On expected signatures and signature cumulants in semimartingale models

• URL: http://arxiv.org/abs/2408.05085v1
• Date: Fri, 9 Aug 2024 14:16:21 GMT
• Title: On expected signatures and signature cumulants in semimartingale models
• Authors: Peter K. Friz, Paul P. Hager, Nikolas Tapia,
• Abstract summary: The concept of signatures and expected signatures is vital in data science, especially for sequential data analysis. A log-transform (expected signatures) leads to log-signatures (signature cumulants)
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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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