Abstract
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.
Highlights
Its compatibility with the quasi-shuffle product together with the fact that the latter can be seen as a deformation of the classical shuffle product [18] suggests to consider ISS(x) as a discrete analog of Chen’s iterated-integrals signature over continuous curves [10, 44]
The latter plays an important role in the theory of controlled ordinary differential equations (ODEs), stochastic analysis and Lyons’ theory of rough paths [19, 37]
Introduced a new set of features for multidimensional time series consisting in iterated sums (Section 3);
Summary
Where N ≥ 1 is some arbitrary time horizon, our foremost, and original, motivation stems from the desire to extract features from x that are invariant to time warping. It is natural to store the iterated-sums invariants of the discrete time series x as a linear map ISS(x) on the quasi-shuffle algebra of compositions, by defining the pairing [1c1 ] · · · [1ck ], ISS(x) :=. The commutative algebra of quasisymmetric functions is the free quasi-shuffle algebra over one generator and it is - as we just saw - the correct framework to store iterated-sums for a one-dimensional time-series. Its compatibility with the quasi-shuffle product together with the fact that the latter can be seen as a deformation of the classical shuffle product [18] suggests to consider ISS(x) as a discrete analog of Chen’s iterated-integrals signature over continuous curves [10, 44] The latter plays an important role in the theory of controlled ordinary differential equations (ODEs), stochastic analysis and Lyons’ theory of rough paths [19, 37]. For details on Hopf algebras the reader is referred to [9, 26, 36, 40, 46]
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