CKTmatrix.kernel {CondCopulas}

Estimate the conditional Kendall's tau matrix at different conditioning points

Description

Assume that we are interested in a random vector (X, Z), where X is of dimension d > 2 and Z is of dimension 1. We want to estimate the dependence across the elements of the conditioned vector X given Z=z. This function takes in parameter observations of (X,Z) and returns kernel-based estimators of

\tau_{i,j | Z=zk}

which is the conditional Kendall's tau between X_i and X_j given to Z=zk, for every conditioning point zk in gridZ. If the conditional Kendall's tau matrix has a block structure, then improved estimation is possible by averaging over the kernel-based estimators of pairwise conditional Kendall's taus. Groups of variables composing the same blocks can be defined using the parameter blockStructure, and the averaging can be set on using the parameter averaging=all, or averaging=diag for faster estimation by averaging only over diagonal elements of each block.

Usage

CKTmatrix.kernel(
  dataMatrix,
  observedZ,
  gridZ,
  averaging = "no",
  blockStructure = NULL,
  h,
  kernel.name = "Epa",
  typeEstCKT = "wdm"
)

Arguments

dataMatrix

a matrix of size (n,d) containing n observations of a d-dimensional random vector X.

observedZ

vector of observed points of a conditioning variable Z. It must have the same length as the number of rows of dataMatrix.

gridZ

points at which the conditional Kendall's tau is computed.

averaging

type of averaging used for fast estimation. Possible choices are

• no: no averaging;

• all: averaging all Kendall's taus in each block;

• diag: averaging along diagonal blocks elements.

blockStructure

list of vectors. Each vector corresponds to one group of variables and contains the indexes of the variables that belongs to this group. blockStructure must be a partition of 1:d, where d is the number of columns in dataMatrix.

h

bandwidth. It can be a real, in this case the same h will be used for every element of gridZ. If h is a vector then its elements are recycled to match the length of gridZ.

kernel.name

name of the kernel used for smoothing. Possible choices are: "Gaussian" (Gaussian kernel) and "Epa" (Epanechnikov kernel).

typeEstCKT

type of estimation of the conditional Kendall's tau.

Value

array with dimensions depending on averaging:

• If averaging = "no": it returns an array of dimensions (n, n, length(gridZ)), containing the estimated conditional Kendall's tau matrix given Z = z. Here, n is the number of rows in dataMatrix.

• If averaging = "all" or "diag": it returns an array of dimensions (length(blockStructure), length(blockStructure), length(gridZ)), containing the block estimates of the conditional Kendall's tau given Z = z with ones on the diagonal.

Author(s)

Rutger van der Spek, Alexis Derumigny

References

van der Spek, R., & Derumigny, A. (2022). Fast estimation of Kendall's Tau and conditional Kendall's Tau matrices under structural assumptions. arxiv:2204.03285.

See Also

CKT.kernel for kernel-based estimation of conditional Kendall's tau between two variables (i.e. the equivalent of this function when X is bivariate and d=2).

Examples

# Data simulation
n = 100
Z = runif(n)
d = 5
CKT_11 = 0.8
CKT_22 = 0.9
CKT_12 = 0.1 + 0.5 * cos(pi * Z)
data_X = matrix(nrow = n, ncol = d)
for (i in 1:n){
  CKT_matrix = matrix(data =
    c(  1      , CKT_11   , CKT_11   , CKT_12[i], CKT_12[i] ,
      CKT_11   ,   1      , CKT_11   , CKT_12[i], CKT_12[i] ,
      CKT_11   , CKT_11   ,    1     , CKT_12[i], CKT_12[i] ,
      CKT_12[i], CKT_12[i], CKT_12[i],   1      , CKT_22    ,
      CKT_12[i], CKT_12[i], CKT_12[i], CKT_22   ,   1
      ) ,
     nrow = 5, ncol = 5)
  sigma = sin(pi * CKT_matrix/2)
  data_X[i, ] = mvtnorm::rmvnorm(n = 1, sigma = sigma)
}
plot(as.data.frame.matrix(data_X))

# Estimation of CKT matrix
h = 1.06 * sd(Z) * n^{-1/5}
gridZ = c(0.2, 0.8)
estMatrixAll <- CKTmatrix.kernel(
  dataMatrix = data_X, observedZ = Z, gridZ = gridZ, h = h)
# Averaging estimator
estMatrixAve <- CKTmatrix.kernel(
  dataMatrix = data_X, observedZ = Z, gridZ = gridZ,
  averaging = "diag", blockStructure = list(1:3,4:5), h = h)

# The estimated CKT matrix conditionally to Z=0.2 is:
estMatrixAll[ , , 1]
# Using the averaging estimator,
# the estimated CKT between the first group (variables 1 to 3)
# and the second group (variables 4 and 5) is
estMatrixAve[1, 2, 1]

# True value (of CKT between variables in block 1 and 2 given Z = 0.2):
0.1 + 0.5 * cos(pi * 0.2)