# Time-warping invariants of multidimensional time series

## TL;DR

This paper explores time-warping invariants of multidimensional time series, linking them to iterated sums called quasisymmetric functions, and develops an algebraic framework for these features.

## Contribution

It introduces a novel algebraic approach to characterize time-warping invariants using quasisymmetric functions, providing foundational properties for feature extraction.

## Key findings

- Identifies quasisymmetric functions as invariants under time-warping.
- Provides an algebraic framework for these invariants.
- Lays groundwork for invariant feature extraction in time series analysis.

## Abstract

In data science, one is often confronted with a time series representing measurements of some quantity of interest. Usually, as a first step, features of the time series need to be extracted. These are numerical quantities that aim to succinctly describe the data and to dampen the influence of noise. In some applications, these features are also required to satisfy some invariance properties. In this paper, we concentrate on time-warping invariants. We show that these correspond to a certain family of iterated sums of the increments of the time series, known as quasisymmetric functions in the mathematics literature. We present these invariant features in an algebraic framework, and we develop some of their basic properties.