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Nonparametric Regression of Stochastic Processes via Signatures

by A. Schell and R. Alaifari

(Report number 2023-45)

Abstract
Nonparametric regression of stochastic processes estimates statistical relationships between multidimensional, time-dependent data without relying on specific parametric assumptions. We propose a novel approach to this classical estimation problem by using the signature transform from rough path theory to encode the information of a stochastic process into a sequence of iterated integrals, capturing its statistical properties in a time-global and hierarchical manner. Viewing statistical regression as an operator learning problem, this signature-based discretisation allows us to characterise the conditional statistical dependence of a stochastic process on another stochastic process as the solution to a convex semi-infinite linear least squares problem. This result is based on a functional monotone class argument involving the bounded signature of the conditioning process and allows for the efficient and provably consistent nonparametric estimation of regression functions and conditional distributions for very general classes of jointly distributed stochastic processes as solutions to convex optimisation problems. The structural insights of this approach are summarised in two universally consistent regression estimators that are computable with practical algorithms and supported by broad theoretical guarantees.

Keywords: conditional expectation, conditional distribution, conditional probability, supervised learning, nonparametric regression, functional regression, function approximation

@Techreport{SA23_1082,
  author = {A. Schell and R. Alaifari},
  title = {Nonparametric Regression of Stochastic Processes via Signatures},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2023-45},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2023/2023-45.pdf },
  year = {2023}
}

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