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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. If data is provided, the covariance matrix is calculated based on the the sample covariance of the centers and ranges and the sample covariance between centers and ranges, and the parameters of the latent variables contained in the intData object. If data is not provided, the covariance matrix is calculated based on sigma_cc, sigma_rr, sigma_cr, LatentParam, and LatentCase.

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. Expects a list with a single number if LatentCase is "U_id_symmetric", a list of two numbers if LatentCase is "U_id", and a list of two matrices if LatentCase is "General".

LatentCase

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

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

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

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

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:

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

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

    • \([\boldsymbol{\mathfrak{E}}_{UU}]_{jj}=\mathbb{E}(U_j^2)\), \(j,\ell=1,\dots,p\),

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

The covariance matrix can be calculated either based on the covariance matrices of the centers and ranges or based on the data. If the data is provided, the covariance matrices are calculated using the sample covariance of the centers and ranges and the sample covariance between centers and ranges. For the robust estimation of the covariance matrix, see IMCD.

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)