Learning the Evolutionary and Multi-scale Graph Structure for Multivariate Time Series Forecasting

1 INTRODUCTION

• The paper positions graph neural networks as a way to model variable relationships, while noting that classical autoregressive models assume linear dependencies and are mainly used for univariate forecasting.CNN-based global aggregation can also struggle to capture pair-wise correlations precisely.Source: §1 · p. 1
• Fixed adjacency matrices miss both evolving relationships and correlations that differ across observation scales, two gaps in existing graph-based forecasting methods.The paper notes that short- and long-term correlations can differ.Source: §1 · p. 2
• The design challenges include modeling dependence on the previous graph, avoiding parameter-heavy per-step graph generation, and extracting scale-specific structure from noisy relationships.The paper links per-step graph generation to convergence difficulty.Source: §1 · p. 2
• ESG combines a hierarchical architecture and dilated convolution for scale-specific correlations with recurrent adjacency matrices and a unified forecasting model.The adjacency matrices are constructed with a gated recurrent unit, and the unified model fuses multi-scale representations.Source: §1 · p. 2
• Experiments on real-world datasets are reported to validate the approach and illustrate evolving, multi-scale interactions; the stated goal is forecasting accuracy, not graph recovery or causal inference.The paper says learned graph structures serve as external factors for extracting global and precise forecasting signals.Source: §1 · p. 2

2 PRELIMINARY

• The graph representation uses variables as nodes and relationships as edges, with an adjacency matrix encoding connectivity.An adjacency entry is nonzero for an edge and zero otherwise.Source: §2 · pp. 2–3
• The forecasting setup takes historical observations of N variables, each with feature dimension C, to predict future states.The history has fixed length P; the mapping functions are denoted F1 and F2 for the single- and multi-step settings.Source: §2 · p. 3
• Single-step forecasting predicts a value at a future time step, whereas multi-step forecasting predicts a sequence of future values.Both settings use the same fixed-length historical look-back window.Source: §2 · p. 3

3.1 Overview

• Each ESG layer applies temporal convolution, evolving-graph learning, and graph convolution; residual and skip connections carry representations toward an end-to-end prediction.The model adds each layer's graph-convolution output to its input and uses skip connections to deliver information to the final output.Source: §3.1 · p. 3

3.2 Learning Evolving Graph Structure

• EGL updates node states recurrently with a GRU from segmented temporal features, initializes them with static node representations, then maps node pairs through an MLP and mask to adjacency matrices.Graphs are assumed unchanged within a time interval but evolutionary across adjacent intervals.Source: §3.2 · p. 4

3.3 Temporal Convolution Module

• Dilated inception uses different-sized filters to capture varied temporal ranges; stacked layers progress from short-term bottom-layer patterns to long-term top-layer patterns.The outputs of two dilated inception layers are gated to control how much information passes to the next module.Source: §3.3 · pp. 4–5

3.4 Evolving Graph Convolution Module

• At each scale, the graph convolution uses the corresponding evolving adjacency matrix for mix-hop propagation, retaining original node information while exploring deeper neighborhoods and selecting useful hops.Retaining a proportion of original states also relieves oversmoothing to a certain extent.Source: §3.4 · p. 5

4.1 Datasets & Setup

• The evaluation covers six real-world datasets; single-step forecasting uses RSE and CORR, while multi-step forecasting uses RMSE, MAE, and CORR.Lower is better for all listed metrics except CORR.Source: §4.1 · p. 5, Table 1 · p. 5

4.2 Baselines

• The study compares ESG with statistical and deep-learning baselines for single-step forecasting and with eight forecasting baselines for multi-step forecasting.The single-step set includes AR and GP; multi-step baselines include XGBoost, graph-convolution methods, and adaptive-graph methods.Source: §4.2 · p. 5

4.3 Main Results

• ESG achieves state-of-the-art results across both forecasting tasks; on NYC-taxi, it lowers RMSE by 7.8%, 7.8%, 10.7% and MAE by 10.7%, 10.4%, 12.0% at horizons 3, 6, 12.The results are compared with baselines in Tables 2 and 3.Source: Table 3 · p. 6, Table 2 · p. 6, §4.3 · p. 6

4.3.1 Single-step Forecasting.

• On Solar-Energy, ESG improves RSE by 3.94% and 3.96% over the state-of-the-art methods at horizons 3 and 24, respectively, and performs optimally on more than half the metrics on Exchange-rate and Wind.The authors suggest smaller graph size and fewer training examples may explain the weaker results on Exchange-rate and Wind.Source: §4.3.1 · p. 6

4.4 Ablation Study

• Ablations indicate that scale-agnostic evolving graphs perform worst, while removing the evolving learner increases result variability; the authors conclude each component contributes to performance.Experiments were repeated 10 times, with average values and standard deviations reported.Source: §4.4 · p. 7, Table 4 · p. 7
• Fusing multiple scales outperforms using any single scale; scale 2 ranks second, and higher scale levels do not yield monotonically greater contributions.Scales 1–3 correspond to successive temporal-convolution outputs from short- to long-term views.Source: §4.4 · p. 7, Figure 4 · p. 7

