom.space2(text=my.text) # plot the parameter space