In ajmcneil/KendallSignature: Kendall Concordance Signatures

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

Estimating concordance signature from data

Consider a Gaussian copula with the following correlation matrix.

P <- copula::p2P((1:10) / 11)
P

This is the true concordance signature.

consignature(P)

We can generate some data from the copula and estimate the signature from the data.

set.seed(17)
data <- rmvnorm(1000, sigma = P)
kappa <- estsignature(data)
kappa

Note how this method gives the same Kendall's tau matrix as the standard estimator. This is only true in the absence of ties.

tau <- c((2*kappa - 1)[2 : 11], (8*kappa - 1)[12 : 16]/7)
tau
cor(data, method = "kendall")

Estimating concordance signature for real data

In the next example we take some cryptocurrency data. The dataset is 4-dimensional and contains the prices of Bitcoin, Ethereum, Litecoin and Ripple is US dollars. We look at the daily log-returns for the year 2017.

data(crypto)
data <- crypto['2016-12-31/2017-12-31']
X <- (diff(log(data))[-1]) * 100
plot(X, multi.panel=TRUE)

Now we estimate the concordance signature and find an extremal mixture copula that matches the estimated signature.

fullsig <- estsignature(as.matrix(X))
fullsig
findextremal(fullsig)

Inferring Kendall correlations for missing variables

Let us suppose that we had no information about the fourth cryptocurrency, Ripple, and had to infer possible rank correlation values from knowledge of the other three.

We want to find the concordance signatures that are consistent with our existing knowledge. We first find the vertices of the polytope of possible weight vectors.

sig_3missing <- fullsig[c("empty", "{1,2}", "{1,3}", "{2,3}")]
sig_3missing
d <- 4
vert <- findpolytope(sig_3missing, d = d)
nvert <- dim(vert)[1]
vert

We can check that each of these vertices is a concordance signature that matches the given values of Kendall's tau. Here we consider the final vertex.

output <- vector("list", length = nvert)
for (i in 1:nvert)
  output[[i]] <- extmixcorr(vert[i,])
output[[nvert]] 
2 * sig_3missing - 1

We now find the set of possible values for the Kendall rank correlations of the pairs ${1,4}$, ${2,3}$ and ${3,4}$.

kappas <- Amatrix(d) %*% t(vert)
kappas3 <- t(kappas[c("{1,4}", "{2,4}", "{3,4}"), ])
taus3 <- 2*kappas3 -1
taus3_hull <- taus3[convexHull(taus3)$pts[,"vtx"],]
taus3_hull <- round(taus3_hull, 10)
plot3d(taus3_hull)
plotHull3D(taus3_hull, drawPoints = TRUE, drawLines=FALSE, drawPolygons=TRUE)

If an expert gave his opinion on the values of the Kendall rank correlations between Ripple and the other cryptocurrencies, the resulting point would have to lie within this 3-dimensional convex polytope.

Now suppose that we are told that $\tau_{{1,4}} =$ 0.5978323 (the actual estimated value). This is possible because the plane it specifies intersects with the polytope. The set of attainable values for the remaining rank correlations is a convex planar set as shown below and, unsuprisingly, the true estimated values lie within it.

sig_2missing <- fullsig[c("empty", "{1,2}", "{1,3}", "{1,4}", "{2,3}")]
sig_2missing
vert <- findpolytope(sig_2missing, d = d)
nvert <- dim(vert)[1]
kappas <- Amatrix(d) %*% t(vert)
kappas2 <- t(kappas[c("{2,4}", "{3,4}"), ])
taus2 <- 2*kappas2 -1
taus2_hull <- taus2[convexHull(taus2)$pts[,"vtx"],]
plot(taus2_hull, col = 2, xlim = c(-1,1), ylim = c(-1,1))
lines(rbind(taus2_hull, taus2_hull[1,]))
points(2*fullsig["{2,4}"]-1, 2*fullsig["{3,4}"]-1)

We can add this set to our 3-dimensional plot.

component14 <- rep(as.numeric(2*fullsig["{1,4}"]-1), dim(taus2_hull)[1])
plane_pts <- cbind(component14, taus2_hull)
dimnames(plane_pts)[[2]] <- c("{1,4}", "{2,4}", "{3,4}")
plotHull3D(plane_pts)