In ajmcneil/extremalcopula: Kendall Concordance Signatures

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

Polytope of weights

We will find the cut polytope of possible concordance probabilities in 3 dimensions. First, we find the vertices of the polytope of weights. These take a trivial form: they are the 4 vertices of the 4-dimensional simplex.

sig <- 1
names(sig) <- "empty"
weights <- findpolytope(sig, d = 3)
weights

Polytope of concordance probabilities

Now we determine the correponding vertices of the polytope of concordance probabilities for the 3 pairs of random variables. There are 4 vertices: 3 vertices in which a single pair has concordance probability one and the others have concordance probability zero; 1 vertex in which all pairs have concordance probability 1.

allkappa <- Amatrix(3) %*% t(weights)
k.vertices <- t(allkappa[-1, ])
k.vertices

We can display the polytope.

plot3d(k.vertices)
plotHull3D(k.vertices)

Polytope of Kendall rank correlations (cut polytope)

We repeat the exercise for the polytope that describes the possible Kendall rank correlation matrices. This is the cut polytope.

alltau <- 2*allkappa -1
t.vertices <- t(alltau[-1, ])
t.vertices
plot3d(t.vertices)
plotHull3D(t.vertices)

An unattainable matrix

Here is a positive-definite correlation matrix.

rho <- -5/12
P <- matrix(rho, nrow = 3, ncol = 3) + (1-rho)*diag(3)
P
eigen(P)$values

If this is a Kendall's tau matrix then the concordance signature is given by.

kappa <- c(1, (1 + P[lower.tri(P)])/2)
kappa

The A matrix in 3 dimensions is easily computed.

Amatrix(3)

The extremal mixture weights must be:

solve(Amatrix(3), kappa)

But unfortunately this yields a negative weight, which is a contradiction. Observe how the point $(-5/12, -5/12, -5/12)$ is not in the cut polytope.

plotHull3D(c(-5/12, -5/12, -5/12))

An attainable Kendall matrix

Let's do the whole thing again but with a different rho value.

rho <- -1/3
P <- matrix(rho, nrow = 3, ncol = 3) + (1-rho)*diag(3)
P
eigen(P)$values

If this is a Kendall's tau matrix then the concordance signature is given by.

kappa <- c(1, (1 + P[lower.tri(P)])/2)
kappa

The A matrix in 3 dimensions is easily computed.

Amatrix(3)

The extremal mixture weights must be:

solve(Amatrix(3), kappa)

This works fine. Note how the solution attaches zero weight to the first extremal copula and equal weight to the others. It is right on the boundary of the cut polytope.