AI 中文总结
本文针对相关矩阵提出商仿射几何下的黎曼高斯模型，推导精确似然方程，开发归一化因子评估与采样方法，经数值及金融应用验证，中小维度下表现可靠。
AI 中文摘要
相关矩阵是协方差矩阵去除边缘尺度后得到的，但归一化似然必须同时考虑商距离和商体积。我们在商仿射几何下针对满秩相关矩阵提出了黎曼高斯模型，该分布是正则的且具有有限径向矩。我们推导了精确的得分方程和轮廓尺度方程，并在二维矩阵上恢复了费希尔变换后的高斯推断。在更高维度下，曲率计算表明归一化常数会随中心变化，因此精确最大似然估计和弗雷歇估计可能具有不同的总体目标。我们开发了基于图的方法来评估归一化因子、拟合似然和采样。数值研究验证了解析情况，并在不同维度和离散度 regime 下对比了积分、估计和采样程序。滚动金融应用和受控先验研究既说明了该模型的价值，也展示了其计算成本。该方法在中小维度下最可靠，而在边界附近和更大离散度时，提议效率和数值条件会恶化。
英文摘要
Correlation matrices arise when marginal scales are removed from covariance matrices, yet a normalized likelihood must account for both quotient distance and quotient volume. We propose a Riemannian Gaussian model for full-rank correlation matrices under quotient-affine geometry. The distribution is proper and has finite radial moments. We derive exact score and profiled-scale equations and recover Fisher-transformed Gaussian inference for two-dimensional matrices. In higher dimension, a curvature calculation shows that the normalizing constant can vary with the center. Exact maximum likelihood and Fréchet estimation may therefore have different population targets. We develop chart-based methods for evaluating the normalizer, fitting the likelihood, and sampling. Numerical studies verify the analytic case and compare integration, estimation, and sampling procedures across dimensions and dispersion regimes. A rolling-finance application and a controlled prior study illustrate both the value and computational cost of the model. The method is most reliable in small to moderate dimensions, while proposal efficiency and numerical conditioning deteriorate near the boundary and at larger dispersion.
