AI 中文总结
该研究提出一种基于对称正定锥测地回归的几何湍流指标，可准确识别市场压力事件，其预测性能优于传统欧氏回归，对应的去风险策略能降低最大回撤。
AI 中文摘要
一篮子资产的协方差矩阵是对称正定对象，其在弯曲流形上演化，而非平坦向量空间。标准线性回归对其向量化元素进行拟合时，会忽略该几何特性，在市场压力时期，可能会返回非正定的拟合结果。我们针对对称正定锥$\text{SPD}(n)$在对数欧氏度量和仿射不变度量下开展测地回归研究，采用一篮子对数收益率的滚动样本协方差作为数据。我们将拟合测地线的速度范数提出为一种感知几何的湍流指标。在四个股权篮子的日度数据上——10只股票的历史标普子篮子（2006-2024年）、当前标普100大盘股篮子、约翰内斯堡证券交易所（JSE）的十大上市股票、埃及证券交易所（EGX）的十大交易最活跃标的——该指标无需任何校准，即可清晰地在已知压力事件（2008年全球金融危机、2020年新冠崩盘、2022年美联储加息周期、2019年10-11月埃及政治经济紧张事件）达到峰值。在崩盘窗口的留存评估中，对数欧氏回归的测地均方误差比欧氏普通最小二乘法小150至300倍，因为后者会产生非对称正定的预测结果，在内在度量下会发生爆炸。一种朴素的感知速度的去风险策略可将最大回撤降低2.5个百分点，代价是年化收益率减少1.4个百分点，这与一个缓慢移动的指标一致，该指标可靠地发出压力信号，但转化为战术交易的效果并不完美。所有数据、代码和实验均可从配套仓库复现。
英文摘要
The covariance matrix of a basket of assets is a symmetric positive-definite object that evolves on a curved manifold, not on a flat vector space. Standard linear regression on its vectorised entries ignores that geometry and, in periods of market stress, can return fits that fail to be positive definite. We study geodesic regression on the symmetric positive-definite cone $\SPD(n)$ under the log-Euclidean and affine-invariant metrics, with the rolling sample covariance of a basket of log-returns as data. The fitted geodesic's velocity norm is proposed as a geometry-aware turbulence indicator. On daily data for four equity baskets- a ten-stock historical S\&P sub-basket (2006-2024), a current S\&P~100 mega-cap basket, the top ten listings of the Johannesburg Stock Exchange (JSE), and the ten most-traded Egyptian Exchange (EGX) tickers the indicator peaks cleanly on known stress episodes (the 2008 global financial crisis, the 2020 COVID crash, the 2022 Fed rate-hike cycle, and the October-November 2019 Egyptian political-economic tension episode) without any calibration. On hold-out evaluation in crash windows, log-Euclidean regression achieves 150 to 300 times smaller geodesic mean squared error than Euclidean ordinary least squares, because the latter produces non-SPD forecasts that blow up under the intrinsic metric. A naive velocity-aware de-risking strategy reduces maximum drawdown by 2.5 percentage points at the cost of 1.4 percentage points of annual return, consistent with a slow-moving indicator that signals stress reliably but translates imperfectly into tactical trading. All data, code, and experiments are reproducible from the companion repository.
Comments22 pages, 8 figures
