Functional regression for estimating probability density functions: an application to electricity price forecasting

Probabilistic forecasting of electricity prices in the medium term is highly important for operational scheduling, fuel purchasing, trading and profit management. In this context, fundamental models are frequently used, which obtain a probabilistic forecast based on market equilibrium simulations. While they provide insights when structural and regulatory changes are expected to happen in the market, these are not well calibrated to actual data. That is why hybrid methods are a growing research field, whose objective is to aggregate the fundamental forecasts with statistical methods to increase predictive capability. The proposed hybrid approach is to use a functional regression model that estimates the probability density function of the electricity price for each hour using, as explanatory variables, the probabilistic forecasts from the fundamental model. The functional parameters used in the regression are integral operators in the $L^2$ space and, in this approach, the kernels of the operators are modeled as a linear combination of sigmoid functions. The novelty of the method is that, as the endogenous variable is unobserved (only price realizations are known), the parameters are estimated by maximizing the likelihood of the price realizations over the estimated density functions.