Copula contour plot

Question

I am trying to get some nice contour plots of various Copulas with Gaussian marginals, but somehow I do not at all get what I would expect - what am I doing wrong ?

My R code:

library(copula)
library(mvtnorm)

#Gaussian Density & distribution
G_copula = norm.cop <- normalCopula(0.70)
cp <- contour(G_copula, dCopula, n=200, nlevels=20, delta=0.01)
persp(G_copula, dCopula)
contour(cp$x,cp$y,cp$z)
contour(qnorm(cp$x),qnorm(cp$y),cp$z)

What I would like to get is something like this

But what I get is this

Answer 1

If you want the contours plot of the copula with margins N(0,1) and N(0,1), one option is to "manually" define its density:

library(copula)

cop <- normalCopula(0.7)

# density of the bivariate distribution with copula 'cop' and margins N(0,1), N(0,1)
f <- function(x, y) {
  dCopula(c(pnorm(x), pnorm(y)), cop) * dnorm(x) * dnorm(y)
}

Then you can use the usual contour function.

But it's easier to create this distribution with mvdc:

library(copula)
mv <- mvdc(
  normalCopula(0.7), margins = c("norm", "norm"),
  paramMargins = list(list(mean = 0, sd = 1), list(mean = 0, sd = 1))
)

Then you can directly call contour on this multivariate distribution:

contour(mv, dMvdc, xlim = c(-3, 3), ylim = c(-3, 3), asp = 1)

Answer 2

Based on the example you've provided, this plot you are trying to recreate just looks like the bivariate normal distribution, which is used to create your copula. You can create a contour plot of the bivariate normal distribution, matching the parameters you used for your copula, as follows:

library(ggplot2)

mu <- c(0, 0)
sigma <- matrix(c(1, 0.7, 0.7, 1), 2, 2) # Use a correlation of 0.7, matching your copula
bivariate_normal <- rmvnorm(100000, mean = mu, sigma = sigma)
colnames(bivariate_normal) <- c("x", "y")

df <- tibble::as_tibble(bivariate_normal)

df |>
  ggplot(aes(x = x, y = y)) +
  geom_density_2d() +
  labs(title = "Bivariate Normal Distribution (Correlation = 0.7)",
       x = "X", y = "Y")

The output looks like this: