CKT.KendallReg.LambdaCV: 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(
  X1 = NULL,
  X2 = NULL,
  Z = NULL,
  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",
  observedX1 = NULL,
  observedX2 = NULL,
  observedZ = NULL
)

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.

Arguments

X1

a vector of n observations of the first variable $X_1$.

X2

a vector of n observations of the second variable $X_2$.

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).

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.

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. (2020). On Kendall’s regression. Journal of Multivariate Analysis, 178, 104610.

Examples

Run this code

# 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(X1 = X1, X2 = X2, Z = Z,
                                  ZToEstimate = newZ, h_lambda = 2)

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