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Interval-Mahalanobis Distance

Source: R/int_distances.r
IMah_dist.Rd

Calculate the squared Interval-Mahalanobis distance of all rows in the data and the barycenter.

Usage

IMah_dist(data, z = NULL, mean_c = NULL, mean_r = NULL, cov = NULL)

Arguments

data

An intData object containing the macrodata/interval data

z

A vector of 0 and 1, indicating which observations should be considered for the calculation. You must provide either z or (mean_c, mean_r and cov)

mean_c

The mean vector of the centers

mean_r

The mean vector of the ranges

cov

The symbolic covariance matrix

Value

A vector with the squared Interval-Mahalanobis distance of each observation.

Details

The squared Interval-Mahalanobis distance is defined according to the LatentCase:

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

  • "U_id": The latent variables are identically distributed: $$d_{IMah}(\boldsymbol{x})^2=(\boldsymbol{c}-\boldsymbol{\mu}_C)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{c}-\boldsymbol{\mu}_C)+\delta(\boldsymbol{r}-\boldsymbol{\mu}_R)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{r}-\boldsymbol{\mu}_R)\\ +\dfrac{\mathbb{E}(U)}{2}(\boldsymbol{c}-\boldsymbol{\mu}_C)^\top\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{r}-\boldsymbol{\mu}_R)+\dfrac{\mathbb{E}(U)}{2}(\boldsymbol{r}-\boldsymbol{\mu}_R)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{c}-\boldsymbol{\mu}_C),$$ 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: $$d_{IMah}(\boldsymbol{x})^2=(\boldsymbol{c}-\boldsymbol{\mu}_C)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{c}-\boldsymbol{\mu}_C)+\dfrac{1}{4}(\boldsymbol{r}-\boldsymbol{\mu}_R)^{\top}\left(\boldsymbol{\mathfrak{E}}_{UU}\bullet\boldsymbol{\Sigma}_{B}^{-1}\right)(\boldsymbol{r}-\boldsymbol{\mu}_R)\\ +\dfrac{1}{2}(\boldsymbol{c}-\boldsymbol{\mu}_C)^{\top}\boldsymbol{\Sigma}_{B}^{-1}\boldsymbol{\Psi}(\boldsymbol{r}-\boldsymbol{\mu}_R)+\dfrac{1}{2}(\boldsymbol{r}-\boldsymbol{\mu}_R)^{\top}\boldsymbol{\Psi}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{c}-\boldsymbol{\mu}_C),$$ 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

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

z <- rep(1, nrow(credit_card_int))
credit_card_dist<-IMah_dist(credit_card_int,z)