The energy distance for ensemble and scenario reduction

Philos Trans A Math Phys Eng Sci. 2021 Jul 26;379(2202):20190431. doi: 10.1098/rsta.2019.0431. Epub 2021 Jun 7.


Scenario reduction techniques are widely applied for solving sophisticated dynamic and stochastic programs, especially in energy and power systems, but are also used in probabilistic forecasting, clustering and estimating generative adversarial networks. We propose a new method for ensemble and scenario reduction based on the energy distance which is a special case of the maximum mean discrepancy. We discuss the choice of energy distance in detail, especially in comparison to the popular Wasserstein distance which is dominating the scenario reduction literature. The energy distance is a metric between probability measures that allows for powerful tests for equality of arbitrary multivariate distributions or independence. Thanks to the latter, it is a suitable candidate for ensemble and scenario reduction problems. The theoretical properties and considered examples indicate clearly that the reduced scenario sets tend to exhibit better statistical properties for the energy distance than a corresponding reduction with respect to the Wasserstein distance. We show applications to a Bernoulli random walk and two real data-based examples for electricity demand profiles and day-ahead electricity prices. This article is part of the theme issue 'The mathematics of energy systems'.

Keywords: Kontorovic distance; Wasserstein metric; electricity load; energy score; maximum mean discrepancy; scenario reduction; stochastic programming.