CKT.estimate {CondCopulas}

Estimation of conditional Kendall's tau between two variables X1 and X2 given Z = z

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.

This function can use different estimators for conditional Kendall's tau, see the description of the parameter methodEstimation for a complete list of possibilities.

Usage

CKT.estimate(
  observedX1, observedX2, observedZ,
  newZ = observedZ, methodEstimation, h,
  listPhi = if(methodEstimation == "kendallReg")
               {list( function(x){return(x)}   ,
                      function(x){return(x^2)} ,
                      function(x){return(x^3)} )
               } else {list(identity)} ,
  ...)

Arguments

observedX1

a vector of n observations of the first variable

observedX2

a vector of n observations of the second variable

observedZ

a vector of n observations of the conditioning variable, or a matrix with n rows of observations of the conditioning vector (if Z is multivariate).

newZ

the new values for the conditioning variable Z at which the conditional Kendall's tau should be estimated.

• If observedZ is a vector, then newZ must be a vector as well.

• If observedZ is a matrix, then newZ must be a matrix as well, with the same number of columns ( = the dimension of Z).

methodEstimation

method for estimating the conditional Kendall's tau. Possible estimation methods are:

• "kernel": kernel smoothing, as described in (Derumigny, & Fermanian (2019a))

• "kendallReg": regression-type model, as described in (Derumigny, & Fermanian (2020))

• "tree", "randomForest", "logit", and "neuralNetwork": use the relationship between conditional Kendall's tau and classification problems to use the respective classification algorithms for the estimation of conditional Kendall's tau, as described in (Derumigny, & Fermanian (2019b))

h

the bandwidth

listPhi

the list of transformations to be applied to the conditioning variable Z (in case of regression-type models).

...

other parameters passed to the estimating functions CKT.fit.tree, CKT.fit.randomForest, CKT.fit.GLM, CKT.fit.nNets, CKT.predict.kNN, CKT.kernel and CKT.kendallReg.fit.

Value

the vector of estimated conditional Kendall's tau at each of the observations of newZ.

References

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

Derumigny, A., & Fermanian, J. D. (2019b). On kernel-based estimation of conditional Kendall’s tau: finite-distance bounds and asymptotic behavior. Dependence Modeling, 7(1), 292-321. doi:10.1515/demo-2019-0016

Derumigny, A., & Fermanian, J. D. (2020). On Kendall’s regression. Journal of Multivariate Analysis, 178, 104610. doi:10.1016/j.jmva.2020.104610

See Also

the specialized functions for estimating conditional Kendall's tau for each method: CKT.fit.tree, CKT.fit.randomForest, CKT.fit.GLM, CKT.fit.nNets, CKT.predict.kNN, CKT.fit.randomForest, CKT.kernel and CKT.kendallReg.fit.

See also the nonparametric estimator of conditional copula models estimateNPCondCopula, and the parametric estimators of conditional copula models estimateParCondCopula.

In the case where Z is discrete or in the case of discrete conditioning events, see bCond.treeCKT.

Examples

# We simulate from a conditional copula
set.seed(1)
N = 300
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])

newZ = seq(2,10,by = 0.1)
h = 0.1
estimatedCKT_tree <- CKT.estimate(
  observedX1 = X1, observedX2 = X2, observedZ = Z,
  newZ = newZ,
  methodEstimation = "tree", h = h)

estimatedCKT_rf <- CKT.estimate(
  observedX1 = X1, observedX2 = X2, observedZ = Z,
  newZ = newZ,
  methodEstimation = "randomForest", h = h)

estimatedCKT_GLM <- CKT.estimate(
  observedX1 = X1, observedX2 = X2, observedZ = Z,
  newZ = newZ,
  methodEstimation = "logit", h = h,
  listPhi = list(function(x){return(x)}, function(x){return(x^2)},
                 function(x){return(x^3)}) )

estimatedCKT_kNN <- CKT.estimate(
  observedX1 = X1, observedX2 = X2, observedZ = Z,
  newZ = newZ,
  methodEstimation = "nearestNeighbors", h = h,
  number_nn = c(50,80, 100, 120,200),
  partition = 4
  )

estimatedCKT_nNet <- CKT.estimate(
  observedX1 = X1, observedX2 = X2, observedZ = Z,
  newZ = newZ,
  methodEstimation = "neuralNetwork", h = h,
  )

estimatedCKT_kernel <- CKT.estimate(
  observedX1 = X1, observedX2 = X2, observedZ = Z,
  newZ = newZ,
  methodEstimation = "kernel", h = h,
  )

estimatedCKT_kendallReg <- CKT.estimate(
   observedX1 = X1, observedX2 = X2, observedZ = Z,
   newZ = newZ,
   methodEstimation = "kendallReg", h = h)

# Comparison between true Kendall's tau (in black)
# and estimated Kendall's tau (in other colors)
trueConditionalTau = -0.9 + 1.8 * pnorm(newZ, mean = 5, sd = 2)
plot(newZ, trueConditionalTau , col="black",
   type = "l", ylim = c(-1, 1))
lines(newZ, estimatedCKT_tree, col = "red")
lines(newZ, estimatedCKT_rf, col = "blue")
lines(newZ, estimatedCKT_GLM, col = "green")
lines(newZ, estimatedCKT_kNN, col = "purple")
lines(newZ, estimatedCKT_nNet, col = "coral")
lines(newZ, estimatedCKT_kernel, col = "skyblue")
lines(newZ, estimatedCKT_kendallReg, col = "darkgreen")