Overview
This research focuses on the computational aspects of the Volterra signature, a mathematical construct extending the classical path signature for time series analysis. The Volterra signature incorporates a general matrix-valued kernel into its iterated integral structure, providing a flexible means to represent memory within time series data. Its components are interpreted as successive Picard iterates derived from linear controlled Volterra equations.
Research Context
The classical path signature is a tool used in time series analysis. The Volterra signature expands upon this by including a matrix-valued kernel, which introduces a notion of memory into the time series representation. While the exact computation of its components holds mathematical interest due to their interpretation as Picard iterates of controlled Volterra equations, the inclusion of the kernel introduces notable algorithmic complexities. Prior work, specifically arXiv:2603.04525, established a Chen-type convolution relation for the Volterra signature.
Approach
The research first decomposes the Chen-type convolution relation into distinct analytic and arithmetic components. This decomposition facilitates the development of multiple computational algorithms. The approach includes:
- A general approximative scheme.
- An FFT-based acceleration mechanism.
- An exact recursion method.
- A finite-difference predictor-corrector scheme.
These algorithms are designed to address the challenges posed by the kernel within the Volterra signature's structure.
Findings
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The Chen-type convolution relation established in arXiv:2603.04525 can be decomposed into analytic and arithmetic parts.
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An efficient approximative scheme for computing Volterra signature components demonstrates a quadratic complexity of $O(J^2)$ with respect to $J$, the number of time steps.
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For convolution kernels defined on uniform grids, an FFT-based acceleration algorithm achieves a complexity of $O(J\log J)$.
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An exact recursion algorithm is developed for kernels that permit a state-space representation of dimension $R$, exhibiting a complexity of $O(JR^2)$. This method maintains the standard signature complexity concerning the path dimension and truncation level $N$.
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For matrix-valued kernels structured as $K(t,s)=\sum_p k_p(t-s)A_p$, the number of factors ($p$) does not increase the asymptotic complexity in terms of $J$ and $N$.
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A finite-difference predictor-corrector scheme is derived specifically for the associated Volterra signature kernel.
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All developed algorithms are implemented within a publicly available JAX-based software package named "tensordev."
Potential Applications
The developed algorithms and the JAX-based package "tensordev" provide computational tools for working with the Volterra signature. This could facilitate further mathematical exploration and application of Volterra signatures in areas where time series memory is relevant.