CKT.estimate: 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(
  X1 = NULL, X2 = NULL, Z = NULL,
  newZ = Z, methodEstimation, h,
  listPhi = if(methodEstimation == "kendallReg")
               {list( function(x){return(x)}   ,
                      function(x){return(x^2)} ,
                      function(x){return(x^3)} )
               } else {list(identity)} ,
  ... ,
  observedX1 = NULL, observedX2 = NULL, observedZ = NULL )

Value

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

Arguments

X1

a vector of $n$ observations of the first variable

X2

a vector of $n$ observations of the second variable

Z

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.

observedX1, observedX2, observedZ

old parameter names for X1, X2, Z. Support for this will be removed at a later version.

References

Derumigny, A., & Fermanian, J. D. (2019a). A classification point-of-view about conditional Kendall’s tau. Computational Statistics & Data Analysis, 135, 70-94. tools:::Rd_expr_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. tools:::Rd_expr_doi("10.1515/demo-2019-0016")

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

Examples

Run this code

# 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(2,8,by = 0.1)
h = 0.1
estimatedCKT_tree <- CKT.estimate(
  X1 = X1, X2 = X2, Z = Z,
  newZ = newZ,
  methodEstimation = "tree", h = h)

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

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

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

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

estimatedCKT_kendallReg <- CKT.estimate(
   X1 = X1, X2 = X2, Z = 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")