Quantifying uncertainty with a derivative tracking SDE model and application to wind power forecast data

Abstract

We develop a data-driven methodology based on parametric Itô’s Stochastic Differential Equations (SDEs) to capture the real asymmetric dynamics of forecast errors, including the uncertainty of the forecast at time zero. Our SDE framework features time-derivative tracking of the forecast, time-varying mean-reversion parameter, and an improved state-dependent diffusion term. Proofs of the existence, strong uniqueness, and boundedness of the SDE solutions are shown by imposing conditions on the time-varying mean-reversion parameter. We develop the structure of the drift term based on sound mathematical theory. A truncation procedure regularizes the prediction function to ensure that the trajectories do not reach the boundaries almost surely in a finite time. Inference based on approximate likelihood, constructed through the moment-matching technique both in the original forecast error space and in the Lamperti space, is performed through numerical optimization procedures. We propose a fixed-point likelihood optimization approach in the Lamperti space. Another novel contribution is the characterization of the uncertainty of the forecast at time zero, which turns out to be crucial in practice. We extend the model specification by considering the length of the unknown time interval preceding the first time a forecast is provided through an additional parameter in the density of the initial transition. All the procedures are agnostic of the forecasting technology, and they enable comparisons between different forecast providers. We apply our SDE framework to model historical Uruguayan normalized wind power production and forecast data between April and December 2019. Sharp empirical confidence bands of wind power production forecast error are obtained for the best-selected model.