publication . Preprint . 2017

Calculation of Iterated-Integral Signatures and Log Signatures

Reizenstein, Jeremy;
Open Access English
  • Published: 07 Dec 2017
Abstract
We explain the algebra needed to make sense of the log signature of a path, with plenty of examples. We show how the log signature can be calculated numerically, and explain some software tools which demonstrate it.
Subjects
arXiv: Computer Science::Databases
free text keywords: Mathematics - Rings and Algebras
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[1] Fernando Casas and Ander Murua. “An efficient algorithm for computing the Baker-CampbellHausdorff series and some of its applications”. In: Journal of Mathematical Physics 50.3 (Mar. 2009), p. 033513. eprint: http://arxiv.org/abs/0810.2656.

[2] Fernando Casas and Ander Murua. The BCH formula and the symmetric BCH formula up to terms of degree 20. url: http://www.ehu.eus/ccwmuura/bch.html.

[3] Kuo-Tsai Chen. “Integration of Paths - A Faithful Representation of Paths by Noncommutative Formal Power Series”. In: Transactions of the American Mathematical Society 89.2 (1958), pp. 395- 407. url: https://www.jstor.org/stable/1993193. [OpenAIRE]

[4] Ilya Chevyrev and Andrey Kormilitzin. “A Primer on the Signature Method in Machine Learning”. 2016. url: http://arxiv.org/abs/1603.03788. [OpenAIRE]

[8] Terry Lyons, Michael Caruana, and Thierry Lévy. Differential Equations Driven by Rough Paths. 2007.

[9] Terry Lyons, Stephen Buckley, et al. CoRoPa Computational Rough Paths (software library). 2010. url: http://coropa.sourceforge.net/.

[10] Christophe Reutenauer. Free Lie Algebras. 1994.

[11] Wikipedia. Necklace polynomial. 2015. url: http://en.wikipedia.org/wiki/Necklace_polynomial.

Abstract
We explain the algebra needed to make sense of the log signature of a path, with plenty of examples. We show how the log signature can be calculated numerically, and explain some software tools which demonstrate it.
Subjects
arXiv: Computer Science::Databases
free text keywords: Mathematics - Rings and Algebras
Download from

[1] Fernando Casas and Ander Murua. “An efficient algorithm for computing the Baker-CampbellHausdorff series and some of its applications”. In: Journal of Mathematical Physics 50.3 (Mar. 2009), p. 033513. eprint: http://arxiv.org/abs/0810.2656.

[2] Fernando Casas and Ander Murua. The BCH formula and the symmetric BCH formula up to terms of degree 20. url: http://www.ehu.eus/ccwmuura/bch.html.

[3] Kuo-Tsai Chen. “Integration of Paths - A Faithful Representation of Paths by Noncommutative Formal Power Series”. In: Transactions of the American Mathematical Society 89.2 (1958), pp. 395- 407. url: https://www.jstor.org/stable/1993193. [OpenAIRE]

[4] Ilya Chevyrev and Andrey Kormilitzin. “A Primer on the Signature Method in Machine Learning”. 2016. url: http://arxiv.org/abs/1603.03788. [OpenAIRE]

[8] Terry Lyons, Michael Caruana, and Thierry Lévy. Differential Equations Driven by Rough Paths. 2007.

[9] Terry Lyons, Stephen Buckley, et al. CoRoPa Computational Rough Paths (software library). 2010. url: http://coropa.sourceforge.net/.

[10] Christophe Reutenauer. Free Lie Algebras. 1994.

[11] Wikipedia. Necklace polynomial. 2015. url: http://en.wikipedia.org/wiki/Necklace_polynomial.

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