In ajmcneil/KendallSignature: Kendall Concordance Signatures

library(KendallSignature)
library(rgl)
library(gMOIP)
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
knitr::knit_hooks$set(webgl = hook_webgl)

Matrix $B_d$ for equiconcordant copulas

We generate the matrix $B_d$ when $d=7$.

d <- 7
Bmatrix(d)

We can display the polytope of attainable skeletal signatures for equiconcordant (or exchangeable) copulas.

v <- t(Bmatrix(d)[-1,])
plot3d(v,  xlim = c(0,1))
plotHull3D(v, drawPoints = TRUE, drawLines=TRUE, drawPolygons=TRUE)

We can add a curve to show the possible signatures of exchangeable elliptical copulas when $d=7$.

rho <- seq(from = -1/6, to = 1-1e-10, length=60)
output <- matrix(NA, nrow = length(rho), ncol = floor(d/2))
for (i in 1:length(rho)){
  P <- matrix(rho[i], nrow = d, ncol = d)
  diag(P) <- rep(1,d)
  kappa <- consignature(P)
  kappas <- 2*(0:(floor(d/2)))
  kapparows <- cumsum(choose(d, kappas))
  output[i,] <- kappa[kapparows[-1]]
}
points3d(output, lty = 2)
lines3d(output)