Time series forecasting has been a widely explored task of great importance in many applications.

Contrastive learning underpins most current self-supervised time series representation methods. The strategy for constructing positive and negative sample pairs significantly affects the final representation quality. However, due to the continuous nature of time series semantics, the modeling approach of contrastive learning struggles to accommodate the characteristics of time series data. This results in issues such as difficulties in constructing hard negative samples and the potential introduction of inappropriate biases during positive sample construction. Although some recent works have developed several scientific strategies for constructing positive and negative sample pairs with improved effectiveness, they remain constrained by the contrastive learning framework. To fundamentally overcome the limitations of contrastive learning, this paper introduces Frequency-masked Embedding Inference (FEI), a novel non-contrastive method that completely eliminates the need for positive and negative samples. The proposed FEI constructs 2 inference branches based on a prompting strategy: 1) Using frequency masking as prompts to infer the embedding representation of the target series with missing frequency bands in the embedding space, and 2) Using the target series as prompts to infer its frequency masking embedding. In this way, FEI enables continuous semantic relationship modeling for time series. Experiments on 8 widely used time series datasets for classification and regression tasks, using linear evaluation and end-to-end fine-tuning, show that FEI significantly outperforms existing contrastive-based methods in terms of generalization. This study provides new insights into self-supervised representation learning for time series. The code is available at https://github.com/USTBInnovationPark/Frequency-masked-Embedding-Inference.

Discovering causal relationships in time series data is central in many scientific areas, ranging from economics to climate science. Granger causality is a powerful tool for causality detection. However, its original formulation is limited by its linear form and only recently nonlinear machine-learning generalizations have been introduced. This study contributes to the definition of neural Granger causality models by investigating the application of Kolmogorov-Arnold networks (KANs) in Granger causality detection and comparing their capabilities against multilayer perceptrons (MLP). In this work, we develop a framework called Granger Causality KAN (GC-KAN) along with a tailored training approach designed specifically for Granger causality detection. We test this framework on both Vector Autoregressive (VAR) models and chaotic Lorenz-96 systems, analysing the ability of KANs to sparsify input features by identifying Granger causal relationships, providing a concise yet accurate model for Granger causality detection. Our findings show the potential of KANs to outperform MLPs in discerning interpretable Granger causal relationships, particularly for the ability of identifying sparse Granger causality patterns in high-dimensional settings, and more generally, the potential of AI in causality discovery for the dynamical laws in physical systems.

Deep models for Multivariate Time Series (MTS) forecasting have recently demonstrated significant success. Channel-dependent models capture complex dependencies that channel-independent models cannot capture. However, the number of channels in real-world applications outpaces the capabilities of existing channel-dependent models, and contrary to common expectations, some models underperform the channel-independent models in handling high-dimensional data, which raises questions about the performance of channel-dependent models. To address this, our study first investigates the reasons behind the suboptimal performance of these channel-dependent models on high-dimensional MTS data. Our analysis reveals that two primary issues lie in the introduced noise from unrelated series that increases the difficulty of capturing the crucial inter-channel dependencies, and challenges in training strategies due to high-dimensional data. To address these issues, we propose STHD, the Scalable Transformer for High-Dimensional Multivariate Time Series Forecasting. STHD has three components: a) Relation Matrix Sparsity that limits the noise introduced and alleviates the memory issue; b) ReIndex applied as a training strategy to enable a more flexible batch size setting and increase the diversity of training data; and c) Transformer that handles 2-D inputs and captures channel dependencies. These components jointly enable STHD to manage the high-dimensional MTS while maintaining computational feasibility. Furthermore, experimental results show STHD's considerable improvement on three high-dimensional datasets: Crime-Chicago, Wiki-People, and Traffic. The source code and dataset are publicly available https://github.com/xinzzzhou/ScalableTransformer4HighDimensionMTSF.git.

Real-world time series often exhibit complex interdependencies that cannot be captured in isolation. Global models that model past data from multiple related time series globally while producing series-specific forecasts locally are now common. However, their forecasts for each individual series remain isolated, failing to account for the current state of its neighbouring series. Multivariate models like multivariate attention and graph neural networks can explicitly incorporate inter-series information, thus addressing the shortcomings of global models. However, these techniques exhibit quadratic complexity per timestep, limiting scalability. This paper introduces the Context Neural Network, an efficient linear complexity approach for augmenting time series models with relevant contextual insights from neighbouring time series without significant computational overhead. The proposed method enriches predictive models by providing the target series with real-time information from its neighbours, addressing the limitations of global models, yet remaining computationally tractable for large datasets.

