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.

翻译：本文针对具有可分离协方差结构的多元函数型数据，解决了稳健协方差估计与可解释异常检测的挑战。我们通过建立具有可分离协方差结构的随机过程与其基表示对应的矩阵变量分布之间的联系，开发了一种同时提升稳健性与可解释性的方法。利用这一联系，我们将最近发展的矩阵变量最小协方差行列式估计量（MMCD）与截断多元函数型马氏半距离相结合，对多元函数型数据的均值与协方差进行稳健估计。为实现可解释异常检测，我们基于Shapley值将多元异常解释泛化，将整体多元函数型异常度分解为时间坐标特定的贡献。关键的是，我们将原本指数级（关于分量数量）的计算复杂度降低至线性复杂度，同时保留Shapley值的核心性质。该集成框架融合了稳健马氏距离、MMCD估计量及基于Shapley值的异常度分解，为分析具有可分离协方差结构的多元函数型数据提供了稳健且可解释的方法。通过理论分析及包含模拟与实际案例的应用验证，证明了该方法的有效性。