R: Fit Kendall's regression, a GLM-type model for conditional...

CKT.kendallReg.fit {CondCopulas}

R Documentation

Fit Kendall's regression, a GLM-type model for conditional Kendall's tau

Description

The function CKT.kendallReg.fit fits a regression-type model for the conditional Kendall's tau between two variables X_1 and X_2 conditionally to some predictors Z. More precisely, it fits the model

\Lambda(\tau_{X_1, X_2 | Z = z}) = \sum_{j=1}^{p'} \beta_j \psi_j(z),

where \tau_{X_1, X_2 | Z = z} is the conditional Kendall's tau between X_1 and X_2 conditionally to Z=z, \Lambda is a function from ]-1, 1] to R, (\beta_1, \dots, \beta_p) are unknown coefficients to be estimated and \psi_1, \dots, \psi_{p'}) are a dictionary of functions. To estimate beta, we used the penalized estimator which is defined as the minimizer of the following criteria

\frac{1}{2n'} \sum_{i=1}^{n'} [\Lambda(\hat\tau_{X_1, X_2 | Z = z_i}) - \sum_{j=1}^{p'} \beta_j \psi_j(z_i)]^2 + \lambda * |\beta|_1,

where the z_i are a second sample (here denoted by ZToEstimate).

The function CKT.kendallReg.predict predicts the conditional Kendall's tau between two variables X_1 and X_2 given Z=z for some new values of z.

Usage

CKT.kendallReg.fit(
  observedX1,
  observedX2,
  observedZ,
  ZToEstimate,
  designMatrixZ = cbind(ZToEstimate, ZToEstimate^2, ZToEstimate^3),
  newZ = designMatrixZ,
  h_kernel,
  Lambda = identity,
  Lambda_inv = identity,
  lambda = NULL,
  Kfolds_lambda = 10,
  l_norm = 1,
  h_lambda = h_kernel,
  ...
)

CKT.kendallReg.predict(fit, newZ, lambda = NULL, Lambda_inv = identity)

Arguments

`observedX1`	a vector of `n` observations of the first variable `X_1`.
`observedX2`	a vector of `n` observations of the second variable `X_2`.
`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).
`ZToEstimate`	the intermediary dataset of observations of `Z` at which the conditional Kendall's tau should be estimated.
`designMatrixZ`	the transformation of the `ZToEstimate` that will be used as predictors. By default, no transformation is applied.
`newZ`	the new observations of the conditioning variable.
`h_kernel`	bandwidth used for the first step of kernel smoothing.
`Lambda`	the function to be applied on conditional Kendall's tau. By default, the identity function is used.
`Lambda_inv`	the functional inverse of `Lambda`. By default, the identity function is used.
`lambda`	the regularization parameter. If `NULL`, then it is chosen by K-fold cross validation. Internally, cross-validation is performed by the function `CKT.KendallReg.LambdaCV`.
`Kfolds_lambda`	the number of folds used in the cross-validation procedure to choose `lambda`.
`l_norm`	type of norm used for selection of the optimal lambda by cross-validation. `l_norm=1` corresponds to the sum of absolute values of differences between predicted and estimated conditional Kendall's tau while `l_norm=2` corresponds to the sum of squares of differences.
`h_lambda`	the smoothing bandwidth used in the cross-validation procedure to choose `lambda`.
`...`	other arguments to be passed to `CKT.kernel` for the first step (kernel-based) estimator of conditional Kendall's tau.
`fit`	the fitted model, obtained by a call to `CKT.kendallReg.fit`.

Value

The function CKT.kendallReg.fit returns a list with the following components:

estimatedCKT: the estimated CKT at the new data points newZ.
fit: the fitted model, of S3 class glmnet (see glmnet::glmnet for more details).
lambda: the value of the penalized parameter used. (i.e. either the one supplied by the user or the one determined by cross-validation)

CKT.kendallReg.predict returns the predicted values of conditional Kendall's tau.

References

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

Examples

# We simulate from a conditional copula
set.seed(1)
N = 400
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)
estimatedCKT_kendallReg <- CKT.kendallReg.fit(
   observedX1 = X1, observedX2 = X2, observedZ = Z,
   ZToEstimate = newZ, h_kernel = 0.07)

coef(estimatedCKT_kendallReg$fit,
     s = estimatedCKT_kendallReg$lambda)

# Comparison between true Kendall's tau (in black)
# and estimated Kendall's tau (in red)
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_kendallReg$estimatedCKT, col = "red")

[Package CondCopulas version 0.1.3 Index]