# Tube Loss: A Novel Approach for Prediction Interval Estimation

arXiv: https://arxiv.org/abs/2412.06853 · HTML: https://mlsanity.com/paper/2412.06853/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 41 references that are in the map.
- First posted on arXiv 8 Dec 2024.

## Builds on (most cited first, 14 of 14)

- [Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks](https://mlsanity.com/paper/1908.10084/) (2019)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks](https://mlsanity.com/paper/1704.04110/) (2017)
- [SemEval-2017 Task 1: Semantic Textual Similarity - Multilingual and Cross-lingual Focused Evaluation](https://mlsanity.com/paper/1708.00055/) (2017)
- [Conformalized Quantile Regression](https://mlsanity.com/paper/1905.03222/) (2019)
- [TimeGPT-1](https://mlsanity.com/paper/2310.03589/) (2023)
- [High-Quality Prediction Intervals for Deep Learning: A Distribution-Free, Ensembled Approach](https://mlsanity.com/paper/1802.07167/) (2018)
- [Beyond Pinball Loss: Quantile Methods for Calibrated Uncertainty Quantification](https://mlsanity.com/paper/2011.09588/) (2020)
- [Prediction Intervals: Split Normal Mixture from Quality-Driven Deep Ensembles](https://mlsanity.com/paper/2007.09670/) (2020)
- [Disentangling Uncertainty in Machine Translation Evaluation](https://mlsanity.com/paper/2204.06546/) (2022)
- [Relaxed Quantile Regression: Prediction Intervals for Asymmetric Noise](https://mlsanity.com/paper/2406.03258/) (2024)
- [Conformalizing Machine Translation Evaluation](https://mlsanity.com/paper/2306.06221/) (2023)

## Built on by (most cited first, 2 of 2)

- [Uncertainty Quantification in SVM prediction](https://mlsanity.com/paper/2505.15429/) (2025)
- [Tube Loss based Deep Networks For Improving the Probabilistic Forecasting of Wind Speed](https://mlsanity.com/paper/2505.18284/) (2025)

## Abstract (arXiv)

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 ∈ (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.