WIAS Preprint No. 2603, (2019)

Time-warping invariants of multidimensional time series


  • Diehl, Joscha
  • Kurusch, Ebrahimi-Fard
  • Tapia, Nikolas
    ORCID: 0000-0003-0018-2492

2010 Mathematics Subject Classification

  • 16T05 68T10 62H30

2008 Physics and Astronomy Classification Scheme

  • 68Q10


  • Time series analysis, time-warping invariants, signature, quasisymmetric functions, quasi-shuffle product, Hoffman's exponential, area-operation, Hopf algebra




In data science, one is often confronted with a time series representing measurements of some quantity of interest. Usually, in 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.

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