CKT.fit.tree: Estimation of conditional Kendall's taus using a classification tree

Description

Let $X_1$ and $X_2$ be two random variables. The goal of this function is to estimate the conditional Kendall's tau (a dependence measure) between $X_1$ and $X_2$ given $Z=z$ for a conditioning variable $Z$. Conditional Kendall's tau between $X_1$ and $X_2$ given $Z=z$ is defined as: $$P( (X_{1,1} - X_{2,1})(X_{1,2} - X_{2,2}) > 0 | Z_1 = Z_2 = z)$$ $$- P( (X_{1,1} - X_{2,1})(X_{1,2} - X_{2,2}) < 0 | Z_1 = Z_2 = z),$$ where $(X_{1,1}, X_{1,2}, Z_1)$ and $(X_{2,1}, X_{2,2}, Z_2)$ are two independent and identically distributed copies of $(X_1, X_2, Z)$. In other words, conditional Kendall's tau is the difference between the probabilities of observing concordant and discordant pairs from the conditional law of $$(X_1, X_2) | Z=z.$$ These functions estimate and predict conditional Kendall's tau using a classification tree. This is possible by the relationship between estimation of conditional Kendall's tau and classification problems (see Derumigny and Fermanian (2019)): estimation of conditional Kendall's tau is equivalent to the prediction of concordance in the space of pairs of observations.

Usage

CKT.fit.tree(datasetPairs, mindev = 0.008, mincut = 0)
CKT.predict.tree(fit, newZ)

Value

CKT.fit.tree returns the fitted tree.

CKT.predict.tree returns a vector of (predicted) conditional Kendall's taus of the same size as the number of rows of newZ.

Arguments

datasetPairs: the matrix of pairs and corresponding values of the kernel as provided by datasetPairs.
mindev: a factor giving the minimum deviation for a node to be splitted. See tree::tree.control() for more details.
mincut: the minimum number of observations (of pairs) in a node See tree::tree.control() for more details.
fit: result of a call to CKT.fit.tree
newZ: new matrix of observations, with the same number of variables. and same names as the designMatrix that was used to fit the tree.

References

Derumigny, A., & Fermanian, J. D. (2019). A classification point-of-view about conditional Kendall’s tau. Computational Statistics & Data Analysis, 135, 70-94. (Section 3.2) tools:::Rd_expr_doi("10.1016/j.csda.2019.01.013")

Examples

Run this code

# We simulate from a conditional copula
set.seed(1)
N = 800
Z = rnorm(n = N, mean = 5, sd = 2)
conditionalTau = -0.9 + 1.8 * pnorm(Z, mean = 5, sd = 2)
simCopula = VineCopula::BiCopSim(N=N , family = 1,
    par = VineCopula::BiCopTau2Par(1 , conditionalTau ))
X1 = qnorm(simCopula[,1])
X2 = qnorm(simCopula[,2])

datasetP = datasetPairs(X1 = X1, X2 = X2, Z = Z, h = 0.07, cut = 0.9)
est_Tree = CKT.fit.tree(datasetPairs = datasetP, mindev = 0.008)
print(est_Tree)

newZ = seq(1,10,by = 0.1)
prediction = CKT.predict.tree(fit = est_Tree, newZ = data.frame(x=newZ))
# Comparison between true Kendall's tau (in red)
# and estimated Kendall's tau (in black)
plot(newZ, prediction, type = "l", ylim = c(-1,1))
lines(newZ, -0.9 + 1.8 * pnorm(newZ, mean = 5, sd = 2), col="red")