Geometry of nonlinear forecast reconciliation
It addresses a gap in the forecasting literature for researchers and practitioners needing theoretical guarantees in nonlinear probabilistic settings, though it is incremental as it extends existing linear results.
This paper tackles the lack of formal error reduction theorems for forecast reconciliation in nonlinear contexts by establishing such theorems for various classes of nonlinear hypersurfaces and vector-valued functions, including an exact analog of a prior theorem for constant-sign curvature cases and probabilistic guarantees for broader scenarios.
Forecast reconciliation, an ex-post technique applied to forecasts that must satisfy constraints, has been a prominent topic in the forecasting literature over the past two decades. Recently, several efforts have sought to extend reconciliation methods to the probabilistic settings. Nevertheless, formal theorems demonstrating error reduction in nonlinear contexts, analogous to those presented in Panagiotelis et al.(2021), are still lacking. This paper addresses that gap by establishing such theorems for various classes of nonlinear hypersurfaces and vector-valued functions. Specifically, we derive an exact analog of Theorem 3.1 from Panagiotelis et al.(2021) for hypersurfaces with constant-sign curvature. Additionally, we provide probabilistic guarantees for the broader case of hypersurfaces with non-constant-sign curvature and for general vector-valued functions. To support reproducibility and practical adoption, we release a JAX-based Python package, \emph{to be released upon publication}, implementing the presented theorems and reconciliation procedures.