Tube Loss: A Novel Approach for Prediction Interval Estimation and probabilistic forecasting

2024-12-08

arXiv.org

AI Summary

This paper addresses the quality and controllability of prediction interval (PI) estimation for regression tasks. We propose Tube Loss, a novel loss function that jointly optimizes upper and lower bounds within a single-objective framework to simultaneously improve coverage probability, interval width, and distributional calibration. Key contributions include: (1) the first empirical risk minimization formulation guaranteeing asymptotically exact confidence level control; (2) a tunable parameter enabling vertical shift of PIs to handle conditional distribution skew; and (3) inherent balancing of the coverage–width trade-off, full compatibility with gradient-based optimization and diverse models (e.g., kernel methods, deep networks), and seamless integration into conformal prediction. Experiments on benchmark datasets and a wind power forecasting task demonstrate significant improvements over state-of-the-art probabilistic forecasting methods—yielding narrower average PI widths and empirical coverage rates markedly closer to target confidence levels. The code is publicly available.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Probabilistic Inference
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

This paper proposes a novel loss function, called 'Tube Loss', for simultaneous estimation of bounds of a Prediction Interval (PI) in the regression setup. The PIs obtained by minimizing the empirical risk based on the Tube Loss are shown to be of better quality than the PIs obtained by the existing methods in the following sense. First, it yields intervals that attain the prespecified confidence level t $in$ (0,1) asymptotically. A theoretical proof of this fact is given. Secondly, the user is allowed to move the interval up or down by controlling the value of a parameter. This helps the user to choose a PI capturing denser regions of the probability distribution of the response variable inside the interval, and thus, sharpening its width. This is shown to be especially useful when the conditional distribution of the response variable is skewed. Further, the Tube Loss based PI estimation method can trade-off between the coverage and the average width by solving a single optimization problem. It enables further reduction of the average width of PI through re-calibration. Also, unlike a few existing PI estimation methods the gradient descent (GD) method can be used for minimization of empirical risk. Through extensive experiments, we demonstrate the effectiveness of Tube Loss-based PI estimation in both kernel machines and neural networks. Additionally, we show that Tube Loss-based deep probabilistic forecasting models achieve superior performance compared to existing probabilistic forecasting techniques across several benchmark and wind datasets. Finally, we empirically validate the advantages of the Tube loss approach within the conformal prediction framework. Codes are available at https://github.com/ltpritamanand/Tube$_$loss.

Problem

• Estimates prediction intervals with better quality and control
• Enables adjustable intervals for skewed data distributions
• Improves probabilistic forecasting accuracy in neural networks

Innovation

• Novel Tube Loss function for interval estimation
• Adjustable prediction intervals via parameter control
• Single optimization for coverage-width trade-off