ElliptCopulas (version 0.1.4.1)

KTMatrixEst: Fast estimation of Kendall's tau matrix

Description

Estimate Kendall's tau matrix using averaging estimators. Under the structural assumption that Kendall's tau matrix is block-structured with constant values in each off-diagonal block, this function estimates Kendall's tau matrix ``fast'', in the sense that each interblock coefficient is estimated in time \(N \cdot n \cdot log(n)\), where N is the amount of pairs that are averaged.

Usage

KTMatrixEst(dataMatrix, blockStructure = NULL, averaging = "no", N = NULL)

Value

matrix with dimensions depending on averaging.

• If averaging = no, the function returns a matrix of dimension (n,n) which estimates the Kendall's tau matrix.

• Else, the function returns a matrix of dimension (length(blockStructure) , length(blockStructure)) giving the estimates of the Kendall's tau for each block with ones on the diagonal.

Arguments

dataMatrix

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

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.

averaging

type of averaging used for fast estimation. Possible choices are

• no: no averaging;

• all: averaging all Kendall's taus in each block. N is then the number of entries in the block, i.e. the products of both dimensions.

• diag: averaging along diagonal blocks elements. N is then the minimum of the block's dimensions.

• row: averaging Kendall's tau along the smallest block side. N is then the minimum of the block's dimensions.

• random: averaging Kendall's taus along a random sample of N entries of the given block. These entries are chosen uniformly without replacement.

N

number of entries to average (n the random case. By default, N is then the minimum of the block's dimensions.

Author

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.

Examples

# Estimating off-diagonal block Kendall's taus
matrixCor = matrix(c(1  , 0.5, 0.3 ,0.3, 0.3,
                     0.5,   1, 0.3, 0.3, 0.3,
                     0.3, 0.3,   1, 0.5, 0.5,
                     0.3, 0.3, 0.5,   1, 0.5,
                     0.3, 0.3, 0.5, 0.5,   1), ncol = 5 , nrow = 5)
dataMatrix = mvtnorm::rmvnorm(n = 100, mean = rep(0, times = 5), sigma = matrixCor)
blockStructure = list(1:2, 3:5)
estKTMatrix = list()
estKTMatrix$all = KTMatrixEst(dataMatrix = dataMatrix,
                              blockStructure = blockStructure,
                              averaging = "all")
estKTMatrix$row = KTMatrixEst(dataMatrix = dataMatrix,
                              blockStructure = blockStructure,
                              averaging = "row")
estKTMatrix$diag = KTMatrixEst(dataMatrix = dataMatrix,
                               blockStructure = blockStructure,
                               averaging = "diag")
estKTMatrix$random = KTMatrixEst(dataMatrix = dataMatrix,
                                 blockStructure = blockStructure,
                                 averaging = "random", N = 2)
InterBlockCor = lapply(estKTMatrix, FUN = function(x) {sin(x[1,2] * pi / 2)})

# Estimation of the correlation between variables of the first group
# and of the second group
print(unlist(InterBlockCor))
# to be compared with the true value: 0.3.