Quantile Surfaces -- Generalizing Quantile Regression to Multivariate Targets

Abstract

In this article, we present a novel approach to multivariate probabilistic forecasting. Our approach is based on an extension of single-output quantile regression (QR) to multivariate-targets, called quantile surfaces (QS). QS uses a simple yet compelling idea of indexing observations of a probabilistic forecast through direction and vector length to estimate a central tendency. We extend the single-output QR technique to multivariate probabilistic targets. QS efficiently models dependencies in multivariate target variables and represents probability distributions through discrete quantile levels. Therefore, we present a novel two-stage process. In the first stage, we perform a deterministic point forecast (i.e., central tendency estimation). Subsequently, we model the prediction uncertainty using QS involving neural networks called quantile surface regression neural networks (QSNN). Additionally, we introduce new methods for efficient and straightforward evaluation of the reliability and sharpness of the issued probabilistic QS predictions. We complement this by the directional extension of the Continuous Ranked Probability Score (CRPS) score. Finally, we evaluate our novel approach on synthetic data and two currently researched realworld challenges in two different domains: First, probabilistic forecasting for renewable energy power generation, second, shortterm cyclists trajectory forecasting for autonomously driving vehicles. Especially for the latter, our empirical results show that even a simple one-layer QSNN outperforms traditional parametric multivariate forecasting techniques, thus improving the state-of-the-art performance.

Cyclist Intention Detection: A Probabilistic Approach

“…To further improve the sharpness of our forecasts, we will investigate the use of different distribution functions for the specialized models, e.g., skew normal distributions or Gaussian mixtures. We also plan to compare our method to different approaches used to estimate probability distributions, such as Monte Carlo dropout [30] or quantile surfaces [31].…”

Pose and Semantic Map Based Probabilistic Forecast of Vulnerable Road Users' Trajectories

“…Overall, this approach was able to make reliable forecasts for the motion types start, stop, turn left, and turn right, while the forecasts for the motion types move and wait were underconfident. Bieshaar et al [10] extended singleoutput quantile regression to multivariate-targets for trajectory forecasting of cyclists. These so-called quantile surfaces represent star-shaped distributions using discrete quantile levels.…”