SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 2, Page 528-556, January 2021.
Statistical modeling of spatiotemporal phenomena often requires selecting a covariance matrix from a covariance class. Yet standard parametric covariance families can be insufficiently flexible for practical applications, while nonparametric approaches may not easily allow certain kinds of prior knowledge to be incorporated. We propose instead to build covariance families out of geodesic curves. These covariances offer more flexibility for problem-specific tailoring than classical parametric families and are preferable to simple convex combinations. Once the covariance family has been chosen, one typically needs to select a representative member by solving an optimization problem, e.g., by maximizing the likelihood of a data set. We consider instead a differential geometric interpretation of this problem: minimizing the geodesic distance to a sample covariance matrix (``natural projection''). Our approach is consistent with the notion of distance employed to build the covariance family and does not require assuming a particular probability distribution for the data. We show that natural projection and maximum likelihood estimation within the covariance family are locally equivalent up to second order. We also demonstrate that natural projection may yield more accurate estimates with noise-corrupted data.

中文翻译：

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测地参数化协方差估计

SIAM 矩阵分析与应用杂志，第 42 卷，第 2 期，第 528-556 页，2021 年 1 月。
时空现象的统计建模通常需要从协方差类中选择协方差矩阵。然而，标准参数协方差族对于实际应用可能不够灵活，而非参数方法可能不容易允许合并某些类型的先验知识。我们建议从测地曲线中建立协方差族。与经典的参数族相比，这些协方差为特定于问题的裁剪提供了更大的灵活性，并且比简单的凸组合更可取。一旦选择了协方差族，人们通常需要通过解决优化问题来选择具有代表性的成员，例如通过最大化数据集的可能性。我们考虑对这个问题的微分几何解释：最小化到样本协方差矩阵的测地距离（“自然投影”）。我们的方法与用于构建协方差族的距离概念一致，并且不需要为数据假设特定的概率分布。我们表明协方差族内的自然投影和最大似然估计在局部等效于二阶。我们还证明了自然投影可能会对噪声损坏的数据产生更准确的估计。我们表明协方差族内的自然投影和最大似然估计在局部等效于二阶。我们还证明了自然投影可能会对噪声损坏的数据产生更准确的估计。我们表明协方差族内的自然投影和最大似然估计在局部等效于二阶。我们还证明了自然投影可能会对噪声损坏的数据产生更准确的估计。