Multivariate Forecasting Evaluation: On Sensitive and Strictly Proper Scoring Rules
This work addresses the need for better evaluation methods in multivariate forecasting, which is crucial for fields like finance and climate science, though it is incremental as it builds on existing scoring rules.
The paper tackles the problem of evaluating multivariate probabilistic forecasts by analyzing the sensitivity of common scoring rules and proposing new copula-based scoring rules. The results show that the widely used energy score performs comparably well in detecting forecast errors, contradicting other studies, and that a proposed copula score strongly distinguishes models with correct versus incorrect dependency structures.
In recent years, probabilistic forecasting is an emerging topic, which is why there is a growing need of suitable methods for the evaluation of multivariate predictions. We analyze the sensitivity of the most common scoring rules, especially regarding quality of the forecasted dependency structures. Additionally, we propose scoring rules based on the copula, which uniquely describes the dependency structure for every probability distribution with continuous marginal distributions. Efficient estimation of the considered scoring rules and evaluation methods such as the Diebold-Mariano test are discussed. In detailed simulation studies, we compare the performance of the renowned scoring rules and the ones we propose. Besides extended synthetic studies based on recently published results we also consider a real data example. We find that the energy score, which is probably the most widely used multivariate scoring rule, performs comparably well in detecting forecast errors, also regarding dependencies. This contradicts other studies. The results also show that a proposed copula score provides very strong distinction between models with correct and incorrect dependency structure. We close with a comprehensive discussion on the proposed methodology.