AI 中文总结
研究路径值协变量下基于签名的路径回归，建立伊藤扩散光滑泛函的\(L^2\)逼近率并证明其最优，通过三种统计学习程序传播截断误差并建立一致性，三个实际应用显示签名可改善预测。
AI 中文摘要
运筹学中的许多预测和决策问题涉及路径值协变量，路径签名已成为其规范特征表示。通用逼近定理证明了其合理性，但未量化逼近误差随截断水平增长的下降速度。本文为基于签名的路径回归发展了逼近和统计理论。建立了伊藤扩散光滑泛函的\(L^2\)逼近率并证明其是极小极大最优的。通过三种统计学习程序传播截断误差并建立其一致性。三个实际数据应用表明签名能提供路径值协变量的信息丰富的有限维表示并改善预测。
英文摘要
Many prediction and decision-making problems in operations research involve path-valued covariates -- data that evolve over time -- for which path signatures have become a canonical feature representation. Their use is justified by a universal approximation theorem, but this is an existence result: it guarantees that a finite-level signature can approximate any continuous path functional, without quantifying how fast the approximation error decreases as the truncation level grows. This paper develops approximation and statistical theory for signature-based path regression. We establish an \(L^2\) approximation rate for smooth functionals of Itô diffusions and show that it is minimax optimal. We then propagate the truncation error through three statistical learning procedures -- Signature-OLS, Signature-LASSO, and Signature-Logistic -- and establish their consistency. Three real-data applications show that signatures provide informative finite-dimensional representations of path-valued covariates and can improve prediction relative to handcrafted features, in the context of finance -- foreign exchange realized volatility forecasting from intraday price paths; energy -- battery end-of-life prediction from early diagnostic current-voltage pulse paths; and medicine -- epileptic seizure detection from short electroencephalogram windows.
Comments82 pages, 8 main figures
