16th Brazilian Meeting of Bayesian Statistics and VI Latin American Conference on Statistical Computing Proceedings

Measurable phenomena in many real problems take value among time-dependent boundaries, which are estimable using physics-based information or other sources.
In this work, we extend our data-driven Itô's Stochastic Differential Equations (SDEs) framework for the path-wise assessment of the forecast error, developed in (Caballero et al. 2021), to deal with the case where a time-varying boundary is constraining above both the nonnegative chronological series of historical data and its forecast.
As a reference case study, we focus on the temporal evolution of solar photovoltaics (PV) power production and its forecast error.
We propose a new class of non-linear and time-inhomogeneous SDE models for the observable phenomenon of interest, driven by the forecast and upper bound smooth functions. Using rigorous mathematical theory, we derive the existence and uniqueness of the strong solution to the SDE models, summarized through a condition upon the time-varying parameter that, in the drift term, describes the intensity of the reversion to the mean of the process. Next, the time-varying intensity parameter is kept bounded by truncating the original forecast function through a threshold parameter. As a result, at any finite time interval, the paths of the process modeling the forecast error do not hit almost surely the time-dependent boundaries.
The model parameters calibration, including the threshold truncation parameter of the forecast function, is performed by applying a novel multi-stage robust optimization procedure to an approximation of the unknown likelihood function. In this work, the transition density of the forecast error process is assumed to follow a Beta distribution, whose parameters are determined through the moment matching technique. The SDE model is fitted to the 2019 daily Uruguayan solar PV power production and forecast data, using the computed daily maximum solar PV production. Path-wise bands with the desired confidence are provided by an indirect inference method, estimating the theoretical quantiles of the Beta distribution using the moment equations. Our methodology achieves the complete characterization of the forecast error uncertainty when time-dependent boundaries, as well as forecast, guide the model specification.

Reference
Caballero, R., Kebaier, A., Scavino, M., Tempone, R. (2021). Quantifying uncertainty with a derivative tracking SDE model and application to wind power forecast data, Statistics and Computing, vol.31, 64. https://link.springer.com/article/10.1007/s11222-021-10040-8.