SlowGuess/NeurIPS_2023_OCR huggingface dataset:
To further evaluate the performance of our model FourierGNN in multi-step forecasting, we conduct more experiments on the Wiki, METR-LA, and ECG datasets, respectively. We compare our model FourierGNN with five models (including StemGNN [10], AGCRN [2], GraphWaveNet [17], MTGNN [13], and Informer [9]) on the Wiki dataset under different prediction lengths, and the results are shown in Table 9. From the table, we observe that FourierGNN outperforms other models on MAE, RMSE, and MAPE metrics for all the prediction lengths. On average, FourierGNN improves MAE, RMSE, and MAPE by  $7.4\%$ ,  $3.5\%$ , and  $22.3\%$ , respectively. Among these models, AGCRN shows promising performances since it captures the spatial and temporal correlations adaptively. However, it fails to simultaneously capture spatiotemporal dependencies, limiting its forecasting performance. In contrast, our model captures comprehensive spatiotemporal dependencies simultaneously on a hypervariate graph for multivariate time series forecasting.
Table 9: Accuracy comparison under different prediction lengths on the Wiki dataset.
Length Metrics	3	6	9	12
MAE	RMSE	MAPE(%)	MAE	RMSE	MAPE(%)	MAE	RMSE	MAPE(%)	MAE	RMSE	MAPE(%)
GraphWaveNet [17]	0.061	0.105	138.60	0.061	0.105	135.32	0.061	0.105	132.52	0.061	0.104	136.12
StemGNN [10]	0.157	0.236	89.00	0.159	0.233	98.01	0.232	0.311	142.14	0.220	0.306	125.40
AGCRN [2]	0.043	0.077	73.49	0.044	0.078	80.44	0.045	0.079	81.89	0.044	0.079	78.52
MTGNN [13]	0.102	0.141	123.15	0.091	0.133	91.75	0.074	0.120	85.44	0.101	0.140	122.96
Informer [9]	0.053	0.089	85.31	0.054	0.090	84.46	0.059	0.095	93.80	0.059	0.095	95.09
FourierGNN	0.040	0.075	58.18	0.041	0.075	60.43	0.041	0.076	60.95	0.041	0.076	64.50
Table 10: Accuracy comparison under different prediction lengths on the METR-LA dataset.
Horizon Metrics	3	6	9	12
MAE	RMSE	MAPE(%)	MAE	RMSE	MAPE(%)	MAE	RMSE	MAPE(%)	MAE	RMSE	MAPE(%)
DCRNN [18]	0.160	0.204	80.00	0.191	0.243	83.15	0.216	0.269	85.72	0.241	0.291	88.25
STGCN [1]	0.058	0.133	59.02	0.080	0.177	60.67	0.102	0.209	62.08	0.128	0.238	63.81
GraphWaveNet [17]	0.180	0.366	21.90	0.184	0.375	22.95	0.196	0.382	23.61	0.202	0.386	24.14
MTGNN [13]	0.135	0.294	17.99	0.144	0.307	18.82	0.149	0.328	19.38	0.153	0.316	19.92
StemGNN [10]	0.052	0.115	86.39	0.069	0.141	87.71	0.080	0.162	89.00	0.093	0.175	90.25
AGCRN [2]	0.062	0.131	24.96	0.086	0.165	27.62	0.099	0.188	29.72	0.109	0.204	31.73
Informer [9]	0.076	0.141	69.96	0.088	0.163	70.94	0.096	0.178	72.26	0.100	0.190	72.54
CoST [36]	0.064	0.118	88.44	0.077	0.141	89.63	0.088	0.159	90.56	0.097	0.171	91.42
FourierGNN	0.050	0.113	86.30	0.066	0.140	87.97	0.076	0.159	88.99	0.084	0.165	89.69
Furthermore, we compare our model FourierGNN with seven MTS models (including STGCN [1], DCRNN [18], StemGNN [10], AGCRN [2], GraphWaveNet [17], MTGNN [13], Informer [9], and CoST [36]) on the METR-LA dataset which has a predefined graph topology in the data, and the results are shown in Table 10. On average, we improve  $5.7\%$  on MAE and  $1.5\%$  on RMSE. Among these models, StemGNN achieves competitive performance because it combines GFT to capture the spatial dependencies and DFT to capture the temporal dependencies. However, it is also limited to simultaneously capturing spatiotemporal dependencies. CoST learns disentangled seasonal-trend representations for time series forecasting via contrastive learning and obtains competitive results. But, our model still outperforms CoST. Because, compared with CoST, our model not only can learn the dynamic temporal representations, but also capture the discriminative spatial representations. Besides, STGCN and DCRNN require pre-defined graph structures. But StemGNN and our model outperform them for all steps, and AGCRN outperforms them when the prediction lengths are 9 and 12. This also shows that a novel adaptive graph learning can precisely capture the hidden spatial dependency. In addition, we compare FourierGNN with the baseline models under the different prediction lengths on the ECG dataset, as shown in Figure 5. It reports that FourierGNN achieves the best performances (MAE, RMSE, and MAPE) for all prediction lengths.
(a) MAE
(b) RMSE
Figure 5: Performance comparison in multi-step prediction on the ECG dataset.
