CKT.fit.nNets: Estimation of conditional Kendall's taus by model averaging of neural networks

Description

Let $X_1$ and $X_2$ be two random variables. The goal of this function is to estimate the conditional Kendall's tau (a dependence measure) between $X_1$ and $X_2$ given $Z=z$ for a conditioning variable $Z$. Conditional Kendall's tau between $X_1$ and $X_2$ given $Z=z$ is defined as: $$P( (X_{1,1} - X_{2,1})(X_{1,2} - X_{2,2}) > 0 | Z_1 = Z_2 = z)$$ $$- P( (X_{1,1} - X_{2,1})(X_{1,2} - X_{2,2}) < 0 | Z_1 = Z_2 = z),$$ where $(X_{1,1}, X_{1,2}, Z_1)$ and $(X_{2,1}, X_{2,2}, Z_2)$ are two independent and identically distributed copies of $(X_1, X_2, Z)$. In other words, conditional Kendall's tau is the difference between the probabilities of observing concordant and discordant pairs from the conditional law of $$(X_1, X_2) | Z=z.$$ This function estimates conditional Kendall's tau using neural networks. This is possible by the relationship between estimation of conditional Kendall's tau and classification problems (see Derumigny and Fermanian (2019)): estimation of conditional Kendall's tau is equivalent to the prediction of concordance in the space of pairs of observations.

Usage

CKT.fit.nNets(
  datasetPairs,
  designMatrix = datasetPairs[, 2:(ncol(datasetPairs) - 3), drop = FALSE],
  vecSize = rep(3, times = 10),
  nObs_per_NN = 0.9 * nrow(designMatrix),
  verbose = 1
)

Value

CKT.fit.nNets returns a list of the fitted neural networks

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 tree

vecSize

vector with the number of neurons for each network

nObs_per_NN

number of observations used for each neural network.

verbose

a number indicated what to print

0: nothing printed at all.
1: a message is printed at the convergence of each neural network.
2: details are printed for each optimization of each network.

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 7) 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 = 800
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)
datasetP = datasetPairs(X1 = X1, X2 = X2, Z = Z, h = 0.07, cut = 0.9)

fitCKT_nets <- CKT.fit.nNets(datasetPairs = datasetP)
estimatedCKT_nNets <- CKT.predict.nNets(
  fit = fitCKT_nets, newZ = matrix(newZ, ncol = 1))

# 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_nNets, col = "red")