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.

翻译：签名与期望签名的概念在数据科学中至关重要，尤其在序列数据分析领域。签名变换作为一种Cartan型展开，将路径映射为高维特征向量，从而捕捉其内在特性。在自然条件下，签名的期望决定了签名的分布律，为数据分布提供了统计摘要。这一性质促进了机器学习与随机过程中稳健的建模与推断。基于作者先前的工作[《统一签名累积量与广义Magnus展开》，FoM Sigma '22]，本文在半鞅的一般框架下重新审视期望签名的实际计算问题，给出了若干新公式。对（期望）签名进行对数变换可导出对数签名（即签名累积量），这能显著降低计算复杂度。