Introduction to the signature method

TL;DR

The paper introduces the signature method, a mathematical technique for effectively representing and analyzing sequential data in earth sciences, enabling improved learning and data assimilation by capturing path information.

Contribution

It presents the signature method as a powerful tool for converting paths into numerical sequences, facilitating high-performance nonlinear function approximation in machine learning.

Findings

Signature can faithfully describe path order and nonlinearity.

Linear regression on signatures can approximate nonlinear functions.

Method enhances data analysis in earth science applications.

Abstract

The sequential data observed in earth science can be regarded as paths in multidimensional space. To read the path effectively, it is useful to convert it into a sequence of numbers called the signature, which can faithfully describe the order of points and nonlinearity in the path. In particular, a linear combination of the terms in a signature can be used to approximate any nonlinear function defined on a set of paths. Thereby, when one learns a set of sequential data with labels attached to it, linear regression can be applied to the pairs of signature and label, which will achieve high performance learning even when the labels are determined by a nonlinear function. By incorporating the signature methods into machine learning and data assimilation utilizing sequential data, it is expected that we can extract information that has previously been overlooked.