CKT.fit.tree {CondCopulas}

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)

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.

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.

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) doi:10.1016/j.csda.2019.01.013

See Also

See also other estimators of conditional Kendall's tau: CKT.fit.nNets, CKT.fit.randomForest, CKT.fit.GLM, CKT.predict.kNN, CKT.kernel, CKT.kendallReg.fit, and the more general wrapper CKT.estimate.

Examples

# 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")