While exogenous variables have a major impact on performance improvement in time series analysis, inter-series correlation and time dependence among them are rarely considered in the present continuous methods. The dynamical systems of multivariate time series could be modelled with complex unknown partial differential equations (PDEs) which play a prominent role in many disciplines of science and engineering. In this paper, we propose a continuous-time model for arbitrary-step prediction to learn an unknown PDE system in multivariate time series whose governing equations are parameterised by self-attention and gated recurrent neural networks. The proposed model, \underline{E}xogenous-\underline{g}uided \underline{P}artial \underline{D}ifferential \underline{E}quation Network (EgPDE-Net), takes account of the relationships among the exogenous variables and their effects on the target series. Importantly, the model can be reduced into a regularised ordinary differential equation (ODE) problem with special designed regularisation guidance, which makes the PDE problem tractable to obtain numerical solutions and feasible to predict multiple future values of the target series at arbitrary time points. Extensive experiments demonstrate that our proposed model could achieve competitive accuracy over strong baselines: on average, it outperforms the best baseline by reducing $9.85\%$ on RMSE and $13.98\%$ on MAE for arbitrary-step prediction.

Time series forecasting is prevalent in various real-world applications. Despite the promising results of deep learning models in time series forecasting, especially the Recurrent Neural Networks (RNNs), the explanations of time series models, which are critical in high-stakes applications, have received little attention. In this paper, we propose a Decomposition-based Linear Explainable LSTM (DeLELSTM) to improve the interpretability of LSTM. Conventionally, the interpretability of RNNs only concentrates on the variable importance and time importance. We additionally distinguish between the instantaneous influence of new coming data and the long-term effects of historical data. Specifically, DeLELSTM consists of two components, i.e., standard LSTM and tensorized LSTM. The tensorized LSTM assigns each variable with a unique hidden state making up a matrix $\mathbf{h}_t$, and the standard LSTM models all the variables with a shared hidden state $\mathbf{H}_t$. By decomposing the $\mathbf{H}_t$ into the linear combination of past information $\mathbf{h}_{t-1}$ and the fresh information $\mathbf{h}_{t}-\mathbf{h}_{t-1}$, we can get the instantaneous influence and the long-term effect of each variable. In addition, the advantage of linear regression also makes the explanation transparent and clear. We demonstrate the effectiveness and interpretability of DeLELSTM on three empirical datasets. Extensive experiments show that the proposed method achieves competitive performance against the baseline methods and provides a reliable explanation relative to domain knowledge.

Structural time series models are state space models for time series data. They are constructed in terms of components each of which has a direct interpretation. For example, one may consider a decomposition in which a series can be seen as the sum of trend and regression components. The multivariate Bayesian structural time series (MBSTS) model (Qiu et al., 2018) is a generalized version of many structural time series models and is constructed as the sum of a trend component, a seasonal component, a cycle component, a regression component, and an error term, where each component provides an independent and additional effect. Users have flexibility in choosing these components and are free to construct their specific forms, for example adding on a regression component with predictors generated through data mining as that in (Jammalamadaka et al., 2019). The MBSTS model uses the Bayes selection technique via Markov chain Monte Carlo (MCMC) methods to select among a set of contemporary predictors, thus one does not need to commit to a fixed set of predictors. Specifically, the variable selection technique uses a "spike and slab" approach, through which a different set of predictors can be selected in each MCMC iteration. Then important predictors will be selected according to their overall frequency of numbers being selected over the total number of MCMC iterations. The multivariate structure and the Bayesian framework allow the model to take advantage of the association structure among target series.

Time series forecasting has been a widely explored task of great importance in many applications. However, it is common that real-world time series data are recorded in a short time period, which results in a big gap between the deep model and the limited and noisy time series. In this work, we propose to address the time series forecasting problem with generative modeling and propose a bidirectional variational auto-encoder (BVAE) equipped with diffusion, denoise, and disentanglement, namely D3VAE. Specifically, a coupled diffusion probabilistic model is proposed to augment the time series data without increasing the aleatoric uncertainty and implement a more tractable inference process with BVAE. To ensure the generated series move toward the true target, we further propose to adapt and integrate the multiscale denoising score matching into the diffusion process for time series forecasting. In addition, to enhance the interpretability and stability of the prediction, we treat the latent variable in a multivariate manner and disentangle them on top of minimizing total correlation. Extensive experiments on synthetic and real-world data show that D3VAE outperforms competitive algorithms with remarkable margins. Our implementation is available at https://github.com/PaddlePaddle/PaddleSpatial/tree/main/research/D3VAE.