counter-plot-MGL-copula • rMGLReg

MGL copula

To illustrate the dependence structure, we simulated copula realizations in scatter plots with \(d=3\) from the MGL copula given with sample size \(n=1,000\) for copula parameter \(\delta=1.2\).
Note the positive dependences and the tail asymmetry features among the three variables.

library(rMGLReg)

set.seed(271)
n <- 1000
delta <- 1.2
d <- 3
U <- rcMGL.multi(n = 1000, d = d, pars = delta)
cor(U, method = "kendall")
#>           [,1]      [,2]      [,3]
#> [1,] 1.0000000 0.2322282 0.2335576
#> [2,] 0.2322282 1.0000000 0.2138058
#> [3,] 0.2335576 0.2138058 1.0000000

par(pty = "s")
pairs(U, gap = 0, cex = 0.5)

Survival MGL copula

set.seed(271)
n <- 1000
delta <- 1.2
d <- 3
U <- rcMGL180.multi(n = 1000, d = d, pars = delta)
cor(U, method = "kendall")
#>           [,1]      [,2]      [,3]
#> [1,] 1.0000000 0.2322282 0.2335576
#> [2,] 0.2322282 1.0000000 0.2138058
#> [3,] 0.2335576 0.2138058 1.0000000

par(pty = "s")
pairs(U, gap = 0, cex = 0.5)

Contour plots of the joint density function of the MGL and survival MGL copula with low, medium and high dependence, along with plots of simuated samples.

library(data.table)
library(ggplot2)
library(latex2exp)
par.copula <- 0.51
Nsim <- 1000
bins <- 30
usim <- (rcMGL.bivar(Nsim, pars = par.copula))
usim <- data.table(u1 = usim[,1], u2 = usim[,2])
n.grid <- 200
xgrid <- ygrid <- seq(0.01, 1, length.out = n.grid)
grid <- expand.grid("u1" = xgrid, "u2" = ygrid)
mtrx3d <- matrix(0, nrow = nrow(grid), ncol = 3)
mtrx3d <- cbind(grid, "cu1u2" = dcMGL.bivar(u1 = grid[,1], u2 = grid[,2], pars = par.copula)) # evaluate W on 'grid'
head(mtrx3d)
mtrx3d <- data.table(u1 = mtrx3d[,1], u2 = mtrx3d[,2], cu1u2 = mtrx3d[,3])
v1p <- ggplot() +
      scale_x_continuous(expand = c(0, 0), limits = c(0, 1)) +
      scale_y_continuous(expand = c(0, 0), limits = c(0, 1)) +
      theme_bw() +
      ggtitle(TeX(sprintf('MGL copula ($\\delta = %g$ )', par.copula))) +
      theme(axis.line = element_line(colour = "black"),
            axis.text.x = element_text(margin = margin(t = 0.25, unit = "cm")),
            axis.text.y = element_text(margin = margin(r = 0.25, unit = "cm"),
                                       size = 10,
                                       vjust = 0.5,
                                       hjust = 0.5),

            plot.title = element_text(hjust = 0.5)) +
      labs(x = TeX("$u_1$"), y = TeX("$u_2$"))  +
      geom_contour(data = mtrx3d,
                   aes(x = u1, y = u2, z = cu1u2),
                   bins = bins,
                   colour = 'black', weight = 3) +
      geom_point(data = usim, aes(x = u1, y = u2),
                 size = 1,
                 color = 'red')

par.copula <- 3.81
Nsim <- 1000
usim <- (rcMGL.bivar(Nsim, pars = par.copula))
usim <- data.table(u1 = usim[,1], u2 = usim[,2])
n.grid <- 200
xgrid <- ygrid <- seq(0.01, 1, length.out = n.grid)
grid <- expand.grid("u1" = xgrid, "u2" = ygrid)
mtrx3d <- matrix(0, nrow = nrow(grid), ncol = 3)
mtrx3d <- cbind(grid, "cu1u2" = dcMGL.bivar(u1 = grid[,1], u2 = grid[,2], pars = par.copula)) # evaluate W on 'grid'
head(mtrx3d)
mtrx3d <- data.table(u1 = mtrx3d[,1], u2 = mtrx3d[,2], cu1u2 = mtrx3d[,3])
v2p <- ggplot() +
      scale_x_continuous(expand = c(0, 0), limits = c(0, 1)) +
      scale_y_continuous(expand = c(0, 0), limits = c(0, 1)) +
      theme_bw() +
      ggtitle(TeX(sprintf('MGL copula ($\\delta = %g$ )', par.copula))) +
      theme(axis.line = element_line(colour = "black"),
            axis.text.x = element_text(margin = margin(t = 0.25, unit = "cm")),
            axis.text.y = element_text(margin = margin(r = 0.25, unit = "cm"),
                                       size = 10,
                                       vjust = 0.5,
                                       hjust = 0.5),

            plot.title = element_text(hjust = 0.5)) +
      labs(x = TeX("$u_1$"), y = TeX("$u_2$"))  +
      geom_contour(data = mtrx3d,
                   aes(x = u1, y = u2, z = cu1u2),
                   bins = bins,
                   colour = 'black', weight = 3) +
      geom_point(data = usim, aes(x = u1, y = u2),
                 size = 1,
                 color = 'red')

par.copula <- 10.56
Nsim <- 1000
usim <- (rcMGL.bivar(Nsim, pars = par.copula))
usim <- data.table(u1 = usim[,1], u2 = usim[,2])
n.grid <- 200
xgrid <- ygrid <- seq(0.01, 1, length.out = n.grid)
grid <- expand.grid("u1" = xgrid, "u2" = ygrid)
mtrx3d <- matrix(0, nrow = nrow(grid), ncol = 3)
mtrx3d <- cbind(grid, "cu1u2" = dcMGL.bivar(u1 = grid[,1], u2 = grid[,2], pars = par.copula)) # evaluate W on 'grid'
head(mtrx3d)
mtrx3d <- data.table(u1 = mtrx3d[,1], u2 = mtrx3d[,2], cu1u2 = mtrx3d[,3])
v3p <- ggplot() +
      scale_x_continuous(expand = c(0, 0), limits = c(0, 1)) +
      scale_y_continuous(expand = c(0, 0), limits = c(0, 1)) +
      theme_bw() +
      ggtitle(TeX(sprintf('MGL copula ($\\delta = %g$ )', par.copula))) +
      theme(axis.line = element_line(colour = "black"),
            axis.text.x = element_text(margin = margin(t = 0.25, unit = "cm")),
            axis.text.y = element_text(margin = margin(r = 0.25, unit = "cm"),
                                       size = 10,
                                       vjust = 0.5,
                                       hjust = 0.5),

            plot.title = element_text(hjust = 0.5)) +
      labs(x = TeX("$u_1$"), y = TeX("$u_2$"))  +
      geom_contour(data = mtrx3d,
                   aes(x = u1, y = u2, z = cu1u2),
                   bins = bins,
                   colour = 'black', weight = 3) +
      geom_point(data = usim, aes(x = u1, y = u2),
                 size = 1,
                 color = 'red')

library(patchwork)

p1 <- v1p + v2p + v3p + plot_layout(ncol = 3)
p1