用期望签名构建时间序列的无模型特征，提升预测性能
Learning with Expected Signatures: Theory and Applications
- 基于期望签名提取时序数据的低维特征表示
- 证明了离散估计与连续理论值的收敛性，增强理论可信度
- 对鞅过程改进估计器，降低误差，适合金融等时序预测
期望签名将数据流映射为低维表示，其特征张量可完全刻画生成分布。这种‘无模型’嵌入已成功用于构建多个领域无关的时间序列与序列学习算法。本文证明了期望签名的离散时间估计量与其理论连续时间值之间的收敛性，完善了基于期望签名的机器学习方法的概率解释。此外，当数据生成过程为鞅时，提出一种简单修正的估计器，显著降低均方误差，并通过实验证明其能有效提升预测性能。
原文摘要 · Abstract (English)
The expected signature maps a collection of data streams to a lower dimensional representation, with a remarkable property: the resulting feature tensor can fully characterize the data generating distribution. This "model-free" embedding has been successfully leveraged to build multiple domain-agnostic machine learning (ML) algorithms for time series and sequential data. The convergence results proved in this paper bridge the gap between the expected signature's empirical discrete-time estimator and its theoretical continuous-time value, allowing for a more complete probabilistic interpretation of expected signature-based ML methods. Moreover, when the data generating process is a martingale, we suggest a simple modification of the expected signature estimator with significantly lower mean squared error and empirically demonstrate how it can be effectively applied to improve predictive performance.
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