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In this article, we present a novel approach to multivariate probabilisticforecasting. Our approach is based on an extension of single-output quantileregression (QR) to multivariate-targets, called quantile surfaces (QS). QS usesa simple yet compelling idea of indexing observations of a probabilisticforecast through direction and vector length to estimate a central tendency. Weextend the single-output QR technique to multivariate probabilistic targets. QSefficiently models dependencies in multivariate target variables and representsprobability distributions through discrete quantile levels. Therefore, wepresent a novel two-stage process. In the first stage, we perform adeterministic point forecast (i.e., central tendency estimation). Subsequently,we model the prediction uncertainty using QS involving neural networks calledquantile surface regression neural networks (QSNN). Additionally, we introducenew methods for efficient and straightforward evaluation of the reliability andsharpness of the issued probabilistic QS predictions. We complement this by thedirectional extension of the Continuous Ranked Probability Score (CRPS) score.Finally, we evaluate our novel approach on synthetic data and two currentlyresearched real-world challenges in two different domains: First, probabilisticforecasting for renewable energy power generation, second, short-term cycliststrajectory forecasting for autonomously driving vehicles. Especially for thelatter, our empirical results show that even a simple one-layer QSNNoutperforms traditional parametric multivariate forecasting techniques, thusimproving the state-of-the-art performance.

Quantile Surfaces -- Generalizing Quantile Regression to Multivariate Targets