Learning with signatures: estimation in a linear model and a generalized signature method for time series classification

We will be concerned with the application of signatures to machine learning. The basic principle of the signature method is to represent multidimensional paths by a graded feature set of their iterated integrals, called the signature. On the one hand, in order to combine signatures with machine learning algorithms, it is necessary to truncate these infinite series. Therefore, we define an estimator of the truncation order and provide theoretical guarantees in a linear functional regression setting. On the other hand, the signature method presents several variations, which can be grouped into "augmentations", "windows", "transforms" and "rescalings". We perform an empirical study on which aspects of this framework typically produce the best results for multivariate time series classification. Combining the top choices produces a canonical pipeline for the generalised signature method, which demonstrates state-of-the-art accuracy.