Tropical time series, iterated-sums signatures and quasisymmetric functions

TL;DR

This paper introduces the iterated-sums signature over semirings, especially the tropical semiring, providing a new feature extraction method for time series that captures chronological information efficiently.

Contribution

It develops a systematic framework for feature extraction from time series using iterated-sums signatures over semirings, with a focus on tropical semirings and quasisymmetric functions.

Findings

Signature extracts chronological aspects of time series

Calculation of signature is achievable in linear time

Framework applies to semiring-valued time series

Abstract

Aiming for a systematic feature-extraction from time series, we introduce the iterated-sums signature over arbitrary commutative semirings. The case of the tropical semiring is a central, and our motivating example. It leads to features of (real-valued) time series that are not easily available using existing signature-type objects. We demonstrate how the signature extracts chronological aspects of a time series, and that its calculation is possible in linear time. We identify quasisymmetric expressions over semirings as the appropriate framework for iterated-sums signatures over semiring-valued time series.