On expected signatures and signature cumulants in semimartingale models
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916727440277504 |
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| author | Friz, Peter K. Hager, Paul P. Tapia, Nikolas |
| author_facet | Friz, Peter K. Hager, Paul P. Tapia, Nikolas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_05085 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On expected signatures and signature cumulants in semimartingale models Friz, Peter K. Hager, Paul P. Tapia, Nikolas Machine Learning Probability 60L10, 60L90, 60E10, 60G44, 60G48, 60G51, 60J76 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. |
| title | On expected signatures and signature cumulants in semimartingale models |
| topic | Machine Learning Probability 60L10, 60L90, 60E10, 60G44, 60G48, 60G51, 60J76 |
| url | https://arxiv.org/abs/2408.05085 |