BiCopGradMMD: Computation of the gradient of the MMD criterion for parametric bivariate copulas models

Description

This function computes a stochastic estimate of the gradient of the MMD criterion for parametric estimation of bidimensional copula family. The main arguments are the two vectors of observations, and the copula family. The family is parametrized as in VineCopula::BiCop(), using the Kendall's tau instead of the first parameter. This function is used by BiCopEstMMD() to perform parameter estimation via MMD minimization.

Usage

BiCopGradMMD(
  u1,
  u2,
  family,
  tau,
  par = NULL,
  par2 = 0,
  kernel = "gaussian.Phi",
  gamma = 0.95,
  alpha = 1,
  epsilon = 1e-04,
  method = "QMCV",
  quasiRNG = "sobol",
  ndrawings = 10
)

Arguments

vector of observations of the first coordinate, in \([0,1]\).

vector of observations of the second coordinate, in \([0,1]\).

family

the chosen family of copulas (see the documentation of the class VineCopula::BiCop() for the available families).

tau

the copula family can be parametrized by the parameter par or by Kendall's tau. This function assumes a Kendall tau parametrization. Thus, the user can choose the value of Kendall tau at which the stochastic gradient should be computed.

par

if different from NULL, the user must instead of tau specify the corresponding parameter par. The value of tau is then ignored.

par2

value for the second parameter, if any. (Works only for Student copula).

kernel

the kernel used in the MMD distance: it can be a function taking in parameter (u1, u2, v1, v2, gamma, alpha) or a name giving the kernel to use in the list:

"gaussian": Gaussian kernel \(k(x,y) = \exp(-\|\frac{x-y}{\gamma}\|_2^2) \)
"exp-l2": \(k(x,y) = \exp(-\|\frac{x-y}{\gamma}\|_2) \)
"exp-l1": \(k(x,y) = \exp(-\|\frac{x-y}{\gamma}\|_1) \)
"inv-l2": \(k(x,y) = 1/(1+\|\frac{x-y}{\gamma}\|_2)^\alpha \)
"inv-l1": \(k(x,y) = 1/(1+\|\frac{x-y}{\gamma}\|_1)^\alpha \)

Each of these names can receive the suffix ".Phi", such as "gaussian.Phi" to indicates that the kernel \(k(x,y)\) is replaced by \(k(\Phi^{-1}(x) , \Phi^{-1}(y))\) where \(\Phi^{-1}\) denotes the quantile function of the standard Normal distribution.

gamma

parameter \(\gamma\) to be used in the kernel.

alpha

parameter \(\alpha\) to be used in the kernel, if any.

epsilon

the differential of VineCopula::BiCopTau2Par() is computed thanks to a finite difference with increment epsilon.

method

the method of computing the stochastic gradient:

MC: classical Monte-Carlo with ndrawings replications.
QMCV: usual Monte-Carlo on U with ndrawings replications, quasi Monte-Carlo on V.

quasiRNG

a function giving the quasi-random points in \([0,1]^2\) or a name giving the method to use in the list:

sobol: use of the Sobol sequence implemented in randtoolbox::sobol
halton: use of the Halton sequence implemented in randtoolbox::halton
torus: use of the Torus sequence implemented in randtoolbox::torus

ndrawings

number of replicas of the stochastic estimate of the gradient drawn at each step. The gradient is computed using the average of these replicas.

Value

the value of the gradient.

References

Alquier, P., Ch<U+00E9>rief-Abdellatif, B.-E., Derumigny, A., and Fermanian, J.D. (2022). Estimation of copulas via Maximum Mean Discrepancy. Journal of the American Statistical Association, 10.1080/01621459.2021.2024836.

Examples

Run this code

# NOT RUN {
# Simulation from a bivariate Gaussian copula with correlation 0.5.
dataSampled = VineCopula::BiCopSim(N = 500, family = 1, par = 0.5)

# computation of the gradient of the MMD criteria at different points
# Gradient is small at the true parameter
BiCopGradMMD(dataSampled[,1], dataSampled[,2], family = 1, par = 0.5)
# Gradient is negative when below the parameter
BiCopGradMMD(dataSampled[,1], dataSampled[,2], family = 1, par = 0.1)
# and positive when above
BiCopGradMMD(dataSampled[,1], dataSampled[,2], family = 1, par = 0.8)

# }