# Simultaneously Reconciled Quantile Forecasting of Hierarchically Related Time Series

arXiv: https://arxiv.org/abs/2102.12612 · HTML: https://mlsanity.com/paper/2102.12612/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 43 references that are in the map.
- First posted on arXiv 25 Feb 2021.

## Builds on (most cited first, 20 of 20)

- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling](https://mlsanity.com/paper/1412.3555/) (2014)
- [Spatio-Temporal Graph Convolutional Networks: A Deep Learning Framework for Traffic Forecasting](https://mlsanity.com/paper/1709.04875/) (2017)
- [DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks](https://mlsanity.com/paper/1704.04110/) (2017)
- [Modeling Long- and Short-Term Temporal Patterns with Deep Neural Networks](https://mlsanity.com/paper/1703.07015/) (2017)
- [N-BEATS: Neural basis expansion analysis for interpretable time series forecasting](https://mlsanity.com/paper/1905.10437/) (2019)
- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [Connecting the Dots: Multivariate Time Series Forecasting with Graph Neural Networks](https://mlsanity.com/paper/2005.11650/) (2020)
- [Weight Uncertainty in Neural Networks](https://mlsanity.com/paper/1505.05424/) (2015)
- [Accurate Uncertainties for Deep Learning Using Calibrated Regression](https://mlsanity.com/paper/1807.00263/) (2018)
- [Deep Learning on Graphs: A Survey](https://mlsanity.com/paper/1812.04202/) (2018)
- [DAG-GNN: DAG Structure Learning with Graph Neural Networks](https://mlsanity.com/paper/1904.10098/) (2019)
- [Learning Discrete Structures for Graph Neural Networks](https://mlsanity.com/paper/1903.11960/) (2019)
- [Gradient-Based Neural DAG Learning](https://mlsanity.com/paper/1906.02226/) (2019)
- [A Multi-Horizon Quantile Recurrent Forecaster](https://mlsanity.com/paper/1711.11053/) (2017)
- [Think Globally, Act Locally: A Deep Neural Network Approach to High-Dimensional Time Series Forecasting](https://mlsanity.com/paper/1905.03806/) (2019)
- [Functional Variational Bayesian Neural Networks](https://mlsanity.com/paper/1903.05779/) (2019)
- [Deep and Confident Prediction for Time Series at Uber](https://mlsanity.com/paper/1709.01907/) (2017)
- [Improving Output Uncertainty Estimation and Generalization in Deep Learning via Neural Network Gaussian Processes](https://mlsanity.com/paper/1707.05922/) (2017)
- [ARMDN: Associative and Recurrent Mixture Density Networks for eRetail Demand Forecasting](https://mlsanity.com/paper/1803.03800/) (2018)

## Built on by (most cited first, 24 of 24)

- [Graph Deep Learning for Time Series Forecasting](https://mlsanity.com/paper/2310.15978/) (2023)
- [Graph-based Time Series Clustering for End-to-End Hierarchical Forecasting](https://mlsanity.com/paper/2305.19183/) (2023)
- [Probabilistic Hierarchical Forecasting with Deep Poisson Mixtures](https://mlsanity.com/paper/2110.13179/) (2021)
- [Deep Learning based Forecasting: a case study from the online fashion industry](https://mlsanity.com/paper/2305.14406/) (2023)
- [Hierarchically Regularized Deep Forecasting](https://mlsanity.com/paper/2106.07630/) (2021)
- [HierarchicalForecast: A Reference Framework for Hierarchical Forecasting in Python](https://mlsanity.com/paper/2207.03517/) (2022)
- [SLOTH: Structured Learning and Task-based Optimization for Time Series Forecasting on Hierarchies](https://mlsanity.com/paper/2302.05650/) (2023)
- [When Rigidity Hurts: Soft Consistency Regularization for Probabilistic Hierarchical Time Series Forecasting](https://mlsanity.com/paper/2206.07940/) (2022)
- [Hierarchical Forecasting at Scale](https://mlsanity.com/paper/2310.12809/) (2023)
- [Scalable Probabilistic Forecasting in Retail with Gradient Boosted Trees: A Practitioner's Approach](https://mlsanity.com/paper/2311.00993/) (2023)
- [Dynamic Combination of Heterogeneous Models for Hierarchical Time Series](https://mlsanity.com/paper/2112.11669/) (2021)
- [Dirichlet Proportions Model for Hierarchically Coherent Probabilistic Forecasting](https://mlsanity.com/paper/2204.10414/) (2022)
- [Large Scale Hierarchical Industrial Demand Time-Series Forecasting incorporating Sparsity](https://mlsanity.com/paper/2407.02657/) (2024)
- [Adaptive Two-Stage Cloud Resource Scaling via Hierarchical Multi-Indicator Forecasting and Bayesian Decision-Making](https://mlsanity.com/paper/2408.01000/) (2024)
- [RHiOTS: A Framework for Evaluating Hierarchical Time Series Forecasting Algorithms](https://mlsanity.com/paper/2408.03399/) (2024)
- [DeepHGNN: Study of Graph Neural Network based Forecasting Methods for Hierarchically Related Multivariate Time Series](https://mlsanity.com/paper/2405.18693/) (2024)
- [GMP-AR: Granularity Message Passing and Adaptive Reconciliation for Temporal Hierarchy Forecasting](https://mlsanity.com/paper/2406.12242/) (2024)
- [Coherent Probabilistic Aggregate Queries on Long-horizon Forecasts](https://mlsanity.com/paper/2111.03394/) (2021)
- [Efficient Forecasting of Large Scale Hierarchical Time Series via Multilevel Clustering](https://mlsanity.com/paper/2205.14104/) (2022)
- [Hierarchically Coherent Multivariate Mixture Networks](https://mlsanity.com/paper/2305.07089/) (2023)
- [When Rigidity Hurts: Soft Consistency Regularization for Probabilistic Hierarchical Time Series Forecasting](https://mlsanity.com/paper/2310.11569/) (2023)
- [CoRe: Coherency Regularization for Hierarchical Time Series](https://mlsanity.com/paper/2502.15983/) (2025)
- [Hierarchical Industrial Demand Forecasting with Temporal and Uncertainty Explanations](https://mlsanity.com/paper/2603.06555/) (2026)
- [Exactness Matters for Physical Rule Enforcement](https://mlsanity.com/paper/2605.08285/) (2026)

## Abstract (arXiv)

Many real-life applications involve simultaneously forecasting multiple time series that are hierarchically related via aggregation or disaggregation operations. For instance, commercial organizations often want to forecast inventories simultaneously at store, city, and state levels for resource planning purposes. In such applications, it is important that the forecasts, in addition to being reasonably accurate, are also consistent w.r.t one another. Although forecasting such hierarchical time series has been pursued by economists and data scientists, the current state-of-the-art models use strong assumptions, e.g., all forecasts being unbiased estimates, noise distribution being Gaussian. Besides, state-of-the-art models have not harnessed the power of modern nonlinear models, especially ones based on deep learning. In this paper, we propose using a flexible nonlinear model that optimizes quantile regression loss coupled with suitable regularization terms to maintain the consistency of forecasts across hierarchies. The theoretical framework introduced herein can be applied to any forecasting model with an underlying differentiable loss function. A proof of optimality of our proposed method is also provided. Simulation studies over a range of datasets highlight the efficacy of our approach.