The non-stationary nature of real-world Multivariate Time Series (MTS) data presents forecasting models with a formidable challenge of the time-variant distribution of time series, referred to as distribution shift. Existing studies on the distribution shift mostly adhere to adaptive normalization techniques for alleviating temporal mean and covariance shifts or time-variant modeling for capturing temporal shifts. Despite improving model generalization, these normalization-based methods often assume a time-invariant transition between outputs and inputs but disregard specific intra-/inter-series correlations, while time-variant models overlook the intrinsic causes of the distribution shift. This limits model expressiveness and interpretability of tackling the distribution shift for MTS forecasting. To mitigate such a dilemma, we present a unified Probabilistic Graphical Model to Jointly capturing intra-/inter-series correlations and modeling the time-variant transitional distribution, and instantiate a neural framework called JointPGM for non-stationary MTS forecasting. Specifically, JointPGM first employs multiple Fourier basis functions to learn dynamic time factors and designs two distinct learners: intra-series and inter-series learners. The intra-series learner effectively captures temporal dynamics by utilizing temporal gates, while the inter-series learner explicitly models spatial dynamics through multi-hop propagation, incorporating Gumbel-softmax sampling. These two types of series dynamics are subsequently fused into a latent variable, which is inversely employed to infer time factors, generate final prediction, and perform reconstruction. We validate the effectiveness and efficiency of JointPGM through extensive experiments on six highly non-stationary MTS datasets, achieving state-of-the-art forecasting performance of MTS forecasting.

Multivariate time series prediction is widely used in daily life, which poses significant challenges due to the complex correlations that exist at multi-grained levels. Unfortunately, the majority of current time series prediction models fail to simultaneously learn the correlations of multivariate time series at multi-grained levels, resulting in suboptimal performance. To address this, we propose a Multi-Grained Correlations-based Prediction (MGCP) Network, which simultaneously considers the correlations at three granularity levels to enhance prediction performance. Specifically, MGCP utilizes Adaptive Fourier Neural Operators and Graph Convolutional Networks to learn the global spatiotemporal correlations and inter-series correlations, enabling the extraction of potential features from multivariate time series at fine-grained and medium-grained levels. Additionally, MGCP employs adversarial training with an attention mechanism-based predictor and conditional discriminator to optimize prediction results at coarse-grained level, ensuring high fidelity between the generated forecast results and the actual data distribution. Finally, we compare MGCP with several state-of-the-art time series prediction algorithms on real-world benchmark datasets, and our results demonstrate the generality and effectiveness of the proposed model.

The challenge of effectively learning inter-series correlations for multivariate time series forecasting remains a substantial and unresolved problem. Traditional deep learning models, which are largely dependent on the Transformer paradigm for modeling long sequences, often fail to integrate information from multiple time series into a coherent and universally applicable model. To bridge this gap, our paper presents ForecastGrapher, a framework reconceptualizes multivariate time series forecasting as a node regression task, providing a unique avenue for capturing the intricate temporal dynamics and inter-series correlations. Our approach is underpinned by three pivotal steps: firstly, generating custom node embeddings to reflect the temporal variations within each series; secondly, constructing an adaptive adjacency matrix to encode the inter-series correlations; and thirdly, augmenting the GNNs' expressive power by diversifying the node feature distribution. To enhance this expressive power, we introduce the Group Feature Convolution GNN (GFC-GNN). This model employs a learnable scaler to segment node features into multiple groups and applies one-dimensional convolutions with different kernel lengths to each group prior to the aggregation phase. Consequently, the GFC-GNN method enriches the diversity of node feature distribution in a fully end-to-end fashion. Through extensive experiments and ablation studies, we show that ForecastGrapher surpasses strong baselines and leading published techniques in the domain of multivariate time series forecasting.

Multivariate time series forecasting poses an ongoing challenge across various disciplines. Time series data often exhibit diverse intra-series and inter-series correlations, contributing to intricate and interwoven dependencies that have been the focus of numerous studies. Nevertheless, a significant research gap remains in comprehending the varying inter-series correlations across different time scales among multiple time series, an area that has received limited attention in the literature. To bridge this gap, this paper introduces MSGNet, an advanced deep learning model designed to capture the varying inter-series correlations across multiple time scales using frequency domain analysis and adaptive graph convolution. By leveraging frequency domain analysis, MSGNet effectively extracts salient periodic patterns and decomposes the time series into distinct time scales. The model incorporates a self-attention mechanism to capture intra-series dependencies, while introducing an adaptive mixhop graph convolution layer to autonomously learn diverse inter-series correlations within each time scale. Extensive experiments are conducted on several real-world datasets to showcase the effectiveness of MSGNet. Furthermore, MSGNet possesses the ability to automatically learn explainable multi-scale inter-series correlations, exhibiting strong generalization capabilities even when applied to out-of-distribution samples.