# Learning Quantile Functions without Quantile Crossing for Distribution-free Time Series Forecasting

arXiv: https://arxiv.org/abs/2111.06581 · HTML: https://mlsanity.com/paper/2111.06581/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 39 references that are in the map.
- First posted on arXiv 12 Nov 2021.

## Builds on (most cited first, 13 of 13)

- [DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks](https://mlsanity.com/paper/1704.04110/) (2017)
- [Temporal Fusion Transformers for Interpretable Multi-horizon Time Series Forecasting](https://mlsanity.com/paper/1912.09363/) (2019)
- [A Multi-Horizon Quantile Recurrent Forecaster](https://mlsanity.com/paper/1711.11053/) (2017)
- [High-Dimensional Multivariate Forecasting with Low-Rank Gaussian Copula Processes](https://mlsanity.com/paper/1910.03002/) (2019)
- [Deep Factors for Forecasting](https://mlsanity.com/paper/1905.12417/) (2019)
- [Domain Adaptation for Time Series Forecasting via Attention Sharing](https://mlsanity.com/paper/2102.06828/) (2021)
- [Probabilistic Forecasting with Temporal Convolutional Neural Network](https://mlsanity.com/paper/1906.04397/) (2019)
- [MQTransformer: Multi-Horizon Forecasts with Context Dependent and Feedback-Aware Attention](https://mlsanity.com/paper/2009.14799/) (2020)
- [Deep Generative Quantile-Copula Models for Probabilistic Forecasting](https://mlsanity.com/paper/1907.10697/) (2019)
- [Foundations of Sequence-to-Sequence Modeling for Time Series](https://mlsanity.com/paper/1805.03714/) (2018)
- [Variance Reduced Training with Stratified Sampling for Forecasting Models](https://mlsanity.com/paper/2103.02062/) (2021)
- [Structured Policy Iteration for Linear Quadratic Regulator](https://mlsanity.com/paper/2007.06202/) (2020)
- [Spliced Binned-Pareto Distribution for Robust Modeling of Heavy-tailed Time Series](https://mlsanity.com/paper/2106.10952/) (2021)

## Built on by (most cited first, 30 of 32)

- [Unified Training of Universal Time Series Forecasting Transformers](https://mlsanity.com/paper/2402.02592/) (2024)
- [GIFT-Eval: A Benchmark For General Time Series Forecasting Model Evaluation](https://mlsanity.com/paper/2410.10393/) (2024)
- [This Time is Different: An Observability Perspective on Time Series Foundation Models](https://mlsanity.com/paper/2505.14766/) (2025)
- [Towards Neural Scaling Laws for Time Series Foundation Models](https://mlsanity.com/paper/2410.12360/) (2024)
- [Copula Conformal Prediction for Multi-step Time Series Forecasting](https://mlsanity.com/paper/2212.03281/) (2022)
- [Pushing the Limits of Pre-training for Time Series Forecasting in the CloudOps Domain](https://mlsanity.com/paper/2310.05063/) (2023)
- [Multivariate Quantile Function Forecaster](https://mlsanity.com/paper/2202.11316/) (2022)
- [Taming Hyperparameter Tuning in Continuous Normalizing Flows Using the JKO Scheme](https://mlsanity.com/paper/2211.16757/) (2022)
- [Robust Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2202.11910/) (2022)
- [VisionTS++: Cross-Modal Time Series Foundation Model with Continual Pre-trained Vision Backbones](https://mlsanity.com/paper/2508.04379/) (2025)
- [End-to-End Probabilistic Framework for Learning with Hard Constraints](https://mlsanity.com/paper/2506.07003/) (2025)
- [Robust Multivariate Time-Series Forecasting: Adversarial Attacks and Defense Mechanisms](https://mlsanity.com/paper/2207.09572/) (2022)
- [Extraction and Recovery of Spatio-Temporal Structure in Latent Dynamics Alignment with Diffusion Models](https://mlsanity.com/paper/2306.06138/) (2023)
- [UniCA: Unified Covariate Adaptation for Time Series Foundation Model](https://mlsanity.com/paper/2506.22039/) (2025)
- [Synapse: Adaptive Arbitration of Complementary Expertise in Time Series Foundational Models](https://mlsanity.com/paper/2511.05460/) (2025)
- [DELPHYNE: A Pre-Trained Model for General and Financial Time Series](https://mlsanity.com/paper/2506.06288/) (2025)
- [Efficiently Generating Correlated Sample Paths from Multi-step Time Series Foundation Models](https://mlsanity.com/paper/2510.02224/) (2025)
- [VQ-AR: Vector Quantized Autoregressive Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2205.15894/) (2022)
- [Neural Spline Search for Quantile Probabilistic Modeling](https://mlsanity.com/paper/2301.04857/) (2023)
- [Any-Quantile Probabilistic Forecasting of Short-Term Electricity Demand](https://mlsanity.com/paper/2404.17451/) (2024)
- [Calibrated Multi-Level Quantile Forecasting](https://mlsanity.com/paper/2512.23671/) (2025)
- [Zeus: Towards Tuning-Free Foundation Model for Time Series Analysis](https://mlsanity.com/paper/2607.01918/) (2026)
- [t0: A Time-Series Foundation Model for Forecasting with Context](https://mlsanity.com/paper/2609.24559/) (2026)
- [Task-based Loss Functions in Computer Vision: A Comprehensive Review](https://mlsanity.com/paper/2504.04242/) (2025)
- [Benchmarking Probabilistic Time Series Forecasting Models on Neural Activity](https://mlsanity.com/paper/2510.18037/) (2025)
- [Auxiliary Quantile Forecasting with Linear Networks](https://mlsanity.com/paper/2212.02578/) (2022)
- [Forecasting Intraday Power Output by a Set of PV Systems using Recurrent Neural Networks and Physical Covariates](https://mlsanity.com/paper/2303.08459/) (2023)
- [Dynamical Implicit Neural Representations](https://mlsanity.com/paper/2511.21787/) (2025)
- [Stabilizing distribution-free probabilistic forecasts](https://mlsanity.com/paper/2605.28531/) (2026)
- [Unicorn: Scaling High-Dimensional Time Series Forecasting via Universal Correlation Modeling](https://mlsanity.com/paper/2605.30376/) (2026)

## Abstract (arXiv)

Quantile regression is an effective technique to quantify uncertainty, fit challenging underlying distributions, and often provide full probabilistic predictions through joint learnings over multiple quantile levels. A common drawback of these joint quantile regressions, however, is textitquantile crossing, which violates the desirable monotone property of the conditional quantile function. In this work, we propose the Incremental (Spline) Quantile Functions I(S)QF, a flexible and efficient distribution-free quantile estimation framework that resolves quantile crossing with a simple neural network layer. Moreover, I(S)QF inter/extrapolate to predict arbitrary quantile levels that differ from the underlying training ones. Equipped with the analytical evaluation of the continuous ranked probability score of I(S)QF representations, we apply our methods to NN-based times series forecasting cases, where the savings of the expensive re-training costs for non-trained quantile levels is particularly significant. We also provide a generalization error analysis of our proposed approaches under the sequence-to-sequence setting. Lastly, extensive experiments demonstrate the improvement of consistency and accuracy errors over other baselines.