Explainable Outlier Detection for Interval-valued Data

2026-06-24

AI Summary

Existing methods for interval-valued anomaly detection suffer from limited interpretability and struggle to pinpoint the root causes of anomalies. To address this, this work proposes a novel approach that decomposes robust interval Mahalanobis distances using Shapley values, grounded in the minimum covariance determinant estimator for interval data. The method derives, for the first time, a closed-form solution for Shapley values, enabling efficient quantification of each variable’s contribution across center, range, and cross-term components. Furthermore, it incorporates Shapley interaction indices to capture synergistic effects among variables that jointly drive anomalous behavior. Experimental results on two real-world datasets demonstrate that the proposed framework effectively identifies cell-level anomalies invisible at the multivariate level, substantially enhancing interpretability while maintaining robust detection performance.

Abstract

Explainability is increasingly recognized as a key aspect of outlier detection. However, for complex data structures such as interval-valued data, it remains largely unexplored. Building on an outlier detection framework based on the Interval Minimum Covariance Determinant estimator, we propose a novel approach to explain the outlyingness of interval-valued observations using the concept of the Shapley value. We derive a closed-form expression for the Shapley value of the squared robust Interval-Mahalanobis distance, enabling efficient computation of variable contributions. This formulation allows for a fine-grained interpretation of outliers, providing a detailed decomposition into contributions from centers, ranges, and cross-terms of the interval-valued observations. Moreover, the Shapley value is closely connected to the concept of cellwise outliers, as it can help identify variable-specific outliers that may not be evident at multivariate level. We further extend the framework through the Shapley interaction index to capture pairwise variable interactions driving atypical behavior. The practical utility of the proposed approach is illustrated through two real-world datasets.

Problem

• Explainable Outlier Detection
• Interval-valued Data
• Shapley Value
• Interpretability
• Cellwise Outliers

Innovation

• Shapley value
• interval-valued data
• explainable outlier detection
• Interval Minimum Covariance Determinant
• Shapley interaction index