Tube Loss: A Novel Approach for High Quality Prediction Interval Estimation

Abstract

This paper proposes a continuous loss function termed 'tube loss' for Prediction Interval (PI) estimation. The minimizer of the proposed tube loss is a pair of functions $\mu_1(x)$ and $\mu_2(x)$ such that the interval $[\mu_1(x),\mu_2(x)]$ contains $t$ fraction of $y_i$ values. The tube loss function also facilitates an upward or downward movement of the PI tube so that the estimated PI may cover the densest regions of response values, thus allowing the sharpening of the width of PI, especially when the distribution of the response is skewed. The tube loss function-based machine learning models also have the privilege of trading off the calibration error and the width of PI by solving a single optimization problem. We have illustrated the use of tube loss functions in kernel machines, neural networks, and sequential deep learning models. Our numerical experiments show that the tube loss function is effective in yielding narrow and more accurate PIs compared to the existing methods.