shapley_interaction {ShapleyOutlier}

Decomposition of squared Mahalanobis distance using Shapley interaction indices.

Description

The shapley_interaction function computes a p \times p matrix containing pairwise outlyingness scores based on Shapley interaction indices. It decomposes the squared Mahalanobis distance of x (with respect to mu and Sigma) into outlyingness contributions of pairs of variables (Mayrhofer and Filzmoser 2022).

Usage

shapley_interaction(x, mu, Sigma, inverted = FALSE)

Arguments

x

Data vector with p entries containing only numeric entries.

mu

Either NULL (default) or mean vector of x. If NULL, method is used for parameter estimation.

Sigma

Either NULL (default) or covariance matrix p \times p of x. If NULL, method is used for parameter estimation.

inverted

Logical. If TRUE, Sigma is supposed to contain the inverse of the covariance matrix.

Value

A p \times p matrix containing the decomposition of the squared Mahalanobis distance of x into outlyingness scores for pairs of variables with respect to mu and Sigma.

References

Mayrhofer M, Filzmoser P (2022). “Multivariate outlier explanations using Shapley values and Mahalanobis distances.” doi:10.48550/ARXIV.2210.10063.

Examples

p <- 5
mu <- rep(0,p)
Sigma <- matrix(0.9, p, p); diag(Sigma) = 1
Sigma_inv <- solve(Sigma)
x <- c(0,1,2,2.3,2.5)
shapley_interaction(x, mu, Sigma)
PHI <- shapley_interaction(x, mu, Sigma_inv, inverted = TRUE)
plot(PHI)