General Signature Kernels
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866914664740290560 |
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| author | Cass, Thomas Lyons, Terry Xu, Xingcheng |
| author_facet | Cass, Thomas Lyons, Terry Xu, Xingcheng |
| contents | Suppose that $γ$ and $σ$ are two continuous bounded variation paths which take values in a finite-dimensional inner product space $V$. Recent papers have introduced the truncated and the untruncated signature kernel of $γ$ and $σ$, and showed how these concepts can be used in classification and prediction tasks involving multivariate time series. In this paper, we introduce general signature kernels and show how they can be interpreted, in many examples, as an average of PDE solutions, and hence how they can be computed using suitable quadrature rules. We extend this analysis to derive closed-form formulae for expressions involving the expected (Stratonovich) signature of Brownian motion. In doing so, we articulate a novel connection between signature kernels and the hyperbolic development map, the latter of which has been a broadly useful tool in the analysis of the signature. As an application we evaluate the use of different general signature kernels as the basis for non-parametric goodness-of-fit tests to Wiener measure on path space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_00447 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | General Signature Kernels Cass, Thomas Lyons, Terry Xu, Xingcheng Probability 60L10, 60L90, 68Q27 Suppose that $γ$ and $σ$ are two continuous bounded variation paths which take values in a finite-dimensional inner product space $V$. Recent papers have introduced the truncated and the untruncated signature kernel of $γ$ and $σ$, and showed how these concepts can be used in classification and prediction tasks involving multivariate time series. In this paper, we introduce general signature kernels and show how they can be interpreted, in many examples, as an average of PDE solutions, and hence how they can be computed using suitable quadrature rules. We extend this analysis to derive closed-form formulae for expressions involving the expected (Stratonovich) signature of Brownian motion. In doing so, we articulate a novel connection between signature kernels and the hyperbolic development map, the latter of which has been a broadly useful tool in the analysis of the signature. As an application we evaluate the use of different general signature kernels as the basis for non-parametric goodness-of-fit tests to Wiener measure on path space. |
| title | General Signature Kernels |
| topic | Probability 60L10, 60L90, 68Q27 |
| url | https://arxiv.org/abs/2107.00447 |