February 2024 Weighted signature kernels
Thomas Cass, Terry Lyons, Xingcheng Xu
Author Affiliations +
Ann. Appl. Probab. 34(1A): 585-626 (February 2024). DOI: 10.1214/23-AAP1973


Suppose that γ and σ are two continuous bounded variation paths, which take values in a finite-dimensional inner product space V. The recent papers (J. Mach. Learn. Res. 20 (2019) 1–45) and (SIAM J. Math. Data Sci. 3 (2021) 873–899), respectively, 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 signature kernels Kϕγ,σ indexed by a weight function ϕ, which generalise the ordinary signature kernel. We show how Kϕγ,σ can be interpreted in many examples as an average of PDE solutions, and thus we show how it can be estimated computationally using suitable quadrature formulae. 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 notion of the hyperbolic development of a path, which has been a broadly useful tool in the recent analysis of the signature; see, for example, (Ann. of Math. (2) 171 (2010) 109–167; J. Funct. Anal. 272 (2017) 2933–2955) and (Trans. Amer. Math. Soc. 372 (2019) 585–614). As applications, we evaluate the use of different general signature kernels as a basis for nonparametric goodness-of-fit tests to Wiener measure on path space.

Funding Statement

The work of all three authors was supported by the EPSRC Programme Grant EP/S026347/1. Terry Lyons was funded in part by the EPSRC [Grant No. EP/S026347/1], in part by The Alan Turing Institute under the EPSRC Grant EP/N510129/1, the Data Centric Engineering Programme (under the Lloyd’s Register Foundation Grant G0095), the Defence and Security Programme (funded by the UK Government) and the Office for National Statistics & The Alan Turing Institute (strategic partnership) and in part by the Hong Kong Innovation and Technology Commission (InnoHK Project CIMDA).


The authors would like to thank Cris Salvi and Bojan Nikolic, respectively, for discussions related to the signature kernel and the problem of RFI mitigation in radio astronomy. The authors are also grateful to the anonymous referees for their helpful comments on the first draft of this paper.


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Thomas Cass. Terry Lyons. Xingcheng Xu. "Weighted signature kernels." Ann. Appl. Probab. 34 (1A) 585 - 626, February 2024. https://doi.org/10.1214/23-AAP1973


Received: 1 July 2021; Revised: 1 April 2023; Published: February 2024
First available in Project Euclid: 28 January 2024

MathSciNet: MR4696286
zbMATH: 07829151
Digital Object Identifier: 10.1214/23-AAP1973

Primary: 60L10

Keywords: contour integration , expected signatures , Gaussian quadrature , general signature kernels , hyperbolic development , kernel methods , The signature

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.34 • No. 1A • February 2024
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