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Compute Shapley Decomposition into contributions of (Centers, Ranges, and CrossCentersRanges) for Interval-valued Data

Source: R/Shapley.r
int_Shapley_decomp.Rd

Decomposes the squared interval-valued Mahalanobis distance of each observation into outlyingness contributions of (Centers, Ranges, and CrossCentersRanges) per variable for interval-valued data.

Usage

int_Shapley_decomp(data, mean_c = NULL, mean_r = NULL, cov = NULL)

Arguments

data

An intData object containing the interval-valued dataset (macrodata).

mean_c

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

mean_r

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

cov

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

Value

A list containing the matrix of Shapley value decomposition into contributions of (Centers, Ranges, and CrossCentersRanges) per variable for each observation.

Details

Let \(\boldsymbol{\mu}_B=(\boldsymbol{\mu}_C^\top,\boldsymbol{\mu}_R^\top)^\top\) be the barycenter and \(\boldsymbol{\Sigma}_B\) the symbolic covariance matrix (see int_cov). Based on the Shapley value (see int_Shapley), we can further decompose the Interval-Mahalanobis distance of an interval-valued observation \(\boldsymbol{x}=(\boldsymbol{c}^\top,\boldsymbol{r}^\top)^\top\) into contributions of the centers, ranges and cross-centers-ranges of the variables. The decomposition is defined according to the LatentCase:

  • "U_id_symmetric": The latent variables are identically distributed and symmetric:

    • Centers contribution: $$(\boldsymbol{c}-\boldsymbol{\mu}_C)\bullet\left[\boldsymbol{\Sigma}_B^{-1}(\boldsymbol{c}-\boldsymbol{\mu}_C)\right],$$

    • Ranges contribution: $$\delta(\boldsymbol{r}-\boldsymbol{\mu}_R)\bullet\left[\boldsymbol{\Sigma}_B^{-1}(\boldsymbol{r}-\boldsymbol{\mu}_R)\right],$$

    where \(\delta=\mathbb{E}(U^2)/4\) is the parameter of the latent variables.

  • "U_id": The latent variables are identically distributed:

    • Centers contribution: $$(\boldsymbol{c}-\boldsymbol{\mu}_C)\bullet\left[\boldsymbol{\Sigma}_B^{-1}(\boldsymbol{c}-\boldsymbol{\mu}_C)\right],$$

    • Ranges contribution: $$\delta(\boldsymbol{r}-\boldsymbol{\mu}_R)\bullet\left[\boldsymbol{\Sigma}_B^{-1}(\boldsymbol{r}-\boldsymbol{\mu}_R)\right],$$

    • CrossCentersRanges contribution: $$\dfrac{\mathbb{E}(U)}{2}(\boldsymbol{c}-\boldsymbol{\mu}_C)\bullet\left[\boldsymbol{\Sigma}_B^{-1}(\boldsymbol{r}-\boldsymbol{\mu}_R)\right]+\dfrac{\mathbb{E}(U)}{2}(\boldsymbol{r}-\boldsymbol{\mu}_R)\bullet\left[\boldsymbol{\Sigma}_B^{-1}(\boldsymbol{c}-\boldsymbol{\mu}_C)\right],$$

    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:

    • Centers contribution: $$(\boldsymbol{c}-\boldsymbol{\mu}_C)\bullet\left[\boldsymbol{\Sigma}_B^{-1}(\boldsymbol{c}-\boldsymbol{\mu}_C)\right],$$

    • Ranges contribution: $$\dfrac{1}{4}(\boldsymbol{r}-\boldsymbol{\mu}_R)\bullet\left[\left(\boldsymbol{\mathfrak{E}}_{UU}\bullet\boldsymbol{\Sigma}_B^{-1}\right)(\boldsymbol{r}-\boldsymbol{\mu}_R)\right],$$

    • CrossCentersRanges contribution: $$\dfrac{1}{2}(\boldsymbol{c}-\boldsymbol{\mu}_C)\bullet\left[\boldsymbol{\Sigma}_B^{-1}\boldsymbol{\Psi}(\boldsymbol{r}-\boldsymbol{\mu}_R)\right]+\dfrac{1}{2}(\boldsymbol{r}-\boldsymbol{\mu}_R)\bullet\left[\boldsymbol{\Psi}\boldsymbol{\Sigma}_B^{-1}(\boldsymbol{c}-\boldsymbol{\mu}_C)\right],$$

    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). Explainable Outlier Detection for Interval-valued Data. arXiv preprint arXiv:2606.26307. https://arxiv.org/abs/2606.26307

Examples

# Create intData object
data(creditcard)
credit_card_int <- creditcard$intData

# Compute Shapley decomposition into contributions of (Centers, Ranges, and CrossCentersRanges) 
# based on IMCD estimates of mean and covariance
credit_card_shap_decomp <- int_Shapley_decomp(credit_card_int)