## 3 Methodology
### Hypergraph-Based Spatiotemporal Forecasting  
The forecasting framework leverages hypergraph structures to model complex multi-station interactions while preserving the decoupled spatiotemporal patterns learned during pretraining. This dual-stream architecture combines feature-driven hypergraph construction with temporal dynamics modeling, enabling the capture of both localized weather phenomena and global atmospheric patterns. **Feature Similarity Hypergraph Construction**  
Let the input multivariate meteorological time series be denoted as \(\textbf{X}\in\mathbb{R}^{T\times N\times F}\), where \(T\) is the number of time steps, \(N\) is the number of stations, and \(F\) is the number of meteorological variables (e.g., temperature, humidity, wind speed). For each station \(i\), we compute a feature vector \(\textbf{f}_{i}\in\mathbb{R}^{F}\) by averaging over the temporal dimension:  
\[\textbf{f}_{i}=\frac{1}{T}\sum_{t=1}^{T}\textbf{X}_{t,i,:} \tag{3}\]We then normalize the station features across each variable:  
\[\tilde{\mathbf{f}}_{i}=\frac{\mathbf{f}_{i}-\boldsymbol{\mu}}{\boldsymbol{\sigma}+ \boldsymbol{\epsilon}},\quad\boldsymbol{\mu},\boldsymbol{\sigma}\in\mathbb{R}^{F} \tag{4}\]  
where \(\boldsymbol{\mu}\) and \(\boldsymbol{\sigma}\) denote the mean and standard deviation over all stations, respectively, and \(\boldsymbol{\epsilon}\) is a small constant for numerical stability. Next, a cosine similarity matrix \(\mathbf{S}\in\mathbb{R}^{N\times N}\) is computed among all normalized station features:  
\[\mathbf{S}_{i,j}=\frac{\tilde{\mathbf{f}}_{i}^{\top}\tilde{\mathbf{f}}_{j}}{ \|\tilde{\mathbf{f}}_{i}\|_{2}\cdot\|\tilde{\mathbf{f}}_{j}\|_{2}} \tag{5}\]  
For each station \(i\), we select its top-\(k\) most similar stations based on \(\mathbf{S}_{i,\cdot}\) to form a hyperedge. Repeating this process for all stations yields a hypergraph \(\mathcal{G}=(\mathcal{V},\mathcal{E})\), where each node represents a station, and each hyperedge \(e\in\mathcal{E}\) connects one node to its \(k\) most similar peers. **Hypergraph-Aware Forecasting**  
The forecasting model utilizes the pretrained spatial and temporal embeddings \(\mathbf{H}_{s}\in\mathbb{R}^{T_{p}\times N\times E}\) and \(\mathbf{H}_{t}\in\mathbb{R}^{T_{p}\times N\times E}\), obtained from Section 3.2, along with the hypergraph \(\mathcal{G}\). A hypergraph convolution is applied to propagate information across high-order spatial neighborhoods:  
\[\mathbf{H}^{(l+1)}=\sigma\Big{(}\mathbf{D}_{v}^{-1/2}\mathbf{H}_{G}\mathbf{W} _{h}\mathbf{D}_{e}^{-1}\mathbf{H}_{G}^{\top}\mathbf{D}_{v}^{-1/2}\mathbf{H}^ {(l)}\mathbf{W}_{g}\Big{)} \tag{6}\]  
where \(\mathbf{H}_{G}\in\mathbb{R}^{N\times|\mathcal{E}|}\) is the incidence matrix of \(\mathcal{G}\), \(\mathbf{D}_{v}\) and \(\mathbf{D}_{e}\) are diagonal matrices of node and hyperedge degrees, and \(\mathbf{W}_{h}\), \(\mathbf{W}_{g}\) are trainable weight matrices. Temporal dynamics are modeled using dilated causal convolutions over the hypergraph-updated spatial features \(\mathbf{H}^{(l+1)}\):  
\[\mathbf{Z}_{t}=\sum_{k=0}^{K-1}\mathbf{W}_{k}*\mathbf{H}_{t-d\cdot k}^{(l+1)} \tag{7}\]  
where \(K=2\) is the kernel size, \(d=\{2^{0},2^{1},\ldots,2^{7}\}\) are the exponentially growing dilation factors across 8 layers, and \(*\) denotes convolution. The final forecast \(\hat{\mathbf{Y}}\) is computed by fusing the hypergraph-updated features with the pretrained embeddings:  
\[\hat{\mathbf{Y}}=\text{FC}\Big{(}\Big{[}\mathbf{Z}_{t}\,\|\,\mathbf{H}^{(l+ 1)}\,\|\,\text{MLP}_{s}(\mathbf{H}_{s})\,\|\,\text{MLP}_{t}(\mathbf{H}_{t}) \Big{]}\Big{)} \tag{8}\]  
where \(\mathbf{H}_{s}\) and \(\mathbf{H}_{t}\) are the spatial and temporal representations extracted via the masked pretraining in Section 3.2, \(\text{MLP}_{s}\) and \(\text{MLP}_{t}\) are multi-layer perceptrons for dimensionality alignment, and \(\|\) denotes feature concatenation. This architecture ensures that both high-order spatial interactions and decoupled temporal trends are jointly considered, enhancing the accuracy of multi-step weather forecasts.