CKT.fit.GLM: 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)

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.

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.

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) 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 = 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)