Learning with signatures: embedding and truncation order selection
Sequential and temporal data arise in many fields of research, such as quantitative finance, medicine, or computer vision. We will be concerned with a novel approach for sequential learning, called the signature method, and rooted in rough path theory. Its basic principle is to represent multidimensional paths by a graded feature set of their iterated integrals, called the signature. On the one hand, this approach relies critically on an embedding principle, which consists in representing discretely sampled data as paths, i.e., functions from [0,1] to R^d. We investigate the influence of embeddings on prediction accuracy with an in-depth study of three recent and challenging datasets. We show that a specific embedding, called lead-lag, is systematically better, whatever the dataset or algorithm used. On the other 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 prove its convergence in the expected signature model.