Joscha Diehl

Joscha Diehl, Richard Krieg

We introduce a pipeline for time series classification that extracts features based on the iterated-sums signature (ISS) and then applies a linear classifier. These features are intrinsically nonlinear, capture chronological information, and, under certain settings, are invariant to time-warping. We are competitive with state-of-the-art methods on the UCR archive, both in terms of accuracy and speed. We make our code available at \url{https://github.com/irkri/fruits}.

Joscha Diehl

Aggregation of time-series or image data over subsets of the domain is a fundamental task in data science. We show that many known aggregation operations can be interpreted as (double) functors on appropriate (double) categories. Such functorial aggregations are amenable to parallel implementation via straightforward extensions of Blelloch's parallel scan algorithm. In addition to providing a unified viewpoint on existing operations, it allows us to propose new aggregation operations for time-series and image data.

Joscha Diehl, Kurusch Ebrahimi-Fard, Nikolas Tapia

We explore the algebraic properties of a generalized version of the iterated-sums signature, inspired by previous work of F.~Király 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.

Joscha Diehl, Kurusch Ebrahimi-Fard, Nikolas Tapia

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.

Joscha Diehl, Jeremy Reizenstein

We introduce a novel class of features for multidimensional time series, that are invariant with respect to transformations of the ambient space. The general linear group, the group of rotations and the group of permutations of the axes are considered. The starting point for their construction is Chen's iterated-integral signature.

Joscha Diehl

We introduce a novel class of rotation invariants of two dimensional curves based on iterated integrals. The invariants we present are in some sense complete and we describe an algorithm to calculate them, giving explicit computations up to order six. We present an application to online (stroke-trajectory based) character recognition. This seems to be the first time in the literature that the use of iterated integrals of a curve is proposed for (invariant) feature extraction in machine learning applications.