(c) MAPE
G Further Analyses
G.1 Scalability Analysis
We further conduct experiments on the Wiki dataset to investigate the performance of FourierGNN under different graph sizes  $(N\times T)$ . The results are shown in Figure 6, where Figure 6(a), Figure 6(b) and Figure 6(c) show MAE, RMSE, and MAPE at the different number of nodes, respectively. From these figures, we observe that FourierGNN keeps a leading edge over the other state-of-the-art MTS models as the number of nodes increases. The results demonstrate the superiority and scalability of FourierGNN on large-scale datasets.
(a) MAE
Figure 6: Scalability analyses in terms of MAE, RMSE, and MAPE under different number of nodes on the Wiki dataset.
(b) RMSE
G.2 Parameter Analysis
We evaluate the forecasting performance of our model FourierGNN under different diffusion steps (layers) on the COVID-19 dataset, as illustrated in Table 11. The table shows that FourierGNN achieves increasingly better performance from  $K = 1$  to  $K = 4$  and achieves the best results when  $K = 3$ . With the further increase of  $K$ , FourierGNN obtains inferior performance. The results indicate that high-order diffusion information
Table 11: Performance at different diffusion steps (layers) on the COVID-19 dataset.
K=1	K=2	K=3	K=4
MAE	0.136	0.133	0.123	0.132
RMSE	0.181	0.177	0.168	0.176
MAPE(%)	72.30	71.80	71.52	72.59
is beneficial for improving forecasting accuracy, but the diffusion information may gradually weaken the effect or even bring noises to forecasting with the increase of the order.
In addition, we conduct additional experiments on the ECG dataset to analyze the effect of the input lookback window length  $T$  and the embedding dimension  $d$ , as shown in Figure 7 and Figure 8, respectively. Figure 7 shows that the performance (including RMSE and MAPE) of FourierGNN gets better as the input lookback window length increases, indicating that FourierGNN can learn a comprehensive hypervariate graph from long MTS inputs to capture the spatial and temporal dependencies. Moreover, Figure 8 shows that the performance (RMSE and MAPE) first increases and then decreases with the increase of the embedding size, which is attributed that a large embedding
Figure 7: Influence of input window.
Figure 8: Influence of embedding size.
size improves the fitting ability of FourierGNN but it may easily lead to the overfitting issue especially when the embedding size is too large.
G.3 Ablation Study
We provide more details about each variant used in this section and Section 5.3.
- w/o Embedding. A variant of FourierGNN feeds the raw MTS input instead of its embeddings into the graph convolution in Fourier space.
- w/o Dynamic FGO. A variant of FourierGNN uses the same FGO for all diffusion steps instead of applying different FGOs in different diffusion steps. It corresponds to a vanilla graph filter.
- w/o Residual. A variant of FourierGNN does not have the  $K = 0$  layer output, i.e.,  $\mathcal{X}_t^{\mathcal{G}}$ , in the summation.
- w/o Summation. A variant of FourierGNN adopts the last order (layer) output as the final frequency output of the FourierGNN.
We conduct another ablation study on the COVID-19 dataset to further investigate the effects of the different components of our FourierGNN. The results are shown in Table 12, which confirms the results in Table 4 and further verifies the effectiveness of each component in FourierGNN. Both Table 12 and Table 4 report that the embedding and dynamic FGOs in FourierGNN contribute more than the design of residual and summation to the state-of-the-art performance of FourierGNN.
Table 12: Ablation studies on the COVID-19 dataset.
Metric	w/o Embedding	w/o Dynamic FGO	w/o Residual	w/o Summation	FourierGNN
MAE	0.157	0.138	0.131	0.134	0.123
RMSE	0.203	0.180	0.174	0.177	0.168
MAPE(%)	76.91	74.01	72.25	72.57	71.52
H Visualizations
H.1 Visualization of the Diffusion Process in FourierGNN
To gain insight into the operation of the FGO, we visualize the frequency output of each layer in our FourierGNN. We select 10 counties from the COVID-19 dataset and visualize their adjacency matrices at two different timestamps, as shown in Figure 9. From left to right, the results correspond to the original spectrum of the input, as well as the outputs of the first, second, and third layers of the FourierGNN. From the top, we can find that as the number of layers increases, some correlation values are reduced, indicating that some correlations are filtered out. In contrast, the bottom case illustrates some correlations are enhanced as the number of layers increases. These results show that FGO can adaptively and effectively capture important patterns while removing noises, enabling the learning of a discriminative model.
Figure 9: The diffusion process of FourierGNN at two timestamps (top and bottom) on COVID-19.
H.2 Visualization of Time-Varying Dependencies Learned by FourierGNN
Furthermore, we explore the capability of FourierGNN in capturing time-varying dependencies among variables. To investigate this, we perform additional experiments to visualize the adjacency matrix of 10 randomly-selected counties over four consecutive days on the COVID-19 dataset. The visualization results, displayed as a heatmap in Figure 10, reveal clear spatial patterns that exhibit continuous evolution in the temporal dimension. This is because FourierGNN can attend to the time-varying variability of the spatiotemporal dependencies. These results verify that our model enjoys the feasibility of exploiting the time-varying dependencies among variables.
Based on the insights gained from these visualization results, we can conclude that the hypervariate graph structure exhibits strong capabilities to encode spatiotemporal dependencies. By incorporating FGOs, FourierGNN can effectively attend to and exploit the time-varying dependencies among variates. The synergy between the hypervariate graph structure and FGOs empowers FourierGNN to capture and model intricate spatiotemporal relationships with remarkable effectiveness.
Figure 10: The adjacency matrix for four consecutive days on the COVID-19 dataset.