Explainable Outlier Detection for Multivariate Functional Data

TL;DR

This paper introduces a robust and interpretable outlier detection method for multivariate functional data with separable covariance structures, combining advanced estimators and Shapley value explanations.

Contribution

It develops a novel framework that integrates robust covariance estimation and interpretable outlier explanations with reduced computational complexity.

Findings

Effective outlier detection demonstrated on simulations and real data.

Reduced computational complexity from exponential to linear in the number of components.

Theoretical analysis confirms robustness and interpretability of the method.

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.