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Interval-Mahalanobis distance for all pairs

Source: R/int_distances.r
IMah_dist_pairs.Rd

Calculate the squared Interval-Mahalanobis distance of all pairs of observations in the data.

Usage

IMah_dist_pairs(data, cov = NULL)

Arguments

data

An intData object containing the macrodata/interval data

cov

(Optional) A covariance matrix. Defaults to NULL, in which case it will be computed using the IMCD function.

Value

A matrix with the squared Interval-Mahalanobis distance of each pair of observations.

Details

The squared Interval-Mahalanobis distance between \(\boldsymbol{x}_1=(\boldsymbol{c}_1^\top,\boldsymbol{r}_1^\top)^\top\) and \(\boldsymbol{x}_2=(\boldsymbol{c}_2^\top,\boldsymbol{r}_2^\top)^\top\) of a population with symbolic covariance matrix \(\boldsymbol{\Sigma}_B\) (see int_cov) is defined according to the LatentCase:

  • "U_id_symmetric": The latent variables are identically distributed and symmetric: $$d_\mathrm{IMah}(\boldsymbol{x}_1,\boldsymbol{x}_2)^2=(\boldsymbol{c}_1-\boldsymbol{c}_2)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{c}_1-\boldsymbol{c}_2)+\delta(\boldsymbol{r}_1-\boldsymbol{r}_2)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{r}_1-\boldsymbol{r}_2),$$ where \(\delta=\mathbb{E}(U^2)/4\) is the parameter of the latent variables.

  • "U_id": The latent variables are identically distributed: $$\begin{aligned} d_\mathrm{IMah}(\boldsymbol{x}_1,\boldsymbol{x}_2)^2&=(\boldsymbol{c}_1-\boldsymbol{c}_2)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{c}_1-\boldsymbol{c}_2)+\delta(\boldsymbol{r}_1-\boldsymbol{r}_2)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{r}_1-\boldsymbol{r}_2)\\ &\quad+\mathbb{E}(U)(\boldsymbol{c}_1-\boldsymbol{c}_2)^\top\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{r}_1-\boldsymbol{r}_2), \end{aligned}$$ where \(\delta=\mathbb{E}(U^2)/4\) and \(\mathbb{E}(U)\) are the parameter of the latent variables.

  • "General": The latent variables do not have any nice properties: $$\begin{aligned} d_\mathrm{IMah}(\boldsymbol{x}_1,\boldsymbol{x}_2)^2&=(\boldsymbol{c}_1-\boldsymbol{c}_2)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{c}_1-\boldsymbol{c}_2)+\dfrac{1}{4}(\boldsymbol{r}_1-\boldsymbol{r}_2)^{\top}\left(\boldsymbol{\mathfrak{E}}_{UU}\bullet\boldsymbol{\Sigma}_{B}^{-1}\right)(\boldsymbol{r}_1-\boldsymbol{r}_2)\\ &\quad+(\boldsymbol{c}_1-\boldsymbol{c}_2)^{\top}\boldsymbol{\Sigma}_{B}^{-1}\boldsymbol{\Psi}(\boldsymbol{r}_1-\boldsymbol{r}_2), \end{aligned}$$ 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.

If cov is not provided, it will be computed using the IMCD function. Additionally, if cov is set as the identity matrix, the computed distance is the Mallows distance between pairs of observations.

References

Loureiro, C. P., Oliveira, M. R., Brito, P., & Oliveira, L. (2026). Minimum Covariance Determinant Estimator and Outlier Detection for Interval-valued Data. arXiv preprint arXiv:2604.26769. https://arxiv.org/abs/2604.26769

Examples

data(creditcard)
credit_card_int <- creditcard$intData

credit_card_dist <- IMah_dist_pairs(credit_card_int)