Path Imputation Strategies for Signature Models of Irregular Time Series

Originally described by Chen [5, 6, 7] and popularised in the theory of rough paths and controlled differential equations [14, 31, 32], the signature transform, also known as the path signature or simply signature, acts on a continuous vector-valued path of bounded variation, and returns a graded sequence of statistics, which determine a path up to a negligible equivalence class. Moreover, every continuous function of a path can be recovered by applying a linear transform to this collection of statistics [3, Proposition A.6]. This'universal nonlinearity' property makes the signature a promising nonparametric feature extractor in both generative and discriminative learning scenarios. Further properties include the signature's uniqueness [20], as well as factorial decay of its higher order terms [32]. These theoretical foundations have been accompanied by outstanding empirical results when applying signatures to clinical time series classification tasks [34, 40]. Due to their similarities, we may hope that tools that apply to continuous paths can also be applied to multivariate time series. But since multivariate time series are not continuous paths, one first needs to construct a continuous path before signature techniques are applicable. Previous work [3, 12, 27] characterised this construction as an embedding problem, and typically considered it a minor technical detail. This is exacerbated by the--perfectly sensible--behaviour of software for computing the signature [22, 39], which commonly considers a continuous piecewise linear path as an input, described by its sequence of knots, i.e. values.