# Multivariate outlier explanations using Shapley values and Mahalanobis distances

arXiv: https://arxiv.org/abs/2210.10063 · HTML: https://mlsanity.com/paper/2210.10063/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 33 references that are in the map.
- First posted on arXiv 18 Oct 2022.

## Builds on (most cited first, 8 of 8)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [Consistent Individualized Feature Attribution for Tree Ensembles](https://mlsanity.com/paper/1802.03888/) (2018)
- [The Shapley Taylor Interaction Index](https://mlsanity.com/paper/1902.05622/) (2019)
- [Detecting deviating data cells](https://mlsanity.com/paper/1601.07251/) (2016)
- [The Cellwise Minimum Covariance Determinant Estimator](https://mlsanity.com/paper/2207.13493/) (2022)
- [Handling cellwise outliers by sparse regression and robust covariance](https://mlsanity.com/paper/1912.12446/) (2019)
- [Outlyingness: why do outliers lie out?](https://mlsanity.com/paper/1708.03761/) (2017)

## Built on by (most cited first, 1 of 1)

- [Introducing 'Inside' Out of Distribution](https://mlsanity.com/paper/2407.04534/) (2024)

## Abstract (arXiv)

For the purpose of explaining multivariate outlyingness, it is shown that the squared Mahalanobis distance of an observation can be decomposed into outlyingness contributions originating from single variables. The decomposition is obtained using the Shapley value, a well-known concept from game theory that became popular in the context of Explainable AI. In addition to outlier explanation, this concept also relates to the recent formulation of cellwise outlyingness, where Shapley values can be employed to obtain variable contributions for outlying observations with respect to their "expected" position given the multivariate data structure. In combination with squared Mahalanobis distances, Shapley values can be calculated at a low numerical cost, making them even more attractive for outlier interpretation. Simulations and real-world data examples demonstrate the usefulness of these concepts.