Skip to contents

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

(Optional) A vector of 0 and 1, indicating which observations should be considered for the calculation. If z is not NULL, mean_c, mean_r, and cov will be computed using only the observations with z=1 (see int_mean_z and int_cov_z). Defaults to NULL.

mean_c

(Optional) A vector specifying the mean of centers. Defaults to NULL, in which case it will be computed using the IMCD function, if z is also NULL.

mean_r

(Optional) A vector specifying the mean of ranges. Defaults to NULL, in which case it will be computed using the IMCD function, if z is also NULL.

cov

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

Value

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

Details

The squared Interval-Mahalanobis distance between \(\boldsymbol{x}=(\boldsymbol{c}^\top,\boldsymbol{r}^\top)^\top\) and the barycenter \(\boldsymbol{\mu}_B=(\boldsymbol{\mu}_C^\top,\boldsymbol{\mu}_R^\top)^\top\) 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: $$d_\mathrm{IMah}(\boldsymbol{x})^2=(\boldsymbol{c}-\boldsymbol{\mu}_C)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{c}-\boldsymbol{\mu}_C)+\delta(\boldsymbol{r}-\boldsymbol{\mu}_R)^{\top}\boldsymbol{\Sigma}_{B}^{-1}(\boldsymbol{r}-\boldsymbol{\mu}_R),$$ where \(\delta=\mathbb{E}(U^2)/4\) is the parameter of the latent variables.

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

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

# Compute squared Interval-Mahalanobis distance using IMCD estimates of mean and covariance
credit_card_dist <- IMah_dist(credit_card_int)