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General Signature Kernels

Thomas Cass, Terry Lyons, Xingcheng Xu

Abstract

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.

General Signature Kernels

Abstract

Suppose that and are two continuous bounded variation paths which take values in a finite-dimensional inner product space . Recent papers have introduced the truncated and the untruncated signature kernel of and , 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.

Paper Structure

This paper contains 19 sections, 24 theorems, 175 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Background on General Signature Kernels
  3. Representing General Signature Kernels
  4. Truncated Signature Kernels
  5. General Signature Kernels by Randomisation
  6. General Signature Kernels by Fourier Series
  7. General Signature Kernels by Integral Transforms
  8. Computing General Signature Kernels
  9. Expected General Signature Kernels
  10. Hyperbolic Development
  11. Signature Kernels and Hyperbolic Development
  12. The Original Kernel for Expected Signatures
  13. Expected signatures for general kernels
  14. Optimal Discrete Measures on Paths
  15. Existence and Uniqueness of Optimal Discrete Measure
  16. ...and 4 more sections

Key Result

Lemma 2.3

Let $\phi:\mathbb{N}\cup\left\{ 0\right\} \rightarrow\mathbb{R}_{+}$ be such that for every $C>0$ the series $\sum_{k\in\mathbb{N}}C^{k}\phi\left(k\right)\left(k!\right)^{-2}$ is summable. Then $\mathcal{S\subset}T_{\phi}\left(V\right).$

Figures (7)

  • Figure 7.1: Boxplots of the factorially-weighted signature kernel. (a) The left panel shows the distribution of the values of the alignment $\cos\angle_{\phi}(\mu^{*},\mathcal{W})$ of the optimal measure and the Wiener measure across 400 samples. The x-axis is the dimension of the Brownian motion, and the y-axis the value of the alignment. (b) The right panel shows the same for the MMD distance $d_{\phi}(\mu^{*},\mathcal{W}).$
  • Figure 7.2: The optimal measure under the original signature kernel
  • Figure 7.3: The similarity under a family of Beta-weighted signature kernels. The left panel is the plot of the distance of these discrete measures and the Wiener measure plotted against different values of $m$ on the horizontal axis. The right panel plots the ratio of the optimal distance and the cubature distance.
  • Figure 7.4: The case for the factorially-weighted signature kernel. (a) and (c) show similarities of discrete measures and the Wiener measure where the horizontal is the value of $\epsilon$ and vertical axis is the value of alignment. (b) and (d) show similarities of discrete measures and the Wiener measure. The solid line is for the optimal measure while the dashed line is for the empirical measure. The upper panel is for the frequency $\nu=2$ and the lower is for $\nu=3$.
  • Figure 7.5: The same example under the original signature kernel
  • ...and 2 more figures

Theorems & Definitions (64)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Proposition 2.8
  • ...and 54 more