# Graph-based Time Series Clustering for End-to-End Hierarchical Forecasting

arXiv: https://arxiv.org/abs/2305.19183 · HTML: https://mlsanity.com/paper/2305.19183/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 81 references that are in the map.
- First posted on arXiv 30 May 2023.

## Builds on (most cited first, 30 of 36)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [On the Properties of Neural Machine Translation: Encoder-Decoder Approaches](https://mlsanity.com/paper/1409.1259/) (2014)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables](https://mlsanity.com/paper/1611.00712/) (2016)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Graph WaveNet for Deep Spatial-Temporal Graph Modeling](https://mlsanity.com/paper/1906.00121/) (2019)
- [Hierarchical Graph Representation Learning with Differentiable Pooling](https://mlsanity.com/paper/1806.08804/) (2018)
- [On the Bottleneck of Graph Neural Networks and its Practical Implications](https://mlsanity.com/paper/2006.05205/) (2020)
- [Graph U-Nets](https://mlsanity.com/paper/1905.05178/) (2019)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Structured Sequence Modeling with Graph Convolutional Recurrent Networks](https://mlsanity.com/paper/1612.07659/) (2016)
- [A Survey on Oversmoothing in Graph Neural Networks](https://mlsanity.com/paper/2303.10993/) (2023)
- [Deep Learning for Time Series Forecasting: Tutorial and Literature Survey](https://mlsanity.com/paper/2004.10240/) (2020)
- [A Gentle Introduction to Deep Learning for Graphs](https://mlsanity.com/paper/1912.12693/) (2019)
- [Filling the G_ap_s: Multivariate Time Series Imputation by Graph Neural Networks](https://mlsanity.com/paper/2108.00298/) (2021)
- [Understanding Pooling in Graph Neural Networks](https://mlsanity.com/paper/2110.05292/) (2021)
- [Scalable Spatiotemporal Graph Neural Networks](https://mlsanity.com/paper/2209.06520/) (2022)
- [Taming Local Effects in Graph-based Spatiotemporal Forecasting](https://mlsanity.com/paper/2302.04071/) (2023)
- [The expressive power of pooling in Graph Neural Networks](https://mlsanity.com/paper/2304.01575/) (2023)
- [Group-Aware Graph Neural Network for Nationwide City Air Quality Forecasting](https://mlsanity.com/paper/2108.12238/) (2021)
- [Hierarchical Representation Learning in Graph Neural Networks with Node Decimation Pooling](https://mlsanity.com/paper/1910.11436/) (2019)
- [Simultaneously Reconciled Quantile Forecasting of Hierarchically Related Time Series](https://mlsanity.com/paper/2102.12612/) (2021)
- [Multivariate Time Series Forecasting with Latent Graph Inference](https://mlsanity.com/paper/2203.03423/) (2022)
- [ST-UNet: A Spatio-Temporal U-Network for Graph-structured Time Series Modeling](https://mlsanity.com/paper/1903.05631/) (2019)
- [Probabilistic forecast reconciliation under the Gaussian framework](https://mlsanity.com/paper/2103.11128/) (2021)

## Built on by (most cited first, 17 of 17)

- [A Survey on Graph Neural Networks for Time Series: Forecasting, Classification, Imputation, and Anomaly Detection](https://mlsanity.com/paper/2307.03759/) (2023)
- [Graph Deep Learning for Time Series Forecasting](https://mlsanity.com/paper/2310.15978/) (2023)
- [Graph-based Forecasting with Missing Data through Spatiotemporal Downsampling](https://mlsanity.com/paper/2402.10634/) (2024)
- [HyperIMTS: Hypergraph Neural Network for Irregular Multivariate Time Series Forecasting](https://mlsanity.com/paper/2505.17431/) (2025)
- [MaxCutPool: differentiable feature-aware Maxcut for pooling in graph neural networks](https://mlsanity.com/paper/2409.05100/) (2024)
- [Interpreting Temporal Graph Neural Networks with Koopman Theory](https://mlsanity.com/paper/2410.13469/) (2024)
- [BN-Pool: Bayesian Nonparametric Pooling for Graphs](https://mlsanity.com/paper/2501.09821/) (2025)
- [ST-Hyper: Learning High-Order Dependencies Across Multiple Spatial-Temporal Scales for Multivariate Time Series Forecasting](https://mlsanity.com/paper/2509.02217/) (2025)
- [SnareNet: Flexible Repair Layers for Neural Networks with Hard Constraints](https://mlsanity.com/paper/2602.09317/) (2026)
- [Mitigating Data Scarcity in Time Series Analysis: A Foundation Model with Series-Symbol Data Generation](https://mlsanity.com/paper/2502.15466/) (2025)
- [Hierarchical Graph Networks for Accurate Weather Forecasting via Lightweight Training](https://mlsanity.com/paper/2510.22094/) (2025)
- [Spatiotemporal Graph Neural Networks in short term load forecasting: Does adding Graph Structure in Consumption Data Improve Predictions?](https://mlsanity.com/paper/2502.12175/) (2025)
- [Reducing Smoothness with Expressive Memory Enhanced Hierarchical Graph Neural Networks](https://mlsanity.com/paper/2504.00349/) (2025)
- [Nonlinear reconciliation: Error reduction theorems](https://mlsanity.com/paper/2507.22500/) (2025)
- [FTSCommDetector: Discovering Behavioral Communities through Temporal Synchronization](https://mlsanity.com/paper/2510.00014/) (2025)
- [STEPS: A Temporal Smooth Error Propagation Solver on the Manifolds for Test-Time Adaptation in Time Series Forecasting](https://mlsanity.com/paper/2605.08005/) (2026)
- [Hierarchical Time Series Forecasting with Operator-Valued Kernels](https://mlsanity.com/paper/2610.18196/) (2026)

## Abstract (arXiv)

Relationships among time series can be exploited as inductive biases in learning effective forecasting models. In hierarchical time series, relationships among subsets of sequences induce hard constraints (hierarchical inductive biases) on the predicted values. In this paper, we propose a graph-based methodology to unify relational and hierarchical inductive biases in the context of deep learning for time series forecasting. In particular, we model both types of relationships as dependencies in a pyramidal graph structure, with each pyramidal layer corresponding to a level of the hierarchy. By exploiting modern - trainable - graph pooling operators we show that the hierarchical structure, if not available as a prior, can be learned directly from data, thus obtaining cluster assignments aligned with the forecasting objective. A differentiable reconciliation stage is incorporated into the processing architecture, allowing hierarchical constraints to act both as an architectural bias as well as a regularization element for predictions. Simulation results on representative datasets show that the proposed method compares favorably against the state of the art.