Learning from the past, predicting the statistics for the future, learning an evolving system

TL;DR

This paper introduces a novel non-parametric regression method for streaming data using rough path theory, specifically the signature transform, enabling efficient, dimension-reduced predictions with applications to time series.

Contribution

It applies the signature of streams within rough path theory to develop a general, computationally efficient regression framework for streaming data, including time series.

Findings

Achieves prediction accuracy comparable to Gaussian Processes

Offers significant computational efficiency for large datasets

Demonstrates effectiveness on stationary time series like AR and ARCH models

Abstract

We bring the theory of rough paths to the study of non-parametric statistics on streamed data. We discuss the problem of regression where the input variable is a stream of information, and the dependent response is also (potentially) a stream. A certain graded feature set of a stream, known in the rough path literature as the signature, has a universality that allows formally, linear regression to be used to characterise the functional relationship between independent explanatory variables and the conditional distribution of the dependent response. This approach, via linear regression on the signature of the stream, is almost totally general, and yet it still allows explicit computation. The grading allows truncation of the feature set and so leads to an efficient local description for streams (rough paths). In the statistical context this method offers potentially significant, even transformational dimension reduction. By way of illustration, our approach is applied to stationary time series including the familiar AR model and ARCH model. In the numerical examples we examined, our predictions achieve similar accuracy to the Gaussian Process (GP) approach with much lower computational cost especially when the sample size is large.