Encoding Time Series Dynamics with Sliding-Window Signatures for Online Forecasting

Accurate forecasting often relies on both past target values and exogenous variables as inputs. In many applications, however, the time series of interest are non-stationary and the relationship between target and inputs evolves over time, requiring forecasters that adapt online. Moreover, inputs affect the target not only through their current values but also through their recent joint dynamics. We address these challenges in three steps. First, to encode the series dynamics, we rely on signatures of the input paths over sliding windows. Motivated by the universality of linear functionals of signatures, we fit a regularized linear regression on these features to capture nonlinear, path-dependent effects. Second, the simplicity of the linear model allows us to update it online as new data arrives, using a Kalman filter to track dynamic effects. Third, to capture dependencies at several time scales, we combine forecasters built on different window lengths through online aggregation. For French national electricity load forecasting, our method outperforms various state-of-the-art approaches. It also remains competitive across several benchmark datasets with exogenous variables.