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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, mean = 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.

mean

(Optional) An intData object containing the barycenter of the data. If provided, it will override mean_c and mean_r.

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)