Archimedan Copulas

Purpose
Plot the density, level curves of the Archimedan Copula

Gumbel Copula

> library(copula)
> n <- 5000
> param.cop <- 10
> dim.cop <- 2
> cop.gumbel <- archmCopula(family = "gumbel", param = param.cop, dim = dim.cop)
> mvd.gumbel <- mvdc(copula = cop.gumbel, margins = c("norm", "norm"), paramMargins = list(list(mean = 0,
+     sd = 2), list(mean = 0, sd = 2)))
> x1 <- rcopula(cop.gumbel, n)
> param.cop <- 2
> dim.cop <- 2
> cop.gumbel <- archmCopula(family = "gumbel", param = param.cop, dim = dim.cop)
> mvd.gumbel <- mvdc(copula = cop.gumbel, margins = c("norm", "norm"), paramMargins = list(list(mean = 0,
+     sd = 2), list(mean = 0, sd = 2)))
> x2 <- rcopula(cop.gumbel, n)
> par(mfrow = c(1, 2))
> plot(x1[, 1], x1[, 2], col = "blue", xlab = "", ylab = "", pch = 16, main = expression(paste("Gumbel ",
+     alpha, "  = 10")))
> plot(x2[, 1], x2[, 2], col = "blue", xlab = "", ylab = "", pch = 16, main = expression(paste("Gumbel ",
+     alpha, "  = 2")))

Gumbel Level Plots

> par(mfrow = c(1, 1))
> contour(mvd.gumbel, dmvdc, xlim = c(-6, 6), ylim = c(-6, 6), main = expression(paste("Gumbel ",
+     alpha, "  = 2")))

Gumbel Density Plots

> par(mfrow = c(1, 1))
> probs <- dmvdc(mvd.gumbel, x2)
> scatterplot3d(x2[, 1], x2[, 2], probs, angle = 220, color = "blue", pch = 16,
+     main = expression(paste("Gumbel ", alpha, "  = 2")))

Clayton Copula

> library(copula)
> n <- 5000
> param.cop <- 10
> dim.cop <- 2
> cop.clayton <- archmCopula(family = "clayton", param = param.cop, dim = dim.cop)
> mvd.clayton <- mvdc(copula = cop.clayton, margins = c("norm", "norm"), paramMargins = list(list(mean = 0,
+     sd = 2), list(mean = 0, sd = 2)))
> x1 <- rcopula(cop.clayton, n)
> param.cop <- 2
> dim.cop <- 2
> cop.clayton <- archmCopula(family = "clayton", param = param.cop, dim = dim.cop)
> mvd.clayton <- mvdc(copula = cop.clayton, margins = c("norm", "norm"), paramMargins = list(list(mean = 0,
+     sd = 2), list(mean = 0, sd = 2)))
> x2 <- rcopula(cop.clayton, n)
> par(mfrow = c(1, 2))
> plot(x1[, 1], x1[, 2], col = "blue", xlab = "", ylab = "", pch = 16, main = expression(paste("Clayton ",
+     alpha, "  = 10")))
> plot(x2[, 1], x2[, 2], col = "blue", xlab = "", ylab = "", pch = 16, main = expression(paste("Clayton ",
+     alpha, "  = 2")))

Clayton Level Plots

> par(mfrow = c(1, 1))
> contour(mvd.clayton, dmvdc, xlim = c(-6, 6), ylim = c(-6, 6), main = expression(paste("Clayton ",
+     alpha, "  = 2")))

Clayton Density Plots

> par(mfrow = c(1, 1))
> probs <- dmvdc(mvd.clayton, x2)
> scatterplot3d(x2[, 1], x2[, 2], probs, angle = 110, color = "blue", pch = 16,
+     main = expression(paste("Clayton ", alpha, "  = 2")))

Frank Copula

> library(copula)
> n <- 5000
> param.cop <- 10
> dim.cop <- 2
> cop.frank <- archmCopula(family = "frank", param = param.cop, dim = dim.cop)
> mvd.frank <- mvdc(copula = cop.frank, margins = c("norm", "norm"), paramMargins = list(list(mean = 0,
+     sd = 2), list(mean = 0, sd = 2)))
> x1 <- rcopula(cop.frank, n)
> param.cop <- 2
> dim.cop <- 2
> cop.frank <- archmCopula(family = "frank", param = param.cop, dim = dim.cop)
> mvd.frank <- mvdc(copula = cop.frank, margins = c("norm", "norm"), paramMargins = list(list(mean = 0,
+     sd = 2), list(mean = 0, sd = 2)))
> x2 <- rcopula(cop.frank, n)
> par(mfrow = c(1, 2))
> plot(x1[, 1], x1[, 2], col = "blue", xlab = "", ylab = "", pch = 16, main = expression(paste("Frank ",
+     alpha, "  = 10")))
> plot(x2[, 1], x2[, 2], col = "blue", xlab = "", ylab = "", pch = 16, main = expression(paste("Frank ",
+     alpha, "  = 2")))

Frank Level Plots

> par(mfrow = c(1, 1))
> contour(mvd.frank, dmvdc, xlim = c(-6, 6), ylim = c(-6, 6), main = expression(paste("Clayton ",
+     alpha, "  = 2")))

Clayton Density Plots

> par(mfrow = c(1, 1))
> probs <- dmvdc(mvd.frank, x2)
> scatterplot3d(x2[, 1], x2[, 2], probs, angle = 110, color = "blue", pch = 16,
+     main = expression(paste("Clayton ", alpha, "  = 2")))

Takeaway

Gaussian - Symmetric and no tail dependence
tCopula - Symmetric and tail dependence
Gumbel - Asymmetric Upper tail dependence
Clayton - Asymmetric Lower tail dependence
Frank - Symmetric and weak tail dependence