Tube Loss: A Novel Approach for Prediction Interval Estimation and probabilistic forecasting
Pritam Anand, Tathagata Bandyopadhyay, Suresh Chandra
TL;DR
Tube Loss introduces a loss function $L_t$ for simultaneous estimation of upper and lower bounds of a Prediction Interval in regression and probabilistic forecasting. The method yields asymptotically valid coverage at the target level $t$ and includes a tunable shift parameter that allows moving the interval to capture denser regions, which is particularly beneficial for skewed conditional distributions. It supports a single optimization to trade off coverage and width, with optional re-calibration, and gradient-descent minimization, with demonstrated applicability to kernel methods, neural networks, and sequential models, and alignment with conformal prediction. Empirical results on synthetic data and benchmark tasks show improved PI quality over baseline approaches such as SVQR and QD-based methods, underscoring practical benefits for calibrated probabilistic forecasting.
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.