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Compute Shapley interaction indices for Interval-valued Data

Source: R/Shapley.r
int_Shapley_interaction.Rd

Obtains a \(p \times p\) matrix containing pairwise outlyingness scores based on Shapley interaction indices for each observation. Decomposes the squared interval-valued Mahalanobis distance of each observation into outlyingness contributions of pairs of variables.

Usage

int_Shapley_interaction(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 interaction indices 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). Let also \(\boldsymbol{\phi}(\boldsymbol{x})\) be the Shapley value of \(\boldsymbol{x}\) (see int_Shapley) and \(\mathrm{diag}(\boldsymbol{v})\) be the diagonal matrix whose main diagonal is the vector \(\boldsymbol{v}\). The Shapley interaction index of an interval-valued observation \(\boldsymbol{x}=(\boldsymbol{c}^\top,\boldsymbol{r}^\top)^\top\), for the Interval-Mahalanobis distance, is defined according to the LatentCase:

  • "U_id_symmetric": The latent variables are identically distributed and symmetric: $$\boldsymbol{\Phi}(\boldsymbol{x})=2(\boldsymbol{c}-\boldsymbol{\mu}_C)(\boldsymbol{c}-\boldsymbol{\mu}_C)^\top\bullet\boldsymbol{\Sigma}_B^{-1} + 2\delta(\boldsymbol{r}-\boldsymbol{\mu}_R)(\boldsymbol{r}-\boldsymbol{\mu}_R)^\top\bullet\boldsymbol{\Sigma}_B^{-1}-\mathrm{diag}\left(\boldsymbol{\phi}(\boldsymbol{x})\right),$$ where \(\delta=\mathbb{E}(U^2)/4\) is the parameter of the latent variables.

  • "U_id": The latent variables are identically distributed: $$\begin{aligned} \boldsymbol{\Phi}(\boldsymbol{x})&=2(\boldsymbol{c}-\boldsymbol{\mu}_C)(\boldsymbol{c}-\boldsymbol{\mu}_C)^\top\bullet\boldsymbol{\Sigma}_B^{-1} + 2\delta(\boldsymbol{r}-\boldsymbol{\mu}_R)(\boldsymbol{r}-\boldsymbol{\mu}_R)^\top\bullet\boldsymbol{\Sigma}_B^{-1}\\ &\quad+\mathbb{E}(U)(\boldsymbol{c}-\boldsymbol{\mu}_C)(\boldsymbol{r}-\boldsymbol{\mu}_R)^\top\bullet\boldsymbol{\Psi} + \mathbb{E}(U)(\boldsymbol{r}-\boldsymbol{\mu}_R)(\boldsymbol{c}-\boldsymbol{\mu}_C)^\top\bullet\boldsymbol{\Sigma}_B^{-1}-\mathrm{diag}\left(\boldsymbol{\phi}(\boldsymbol{x})\right), \end{aligned}$$ 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: $$\begin{aligned} \boldsymbol{\Phi}(\boldsymbol{x})&=2(\boldsymbol{c}-\boldsymbol{\mu}_C)(\boldsymbol{c}-\boldsymbol{\mu}_C)^\top\bullet\boldsymbol{\Sigma}_B^{-1} + \dfrac{1}{2}(\boldsymbol{r}-\boldsymbol{\mu}_R)(\boldsymbol{r}-\boldsymbol{\mu}_R)^\top\bullet\boldsymbol{\mathfrak{E}}_{UU}\bullet\boldsymbol{\Sigma}_B^{-1}\\ &\quad+(\boldsymbol{c}-\boldsymbol{\mu}_C)(\boldsymbol{r}-\boldsymbol{\mu}_R)^\top\bullet\boldsymbol{\Sigma}_B^{-1}\boldsymbol{\Psi} + (\boldsymbol{r}-\boldsymbol{\mu}_R)(\boldsymbol{c}-\boldsymbol{\mu}_C)^\top\bullet\boldsymbol{\Psi}\boldsymbol{\Sigma}_B^{-1}-\mathrm{diag}\left(\boldsymbol{\phi}(\boldsymbol{x})\right), \end{aligned}$$ 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). 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 interaction indices based on the mean and covariance matrix estimated by IMCD
credit_card_shap_inter <- int_Shapley_interaction(credit_card_int)