CKT.fit.GLM {CondCopulas}

Estimation of conditional Kendall's taus by penalized GLM

Description

The function CKT.fit.GLM fits a regression model for the conditional Kendall's tau \tau_{1,2|Z} between two variables X_1 and X_2 conditionally to some predictors Z. More precisely, this function fits the model

\tau_{1,2|Z} = 2 * \Lambda( \beta_0 + \beta_1 \phi_1(Z) + ... + \beta_p \phi_p(Z) )

for a link function \Lambda, and p real-valued functions \phi_1, ..., \phi_p. The function CKT.predict.GLM predicts the values of conditional Kendall's tau for some values of the conditioning variable Z.

Usage

CKT.fit.GLM(
  datasetPairs,
  designMatrix = datasetPairs[, 2:(ncol(datasetPairs) - 3), drop = FALSE],
  link = "logit",
  ...
)

CKT.predict.GLM(fit, newZ)

Arguments

datasetPairs

the matrix of pairs and corresponding values of the kernel as provided by datasetPairs.

designMatrix

the matrix of predictor to be used for the fitting of the model. It should have the same number of rows as the datasetPairs.

link

link function, can be one of logit, probit, cloglog, cauchit).

...

other parameters passed to ordinalNet::ordinalNet().

fit

result of a call to CKT.fit.GLM

newZ

new matrix of observations of the conditioning vector Z, with the same number of variables and same names as the designMatrix that was used to fit the GLM.

Value

CKT.fit.GLM returns the fitted GLM, an object with S3 class ordinalNet.

CKT.predict.GLM returns a vector of (predicted) conditional Kendall's taus of the same size as the number of rows of the matrix 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. (Algorithm 2) doi:10.1016/j.csda.2019.01.013

See Also

See also other estimators of conditional Kendall's tau: CKT.fit.tree, CKT.fit.randomForest, CKT.fit.nNets, 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 = 400
Z = rnorm(n = N, mean = 5, sd = 2)
conditionalTau = 2*plogis(-1 + 0.8*Z - 0.1*Z^2) - 1
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)
designMatrix = cbind(datasetP[,2], datasetP[,2]^2)
fitCKT_GLM <- CKT.fit.GLM(
  datasetPairs = datasetP, designMatrix = designMatrix,
  maxiterOut = 10, maxiterIn = 5)
print(coef(fitCKT_GLM))
# These are rather close to the true coefficients -1, 0.8, -0.1
# used to generate the data above.

newZ = seq(2,10,by = 0.1)
estimatedCKT_GLM = CKT.predict.GLM(
  fit = fitCKT_GLM, newZ = cbind(newZ, newZ^2))

# Comparison between true Kendall's tau (in red)
# and estimated Kendall's tau (in black)
trueConditionalTau = 2*plogis(-1 + 0.8*newZ - 0.1*newZ^2) - 1
plot(newZ, trueConditionalTau , col="red",
   type = "l", ylim = c(-1, 1))
lines(newZ, estimatedCKT_GLM)