The minimum covariance determinant (MCD) estimator is a popular method for robustly estimating the mean and covariance of multivariate data. We extend the MCD to the setting where the observations are matrices rather than vectors and introduce the matrix minimum covariance determinant (MMCD) estimators for robust parameter estimation. These estimators hold equivariance properties, achieve a high breakdown point, and are consistent under elliptical matrix-variate distributions. We have also developed an efficient algorithm with convergence guarantees to compute the MMCD estimators. Using the MMCD estimators, we can compute robust Mahalanobis distances that can be used for outlier detection. Those distances can be decomposed into outlyingness contributions from each cell, row, or column of a matrix-variate observation using Shapley values, a concept for outlier explanation recently introduced in the multivariate setting. Simulations and examples reveal the excellent properties and usefulness of the robust estimators.

翻译：最小协方差行列式（MCD）估计量是稳健估计多元数据均值和协方差的流行方法。我们将MCD扩展到观测值为矩阵而非向量的场景，并引入矩阵最小协方差行列式（MMCD）估计量用于稳健参数估计。这些估计量具有等变性性质，达到高崩溃点，并在椭圆矩阵变量分布下具有一致性。我们还开发了一种具有收敛保证的高效算法来计算MMCD估计量。利用MMCD估计量，可以计算稳健的马氏距离，用于异常值检测。这些距离可通过沙普利值（近期在多元背景下引入的异常解释概念）分解为矩阵变量观测值中每个单元格、行或列的异常贡献。模拟实验和案例分析揭示了该稳健估计量的卓越性质与实用性。