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Mallows Distance

Source: R/int_distances.r
Mallows_dist.Rd

Calculate the squared Mallows distance between all rows in data and the barycenter.

Usage

Mallows_dist(data, mean_c = NULL, mean_r = NULL)

Arguments

data

An intData object containing the macrodata/interval data

mean_c

(Optional) A vector specifying the mean of centers. Defaults to NULL, in which case it will be computed using the sample mean of centers.

mean_r

(Optional) A vector specifying the mean of ranges Defaults to NULL, in which case it will be computed using the sample mean of ranges.

Value

A vector with the squared Mallows distance of each observation.

Details

The squared Mallows distance is defined according to the LatentCase:

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

  • "U_id": The latent variables are identically distributed: $$d_\mathrm{M}(\boldsymbol{x})^2=(\boldsymbol{c}-\boldsymbol{\mu}_C)^{\top}(\boldsymbol{c}-\boldsymbol{\mu}_C)+\delta(\boldsymbol{r}-\boldsymbol{\mu}_R)^{\top}(\boldsymbol{r}-\boldsymbol{\mu}_R) +\mathbb{E}(U)(\boldsymbol{c}-\boldsymbol{\mu}_C)^\top(\boldsymbol{r}-\boldsymbol{\mu}_R),$$ 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: $$d_\mathrm{M}(\boldsymbol{x})^2=(\boldsymbol{c}-\boldsymbol{\mu}_C)^{\top}(\boldsymbol{c}-\boldsymbol{\mu}_C)+(\boldsymbol{r}-\boldsymbol{\mu}_R)^{\top}\boldsymbol{\Delta}(\boldsymbol{r}-\boldsymbol{\mu}_R) +(\boldsymbol{c}-\boldsymbol{\mu}_C)^{\top}\boldsymbol{\Psi}(\boldsymbol{r}-\boldsymbol{\mu}_R),$$ where:

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

    • \(\boldsymbol{\Delta}=\text{diag}(\mathbb{E}(U^2_1),\dots,\mathbb{E}(U^2_p))/4\).

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_dist <- Mallows_dist(credit_card_int)