CKT.KendallReg.LambdaCV {CondCopulas}

Kendall's regression: choice of the penalization parameter by K-folds cross-validation

Description

In this model, three variables X_1, X_2 and Z are observed. We try to model the conditional Kendall's tau between X_1 and X_2 conditionally to Z=z, as follows:

\Lambda(\tau_{X_1, X_2 | Z = z}) = \sum_{i=1}^{p'} \beta_i \psi_i(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}) - \sum_{j=1}^{p'} \beta_j \psi_j(z)]^2 + \lambda * |\beta|_1.

This function chooses the penalization parameter lambda by cross-validation.

Usage

CKT.KendallReg.LambdaCV(
  observedX1,
  observedX2,
  observedZ,
  ZToEstimate,
  designMatrixZ = cbind(ZToEstimate, ZToEstimate^2, ZToEstimate^3),
  typeEstCKT = 4,
  h_lambda,
  Lambda = identity,
  kernel.name = "Epa",
  Kfolds_lambda = 10,
  l_norm = 1,
  matrixSignsPairs = NULL,
  progressBars = "global"
)

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 new data 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.

typeEstCKT

type of estimation of the conditional Kendall's tau.

h_lambda

the smoothing bandwidth used in the cross-validation procedure to choose lambda.

Lambda

the function to be applied on conditional Kendall's tau. By default, the identity function is used.

kernel.name

name of the kernel. Possible choices are "Gaussian" (Gaussian kernel) and "Epa" (Epanechnikov kernel).

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. 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.

matrixSignsPairs

the results of a call to computeMatrixSignPairs (if already computed). If NULL (the default value), the matrixSignsPairs will be computed again from the data.

progressBars

should progress bars be displayed? Possible values are

• "none": no progress bar at all.

• "global": only one global progress bar (default behavior)

• "eachStep": uses a global progress bar + one progress bar for each kernel smoothing step.

Value

A list with the following components

• lambdaCV: the chosen value of the penalization parameters lambda.

• vectorLambda: a vector containing the values of lambda that have been compared.

• vectorMSEMean: the estimated MSE for each value of lambda in vectorLambda

• vectorMSESD: the estimated standard deviation of the MSE for each lambda. It can be used to construct confidence intervals for estimates of the MSE given by vectorMSEMean.

References

Derumigny, A., & Fermanian, J. D. (2020). On Kendall’s regression. Journal of Multivariate Analysis, 178, 104610.

See Also

the main fitting function CKT.kendallReg.fit.

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)
result <- CKT.KendallReg.LambdaCV(
   observedX1 = X1, observedX2 = X2, observedZ = Z,
   ZToEstimate = newZ, h_lambda = 2)

plot(x = result$vectorLambda, y = result$vectorMSEMean,
     type = "l", log = "x")