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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,
  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 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)