contord: Correlations of discretized variables

Description

The function computes the correlation matrix of the \(k\) variables, with given marginal distributions, derived discretizing a \(k\)-variate standard normal or Student's \(t\) variable with given correlation matrix

Usage

contord(marginal, Sigma,
support = list(),
Spearman = FALSE,
df=Inf,
integerdf=TRUE,
prob=FALSE)

Value

the correlation matrix of the discretized variables; if prob=TRUE, it returns a list containing the correlation along with the probability table, in case of a bivariate random variable

Arguments

marginal: a list of \(k\) elements, where \(k\) is the number of variables. The \(i\)-th element of marginal is the vector of the cumulative probabilities defining the marginal distribution of the \(i\)-th component of the multivariate variable. If the \(i\)-th component can take \(k_i\) values, the \(i\)-th element of marginal will contain \(k_i-1\) probabilities (the \(k_i\)-th is obviously 1 and shall not be included).
Sigma: the correlation matrix of the standard multivariate normal variable
support: a list of \(k\) elements, where \(k\) is the number of variables. The \(i\)-th element of support is the vector containing the ordered values of the support of the \(i\)-th variable. By default, the support of the \(i\)-th variable is \(1,2,...,k_i\)
Spearman: if TRUE, the function finds Spearman's correlations (and it is not necessary to provide support), if FALSE (default) Pearson's correlations
df: the degrees of freedom of the multivariate Student's \(t\)
integerdf: if TRUE uses pmvt, which requires an integer value for df; otherwise, uses pmvt.alt, which integrates dmvt with possibly non-integer df
prob: if TRUE, it return also the probability table of the bivariate discrete random variable

Author

Alessandro Barbiero, Pier Alda Ferrari

Examples

Run this code

# Example 1
# consider a bivariate discrete random vector
k <- 2
# with these cumulative margins
marginal <- list(c(1/3, 2/3), c(0.1, 0.3, 0.6))
# generated discretizing a multivariate standard normal variable
# with correlation matrix
Sigma <- matrix(c(1, .75, .75, 1), 2, 2)
# the resulting joint distribution and correlation matrix
# for the bivariate discrete random vector are
res <- contord(marginal, Sigma, prob=TRUE)
res$pij
res$SigmaO
# let's check the margins are those assigned
cumsum(margin.table(res$pij,1))
cumsum(margin.table(res$pij,2))
# -> OK
# Example 2
# consider 4 discrete variables
k <- 4
# with these marginal distributions
marginal <- list(0.4,c(0.3,0.6), c(0.25,0.5,0.75), c(0.1,0.2,0.8,0.9))
# generated discretizing a multivariate standard normal variable
# with correlation matrix
Sigma <- matrix(0.5,4,4)
diag(Sigma) <- 1
# the resulting correlation matrix for the discrete variables is
contord(marginal, Sigma)
# note all the correlations are smaller than the original 0.6
# change Sigma, adding a negative correlation
Sigma[1,2] <- -0.15
Sigma[2,1] <- Sigma[1,2]
Sigma
# checking whether Sigma is still positive definite
eigen(Sigma)$values # all >0, OK
contord(marginal, Sigma)
# Example 2
# the same margins and the same correlation matrix
# but now we consider a 4-variate Students's t with df=3
contord(marginal, Sigma, df=3)
# -> a slight reduction in magnitude for all the correlations