4.5 Study of Evolving Graph Structure

• The NYC-Bike case study shows learned adjacency weights changing alongside station relationships, including a declining 166–141 correlation after 16:30 and a weak 166–217 correlation before then despite their proximity.The paper notes that the adjacency matrices are not constrained to be symmetric.Source: §4.5 · pp. 7–8, Figure 5 · p. 8
• Adjacency matrices differ across observation scales: the short-term example is polarized, whereas the long-term example has more average values, consistent with similar long-term patterns among the four series.The authors interpret these case studies as evidence that ESG captures evolving and multi-scale correlations.Source: §4.5 · p. 8, Figure 5 · p. 8

5 RELATED WORK

• The related-work section reviews time-series forecasting and graph neural networks as its two perspectives.Source: §5 · p. 8

5.1 Time Series Forecasting

• Univariate forecasting treats variables separately, while multivariate methods model their correlations; earlier approaches include ARIMA, LSTNet, TPA-LSTM, and MTGNN.The review also covers FC-LSTM, N-BEATS, spatio-temporal models, DeepGLO, and TLAE.Source: §5.1 · p. 8

5.2 Graph Neural Network

• Graph neural networks use spectral filtering or spatial message passing, but evolving and multi-scale time-series correlations are seldom captured.The review describes GNN applications in traffic forecasting and methods using road-network distance or recent, daily, and weekly representations.Source: §5.2 · pp. 8–9

• ESG recurrently constructs adjacency matrices using current inputs and historical graph states, combines this with a hierarchical architecture for multi-scale inter- and intra-time-series correlations, and outperforms baselines on real-world datasets.The authors identify exploring evolving graph structure in more scenarios as future work.Source: §6 · p. 9

• The section indicates that more details are presented there.Source: A APPENDIX · p. 10

• Four single-step datasets and two multi-step datasets are used, with dataset details and evaluation metrics presented next.Source: §A.1 · p. 10

A.1.1 Single-Step Forecasting.

• Solar-Energy, Electricity, Exchange Rate, and Wind cover solar production, electricity consumption, national exchange rates, and wind potential estimates.The datasets include 137 Alabama PV plants, 321 electricity clients, eight countries, and one area, respectively.Source: §A.1.1 · p. 10
• The setup uses a 168-step look-back, independently trains horizons Q=3, 6, 12, and 24, and sets feature dimension C=1.It evaluates RSE and CORR.Source: §A.1.1 · p. 10
• NYC-Bike and NYC-Taxi [27] provide New York sharing-bike and taxi demand data from April 1 to June 30, 2016.NYC-Bike covers 250 stations.Source: §A.1.2 · p. 10
• For traffic forecasting, the look-back and prediction windows are each 12 steps (6 hours), with two features for pick-up and drop-off demand.Evaluation uses RMSE, MAE, and CORR.Source: §A.1.2 · p. 10

• Baseline hyperparameters are tuned, and each experiment is run 10 times with the average value reported.Source: §A.2 · p. 10

A.2.1 Single-step forecasting.

• Single-step comparisons include eight baselines spanning classical statistics and neural networks, including AR, GP, VARMLP, RNN-GRU, LSTNet, TPA-LSTM, MTGNN, and StemGNN.The listed methods cover linear correlations, recurrent and convolutional models, attention, adaptive adjacency, and Fourier-based pattern discovery.Source: §A.2.1 · p. 10

A.2.2 Multi-step forecasting.

• Multi-step comparisons use eight baselines: XGBoost, DCRNN, STGCN, STG2Seq, STSGCN, MTGNN, CCRNN, and GTS.The group includes methods using hand-crafted, self-learned, and discrete graph structures.Source: §A.2.2 · p. 10

A.2.2 Multi-step forecasting.

• The multi-step baseline group comprises XGBoost, DCRNN, STGCN, STG2Seq, STSGCN, MTGNN, CCRNN, and GTS.They include approaches based on graph convolution, recurrent networks, and graph structure learning.Source: §A.2.2 · p. 10

A.3 Experimental Setup

• Experiments use PyTorch on an Ubuntu machine with Xeon CPUs and a TITAN Xp GPU, averaging 10 runs; training uses Adam, gradient clipping, grid-searched learning rates, dropout, and layer normalization.The setup also specifies mix-hop propagation and static-node representation dimensions.Source: §A.3 · pp. 10–11

A.4 Extra Experiment Result

• ESG achieves the best average ablation results on NYC-Bike and also leads at horizons 3, 6, and 12.Table 5 presents the ablation study.Source: Table 5 · p. 11, §A.4 · p. 11

A.3.1 Single-step forecasting.

• The configuration stacks 3 multi-scale extractors with time-interval sequence d(l), 1, 1, 1; temporal convolutions use dilation factor 1 and filter sizes 2 and 6, with 32 output channels for temporal and graph convolutions.The output module uses 128 channels in its first layer and 12×2 in its second; batch size is 16, and evolving-node representation dimension is 20.Source: §A.3.2 · p. 11