# Beyond expectation: Deep joint mean and quantile regression for spatio-temporal problems

arXiv: https://arxiv.org/abs/1808.08798 · HTML: https://mlsanity.com/paper/1808.08798/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Aug 2018.

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

- [Temporal Latent Auto-Encoder: A Method for Probabilistic Multivariate Time Series Forecasting](https://mlsanity.com/paper/2101.10460/) (2021)
- [Uncertainty Quantification of Sparse Travel Demand Prediction with Spatial-Temporal Graph Neural Networks](https://mlsanity.com/paper/2208.05908/) (2022)
- [Multi-scale Attention Flow for Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2205.07493/) (2022)
- [Uncertainty Quantification of Spatiotemporal Travel Demand with Probabilistic Graph Neural Networks](https://mlsanity.com/paper/2303.04040/) (2023)
- [Regression modelling of spatiotemporal extreme U.S. wildfires via partially-interpretable neural networks](https://mlsanity.com/paper/2208.07581/) (2022)
- [Conditionally Elicitable Dynamic Risk Measures for Deep Reinforcement Learning](https://mlsanity.com/paper/2206.14666/) (2022)
- [Modeling Censored Mobility Demand through Quantile Regression Neural Networks](https://mlsanity.com/paper/2104.01214/) (2021)
- [Deep Spatio-Temporal Forecasting of Electrical Vehicle Charging Demand](https://mlsanity.com/paper/2106.10940/) (2021)
- [Deep Evidential Learning for Bayesian Quantile Regression](https://mlsanity.com/paper/2308.10650/) (2023)
- [Deep Quantile Regression for Uncertainty Estimation in Unsupervised and Supervised Lesion Detection](https://mlsanity.com/paper/2109.09374/) (2021)
- [Matching DNN Compression and Cooperative Training with Resources and Data Availability](https://mlsanity.com/paper/2212.02304/) (2022)
- [Simulation-Based Inference with Quantile Regression](https://mlsanity.com/paper/2401.02413/) (2024)
- [Relative Advantage Debiasing for Watch-Time Prediction in Short-Video Recommendation](https://mlsanity.com/paper/2508.11086/) (2025)
- [Short-term bus travel time prediction for transfer synchronization with intelligent uncertainty handling](https://mlsanity.com/paper/2104.06819/) (2021)
- [Mind the Gap: Modelling Difference Between Censored and Uncensored Electric Vehicle Charging Demand](https://mlsanity.com/paper/2301.06418/) (2023)
- [Contextualizing MLP-Mixers Spatiotemporally for Urban Data Forecast at Scale](https://mlsanity.com/paper/2307.01482/) (2023)
- [Probabilistic forecasting of power system imbalance using neural network-based ensembles](https://mlsanity.com/paper/2404.14836/) (2024)
- [Marginally calibrated response distributions for end-to-end learning in autonomous driving](https://mlsanity.com/paper/2110.01050/) (2021)
- [Uncertainty estimation of machine learning spatial precipitation predictions from satellite data](https://mlsanity.com/paper/2311.07511/) (2023)
- [Statistical Learning of Value-at-Risk and Expected Shortfall](https://mlsanity.com/paper/2209.06476/) (2022)
- [Beyond Prediction: On-street Parking Recommendation using Heterogeneous Graph-based List-wise Ranking](https://mlsanity.com/paper/2305.00162/) (2023)
- [Positional Encoder Graph Quantile Neural Networks for Geographic Data](https://mlsanity.com/paper/2409.18865/) (2024)
- [Nonparametric Probabilistic Regression with Coarse Learners](https://mlsanity.com/paper/2210.16247/) (2022)
- [Deep COVID-19 Forecasting for Multiple States with Data Augmentation](https://mlsanity.com/paper/2302.01155/) (2023)
- [Edge-Assisted ML-Aided Uncertainty-Aware Vehicle Collision Avoidance at Urban Intersections](https://mlsanity.com/paper/2404.14523/) (2024)
- [CONTINA: Confidence Interval for Traffic Demand Prediction with Coverage Guarantee](https://mlsanity.com/paper/2504.13961/) (2025)
- [Forecast-Then-Optimize Deep Learning Methods](https://mlsanity.com/paper/2506.13036/) (2025)
- [QUTCC: Quantile Uncertainty Training and Conformal Calibration for Imaging Inverse Problems](https://mlsanity.com/paper/2507.14760/) (2025)
- [Transformer-Based Predictive Maintenance for Risk-Aware Instrument Calibration](https://mlsanity.com/paper/2603.20297/) (2026)
- [ConquerNet: Convolution-Smoothed Quantile ReLU Neural Networks with Minimax Guarantees](https://mlsanity.com/paper/2605.06265/) (2026)

## Abstract (arXiv)

Spatio-temporal problems are ubiquitous and of vital importance in many research fields. Despite the potential already demonstrated by deep learning methods in modeling spatio-temporal data, typical approaches tend to focus solely on conditional expectations of the output variables being modeled. In this paper, we propose a multi-output multi-quantile deep learning approach for jointly modeling several conditional quantiles together with the conditional expectation as a way to provide a more complete "picture" of the predictive density in spatio-temporal problems. Using two large-scale datasets from the transportation domain, we empirically demonstrate that, by approaching the quantile regression problem from a multi-task learning perspective, it is possible to solve the embarrassing quantile crossings problem, while simultaneously significantly outperforming state-of-the-art quantile regression methods. Moreover, we show that jointly modeling the mean and several conditional quantiles not only provides a rich description about the predictive density that can capture heteroscedastic properties at a neglectable computational overhead, but also leads to improved predictions of the conditional expectation due to the extra information and a regularization effect induced by the added quantiles.