Explainable Outlier Detection for Multivariate Functional Data

2026-05-19

AI Summary

This work proposes a robust and interpretable anomaly detection method for multivariate functional data with separable covariance structure. By establishing a connection between separable covariance stochastic processes and matrix-variate distributions under basis representations, the approach integrates matrix-variate minimum covariance determinant (MMCD) estimation with a truncated multivariate functional Mahalanobis semi-distance to achieve robust mean and covariance estimation. The study innovatively extends Shapley values to functional anomaly interpretation, reducing computational complexity from exponential to linear while preserving their axiomatic properties, thereby decomposing global anomaly scores into localized contributions over the time domain. Theoretical analysis, simulation studies, and real-data applications demonstrate the method’s superior performance in both robust estimation and interpretable anomaly detection.

Abstract

This work addresses the challenges of robust covariance estimation and interpretable outlier detection for multivariate functional data with separable covariance structure. We develop a method that simultaneously improves robustness and interpretability in this context by establishing a connection between stochastic processes with separable covariance structures and the corresponding matrix-variate distribution of their basis representations. Leveraging this connection, we employ the recently developed matrix-variate counterpart of the Minimum Covariance Determinant estimator (MMCD) in conjunction with a truncated multivariate functional Mahalanobis semi-distance to robustly estimate mean and covariance for multivariate functional data. For interpretable outlier detection, we generalize multivariate outlier explanations based on Shapley values to decompose overall multivariate functional outlyingness into time-coordinate-specific contributions. Importantly, we reduce the otherwise exponential computational complexity (relative to the number of components) to linear complexity, while retaining the key properties of the Shapley value. This integrated framework combines robust Mahalanobis distances, MMCD estimators, and Shapley value-based outlyingness decomposition to provide a robust and interpretable approach for analyzing multivariate functional data with separable covariance structures. The effectiveness of this approach is demonstrated through both theoretical analysis and practical applications, including simulations and real-world examples.

Problem

• multivariate functional data
• separable covariance structure
• outlier detection
• explainable AI
• robust covariance estimation

Innovation

• Matrix-variate MCD
• Functional outlier detection
• Shapley values
• Separable covariance
• Mahalanobis distance