用流形学习构建动态因子模型，提升高维变量与响应的联合预测能力。
Data-Driven Dynamic Factor Modeling via Manifold Learning
- 通过各向异性扩散映射学习低维嵌入，保留变量几何结构与预测关系。
- 在美联储压力测试中，预测误差比传统方法降低39%~55%。
- 适合处理高维金融时间序列的联合建模与风险预测场景。
我们提出一种无参数假设的数据驱动动态因子框架，用于建模高维协变量与响应变量的联合演化过程。传统仅作用于协变量的因子模型往往难以解释响应变量。本方法采用各向异性扩散映射（anisotropic diffusion maps）这一流形学习技术，学习能同时保持协变量内在几何结构与对响应变量预测关系的低维嵌入。对于来自欧几里得空间中朗之万扩散过程的时间序列，我们证明其关联图拉普拉斯算子收敛于底层扩散的生成算子。进一步建立了扩散映射坐标与线性扩散过程之间的近似误差界，并在标准谱假设下证明嵌入空间中的遍历平均收敛。这些结果为在扩散映射坐标中使用卡尔曼滤波预测协变量-响应联合演化提供了理论支持。我们将该方法应用于基于美联储监管情景的股票组合压力测试，使用宏观经济与金融变量，相比经典情景分析和主成分分析基准，平均绝对误差分别降低最高达55%和39%。
原文摘要 · Abstract (English)
We introduce a data-driven dynamic factor framework for modeling the joint evolution of high-dimensional covariates and responses without parametric assumptions. Standard factor models applied to covariates alone often lose explanatory power for responses. Our approach uses anisotropic diffusion maps, a manifold learning technique, to learn low-dimensional embeddings that preserve both the intrinsic geometry of the covariates and the predictive relationship with responses. For time series arising from Langevin diffusions in Euclidean space, we show that the associated graph Laplacian converges to the generator of the underlying diffusion. We further establish a bound on the approximation error between the diffusion map coordinates and linear diffusion processes, and we show that ergodic averages in the embedding space converge under standard spectral assumptions. These results justify using Kalman filtering in diffusion-map coordinates for predicting joint covariate-response evolution. We apply this methodology to equity-portfolio stress testing using macroeconomic and financial variables from Federal Reserve supervisory scenarios, achieving mean absolute error improvements of up to 55% over classical scenario analysis and 39% over principal component analysis benchmarks.
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