Generalized iterated-sums signatures

We explore the algebraic properties of a generalized version of the iterated-sums signature, inspired by previous work of F.~Kir\'aly and H.~Oberhauser. In particular, we show how to recover the character property of the associated linear map over the tensor algebra by considering a deformed quasi-shuffle product of words on the latter. We introduce three non-linear transformations on iterated-sums signatures, close in spirit to Machine Learning applications, and show some of their properties.

Tropical time series, iterated-sums signatures and quasisymmetric functions

Driven by the need for principled extraction of features from time series,we introduce the iterated-sums signature over any commutative semiring.The case of the tropical semiring is a central, and our motivating, example,as it leads to features of (real-valued) time series that are not easily availableusing existing signature-type objects.

Time warping invariants of multidimensional time series

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.