半鞅模型中的期望签名与签名累积量
On expected signatures and signature cumulants in semimartingale models
AI总结：
本文在半鞅模型下重新研究期望签名的计算，提出若干新公式，并利用对数变换得到签名累积量以显著降低复杂度。
摘要：
签名与期望签名的概念在数据科学中至关重要，尤其是对于序列数据分析。签名变换是一种嘉当型展开，它将路径转换为高维特征向量，捕捉其内在特征。在自然条件下，签名的期望决定了签名的分布，从而提供数据分布的统计摘要。这一性质有助于在机器学习和随机过程中进行稳健的建模与推断。基于本文作者先前的工作 [Unified signature cumulants and generalized Magnus expansions, FoM Sigma '22]，我们在此重新审视一般半鞅设定下期望签名的实际计算。文中给出了若干新公式。对（期望）签名进行对数变换可得到对数签名（签名累积量），从而显著降低复杂度。
英文摘要：
The concept of signatures and expected signatures is vital in data science, especially for sequential data analysis. The signature transform, a Cartan type development, translates paths into high-dimensional feature vectors, capturing their intrinsic characteristics. Under natural conditions, the expectation of the signature determines the law of the signature, providing a statistical summary of the data distribution. This property facilitates robust modeling and inference in machine learning and stochastic processes. Building on previous work by the present authors [Unified signature cumulants and generalized Magnus expansions, FoM Sigma '22] we here revisit the actual computation of expected signatures, in a general semimartingale setting. Several new formulae are given. A log-transform of (expected) signatures leads to log-signatures (signature cumulants), offering a significant reduction in complexity.
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