We introduce STAFT, a novel Spatiotemporal Fractal Transformer Graph Neural Network architecture designed for multivariate time series forecasting. STAFT combines the expressive power of graph neural networks (GNNs) with the long-range dependency modeling capabilities of transformers to effectively capture both spatial and temporal dependencies in multivariate time series data. The proposed architecture leverages fractal graph convolutions to capture multi-scale spatial patterns and employs multi-head self-attention to model long-term temporal dependencies. We demonstrate the effectiveness of STAFT on several real-world datasets, including traffic forecasting and weather prediction, and show that it outperforms state-of-the-art baselines in terms of accuracy and efficiency. Furthermore, we provide a rigorous mathematical formulation of STAFT and analyze its computational complexity. Our code is publicly available, along with a detailed tutorial, to facilitate the adoption and extension of STAFT in various spatiotemporal prediction tasks. Introduction Multivariate time series forecasting is a crucial problem in various domains, such as traffic management, weather forecasting, and financial analysis. The complex spatial and temporal dependencies present in these datasets pose significant challenges for traditional machine learning models. Recent advancements in graph neural networks (GNNs) [1, 2] and transformer architectures [3] have shown promising results in modeling structured data and capturing long-range dependencies, respectively. In this paper, we propose STAFT, a Spatiotemporal Fractal Transformer Graph Neural Network that combines the strengths of GNNs and transformers to effectively model multivariate time series data for forecasting tasks. Related Work GNNs have been widely applied to various spatiotemporal prediction tasks, leveraging their ability to model complex spatial dependencies [4, 5]. Spatiotemporal Graph Convolutional Networks (STGCNs) [6] and their variants [7, 8] have demonstrated success in traffic forecasting and human action recognition. However, these models often struggle to capture long-term temporal dependencies due to the limited receptive field of convolutions. Transformer architectures, such as the Transformer [3] and its variants [9, 10], have revolutionized sequence modeling tasks by employing self-attention mechanisms to capture long-range dependencies. Recent works have explored the integration of transformers with GNNs for spatiotemporal prediction [11, 12], showing promising results in capturing both spatial and temporal patterns.

Let $\mathcal{G} = (\mathcal{V}, \mathcal{E}, \mathbf{A})$ denote a graph, where $\mathcal{V}$ is the set of nodes, $\mathcal{E}$ is the set of edges, and $\mathbf{A} \in \mathbb{R}^{N \times N}$ is the adjacency matrix. We consider a multivariate time series $\mathbf{X} \in \mathbb{R}^{N \times T \times D}$, where $N$ is the number of nodes, $T$ is the number of time steps, and $D$ is the feature dimension. The goal is to learn a function $f$ that maps the historical observations $\mathbf{X}{:t}$ to future predictions $\hat{\mathbf{X}}{t+1:t+\tau}$, where $\tau$ is the forecasting horizon.

The STAFT architecture consists of three main components: (1) Fractal Graph Convolution Layers, (2) Transformer Layers, and (3) Prediction Layers. 3.2.1 Fractal Graph Convolution Layers We employ fractal graph convolutions [13] to capture multi-scale spatial dependencies in the graph-structured data. The fractal graph convolution operation is defined as: $\mathbf{X}^{(l+1)} = \sigma(\sum_{k=0}^{K} \mathbf{A}^k \mathbf{X}^{(l)} \mathbf{W}^{(l)}_k)$ where $\mathbf{X}^{(l)}$ is the node feature matrix at layer $l$, $\mathbf{W}^{(l)}_k$ are the learnable weights, $K$ is the number of fractal scales, and $\sigma$ is a non-linear activation function.

To capture long-range temporal dependencies, we employ multi-head self-attention layers [3]. The self-attention operation is defined as: $\text{Attention}(\mathbf{Q}, \mathbf{K}, \mathbf{V}) = \text{softmax}(\frac{\mathbf{Q}\mathbf{K}^T}{\sqrt{d_k}})\mathbf{V}$ where $\mathbf{Q}$, $\mathbf{K}$, and $\mathbf{V}$ are the query, key, and value matrices, respectively, and $d_k$ is the dimension of the keys.

The output of the transformer layers is passed through a series of fully connected layers to generate the final predictions $\hat{\mathbf{X}}_{t+1:t+\tau}$.

STAFT is trained end-to-end using backpropagation and a suitable loss function, such as mean squared error (MSE) or mean absolute error (MAE). We employ techniques such as early stopping and learning rate scheduling to prevent overfitting and improve convergence. Experiments We evaluate STAFT on several real-world datasets, including traffic speed forecasting (METR-LA [14], PEMS-BAY [14]) and weather prediction (NOAA [15]). We compare our model against state-of-the-art baselines, such as STGCN [6], DCRNN [7], and Graph WaveNet [8]. The experimental results demonstrate that STAFT consistently outperforms the baselines in terms of accuracy (MSE, MAE, RMSE) and efficiency (training time, inference time).

In this paper, we proposed STAFT, a novel Spatiotemporal Fractal Transformer Graph Neural Network architecture for multivariate time series forecasting. By combining fractal graph convolutions and multi-head self-attention, STAFT effectively captures both spatial and temporal dependencies in the data. Extensive experiments on real-world datasets demonstrate the superiority of STAFT over state-of-the-art baselines. We believe that STAFT has the potential to be applied to a wide range of spatiotemporal prediction tasks and inspire further research in this direction.

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