mlsanityData to 7 Oct 2026

Multivariate Quantile Function Forecaster

arXiv 2202.11316 ↗First posted 23 Feb 2022cs.LG stat.MLcited by 19

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Abstract (arXiv)

We propose Multivariate Quantile Function Forecaster (MQF2), a global probabilistic forecasting method constructed using a multivariate quantile function and investigate its application to multi-horizon forecasting. Prior approaches are either autoregressive, implicitly capturing the dependency structure across time but exhibiting error accumulation with increasing forecast horizons, or multi-horizon sequence-to-sequence models, which do not exhibit error accumulation, but also do typically not model the dependency structure across time steps. MQF2 combines the benefits of both approaches, by directly making predictions in the form of a multivariate quantile function, defined as the gradient of a convex function which we parametrize using input-convex neural networks. By design, the quantile function is monotone with respect to the input quantile levels and hence avoids quantile crossing. We provide two options to train MQF2: with energy score or with maximum likelihood. Experimental results on real-world and synthetic datasets show that our model has comparable performance with state-of-the-art methods in terms of single time step metrics while capturing the time dependency structure.

Key facts. Cited by 19 papers in the map (data through 7 Oct 2026). Builds on 25 of its 59 references that are in the map. First posted on arXiv 23 Feb 2022.

Builds on

25 of 59 references.

References · 25

  1. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
  2. 2017DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networkscited by 1,004
  3. 2019Temporal Fusion Transformers for Interpretable Multi-horizon Time Series Forecastingcited by 748
  4. 2017Masked Autoregressive Flow for Density Estimationcited by 680
  5. 2019Enhancing the Locality and Breaking the Memory Bottleneck of Transformer on Time Series Forecastingcited by 606
  6. 2016Input Convex Neural Networkscited by 407
  7. 2018Implicit Quantile Networks for Distributional Reinforcement Learningcited by 362
  8. 2021Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecastingcited by 332
  9. 2017A Multi-Horizon Quantile Recurrent Forecastercited by 211
  10. 2019Optimal transport mapping via input convex neural networkscited by 202
  11. 2019High-Dimensional Multivariate Forecasting with Low-Rank Gaussian Copula Processescited by 147
  12. 2020OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transportcited by 143
13 more
  1. 2020Multivariate Probabilistic Time Series Forecasting via Conditioned Normalizing Flowscited by 142
  2. 2021An Introduction to Deep Generative Modelingcited by 104
  3. 2020Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimizationcited by 89
  4. 2018Optimal Control Via Neural Networks: A Convex Approachcited by 82
  5. 2021Domain Adaptation for Time Series Forecasting via Attention Sharingcited by 68
  6. 2013RNADE: The real-valued neural autoregressive density-estimatorcited by 65
  7. 2019Sum-of-Squares Polynomial Flowcited by 62
  8. 2021Learning Quantile Functions without Quantile Crossing for Distribution-free Time Series Forecastingcited by 32
  9. 2020MQTransformer: Multi-Horizon Forecasts with Context Dependent and Feedback-Aware Attentioncited by 29
  10. 2021Probabilistic Time Series Forecasting with Implicit Quantile Networkscited by 17
  11. 2021Variance Reduced Training with Stratified Sampling for Forecasting Modelscited by 13
  12. 2020Structured Policy Iteration for Linear Quadratic Regulatorcited by 8
  13. 2014Vector Quantile Regression: An Optimal Transport Approachcited by 2

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 4; 2023: 3; 2024: 3; 2025: 6; 2026: 3.

19 papers.

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  1. 2022Copula Conformal Prediction for Multi-step Time Series Forecastingcited by 28
  2. 2025Multivariate Conformal Prediction using Optimal Transportcited by 21
  3. 2023Deep Learning based Forecasting: a case study from the online fashion industrycited by 12
  4. 2025Relational Conformal Prediction for Correlated Time Seriescited by 11
  5. 2022Robust Multivariate Time-Series Forecasting: Adversarial Attacks and Defense Mechanismscited by 8
  6. 2025Optimal Control for Transformer Architectures: Enhancing Generalization, Robustness and Efficiencycited by 8
  7. 2025Multivariate Standardized Residuals for Conformal Predictioncited by 6
  8. 2022Multi-Target XGBoostLSS Regressioncited by 4
  9. 2022Adaptive Data Fusion for Multi-task Non-smooth Optimizationcited by 3
  10. 2025DoFlow: Flow-based Generative Models for Interventional and Counterfactual Forecasting on Time Seriescited by 3
  11. 2024GDFlow: Anomaly Detection with NCDE-based Normalizing Flow for Advanced Driver Assistance Systemcited by 2
  12. 2024Coherent Hierarchical Probabilistic Forecasting of Electric Vehicle Charging Demandcited by 2
7 more
  1. 2025Multivariate Uncertainty Quantification with Tomographic Quantile Forestscited by 1
  2. 2023ProbPNN: Enhancing Deep Probabilistic Forecasting with Statistical Informationcited by 0
  3. 2023Deep Generative Methods for Producing Forecast Trajectories in Power Systemscited by 0
  4. 2024Can Transformers Predict Vibrations?cited by 0
  5. 2026Signature-Kernel Based Evaluation Metrics for Robust Probabilistic and Tail-Event Forecastingcited by 0
  6. 2026Super-Level-Set Regression: Conditional Quantiles via Volume Minimizationcited by 0
  7. 2026A lift for input-convex neural net trainingcited by 0