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Interval-valued Covariance

Source: R/int_cov.r
int_cov.Rd

Calculate the interval-valued covariance matrix based on the covariance matrices of the centers and ranges or data.

Usage

int_cov(
  data = NULL,
  sigma_cc = NULL,
  sigma_rr = NULL,
  sigma_cr = NULL,
  LatentParam = NULL,
  LatentCase = c("U_id_symmetric", "U_id", "General")
)

Arguments

data

An intData object containing the macrodata/interval data.

sigma_cc

Covariance matrix of the centers.

sigma_rr

Covariance matrix of the ranges.

sigma_cr

Covariance matrix between the centers and ranges.

LatentParam

A list with the parameters of the latent variables.

LatentCase

A string specifying which of the three scenarios applies to the latent variables:

  • "General": The case where the latent variables do not have any nice properties.

  • "U_id": The case where the latent variables are identically distributed.

  • "U_id_symmetric": The case where the latent variables are identically distributed and symmetric.

Defaults to "U_id_symmetric".

Value

The symbolic covariance matrix.

Details

This function calculates the interval-valued covariance matrix, \(\boldsymbol{\Sigma}_B\), based on the covariance matrices of the centers, \(\boldsymbol{\Sigma}_{CC}\), ranges, \(\boldsymbol{\Sigma}_{RR}\), and the covariance matrix between the centers and ranges, \(\boldsymbol{\Sigma}_{CR}=\boldsymbol{\Sigma}_{RC}^\top\). The covariance matrix is defined according to the LatentCase:

  • "U_id_symmetric": The latent variables are identically distributed and symmetric: $$\boldsymbol{\Sigma}_B=\boldsymbol{\Sigma}_{CC}+\delta\boldsymbol{\Sigma}_{RR},$$ where \(\delta=\mathbb{E}(U^2)/4\) is the parameter of the latent variables.

  • "U_id": The latent variables are identically distributed: $$\boldsymbol{\Sigma}_B=\boldsymbol{\Sigma}_{CC}+\delta\boldsymbol{\Sigma}_{RR}+\dfrac{\mathbb{E}(U)}{2}\left(\boldsymbol{\Sigma}_{CR}+\boldsymbol{\Sigma}_{RC}\right),$$ where \(\delta=\mathbb{E}(U^2)/4\) and \(\mathbb{E}(U)\) are the parameters of the latent variables.

  • "General": The latent variables do not have any nice properties: $$\boldsymbol{\Sigma}_B=\boldsymbol{\Sigma}_{CC}+\dfrac{1}{4}\boldsymbol{\mathfrak{E}}_{UU}\bullet\boldsymbol{\Sigma}_{RR}+\dfrac{1}{2}\boldsymbol{\Sigma}_{CR}\boldsymbol{\Psi}+\dfrac{1}{2}\boldsymbol{\Psi}\boldsymbol{\Sigma}_{RC}$$ where:

    • \(\boldsymbol{\Psi}=\text{diag}(\mathbb{E}(U_1),\dots,\mathbb{E}(U_p))\),

    • \([\boldsymbol{\mathfrak{E}}_{UU}]_{ij}=\mathcal{E}(U_i,U_j)\), \(i\neq j\), with \(\mathcal{E}(U_i,U_j)=\int_0^1 F_{U_i}^{-1}(t) F_{U_j}^{-1}(t) \, dt\),

    • \([\boldsymbol{\mathfrak{E}}_{UU}]_{ii}=\mathbb{E}(U_i^2)\), \(i,j=1,\dots,p\),

    • \(\bullet\) denotes the Schur (or entrywise) product of matrices.

References

Oliveira, M. R., Pinheiro, D., & Oliveira, L. (2025). Location and association measures for interval-valued data based on Mallows' distance. arXiv preprint arXiv:2407.05105. https://arxiv.org/abs/2407.05105

Examples

data(creditcard)
credit_card_int <- creditcard$intData

credit_card_cov<-int_cov(credit_card_int)