Ellipsoidal Prediction Regions for Multivariate Uncertainty Characterization

While substantial advances are observed in probabilistic forecasting for power system operation and electricity market applications, most approaches are still developed in a univariate framework. This prevents from informing about the interdependence structure among locations, lead times, and variables of interest. Such dependencies are key in a large share of operational problems involving renewable power generation and electricity prices for instance. The few methods that account for dependencies translate to sampling scenarios based on given marginals and dependence structures. However, for classes of decision-making problems based on robust, interval, and chance-constrained optimization, necessary inputs take the form of multivariate prediction regions rather than scenarios. The current literature is at a very primitive stage of characterizing multivariate prediction regions to be employed in these classes of optimization problems. To address this issue, we introduce a new class of multivariate forecasts, which form as multivariate ellipsoids for non-Gaussian variables. We propose a data-driven systematic framework to readily generate and evaluate ellipsoidal prediction regions, with predefined probability guarantees and minimum conservativeness. A skill score is proposed for quantitative assessment of the quality of prediction ellipsoids. A set of experiments is used to illustrate the discrimination ability of the proposed scoring rule for potential misspecification of ellipsoidal prediction regions. Application results based on three datasets with wind, photovoltaic power, and electricity prices allow us to assess the skill of the resulting ellipsoidal prediction regions, in terms of calibration, sharpness, and overall skill.