Calculate the squared Mallows distance between all rows in data and the barycenter.
Arguments
- data
An intData object containing the macrodata/interval data
- mean_c
The mean vector of the centers
- mean_r
The mean vector of the ranges
Details
The squared Mallows distance is defined according to the LatentCase:
"U_id_symmetric": The latent variables are identically distributed and symmetric: $$d_{M}(\boldsymbol{x})^2=(\boldsymbol{c}-\boldsymbol{\mu}_C)^{\top}(\boldsymbol{c}-\boldsymbol{\mu}_C)+\delta(\boldsymbol{r}-\boldsymbol{\mu}_R)^{\top}(\boldsymbol{r}-\boldsymbol{\mu}_R),$$ where \(\delta=\mathbb{E}(U^2)/4\) is the parameter of the latent variables."U_id": The latent variables are identically distributed: $$d_{M}(\boldsymbol{x})^2=(\boldsymbol{c}-\boldsymbol{\mu}_C)^{\top}(\boldsymbol{c}-\boldsymbol{\mu}_C)+\delta(\boldsymbol{r}-\boldsymbol{\mu}_R)^{\top}(\boldsymbol{r}-\boldsymbol{\mu}_R) +\mathbb{E}(U)(\boldsymbol{c}-\boldsymbol{\mu}_C)^\top(\boldsymbol{r}-\boldsymbol{\mu}_R),$$ 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: $$d_{M}(\boldsymbol{x})^2=(\boldsymbol{c}-\boldsymbol{\mu}_C)^{\top}(\boldsymbol{c}-\boldsymbol{\mu}_C)+(\boldsymbol{r}-\boldsymbol{\mu}_R)^{\top}\boldsymbol{\Delta}(\boldsymbol{r}-\boldsymbol{\mu}_R) +(\boldsymbol{c}-\boldsymbol{\mu}_C)^{\top}\boldsymbol{\Psi}(\boldsymbol{r}-\boldsymbol{\mu}_R),$$ 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)