Deep Coupling Network For Multivariate Time Series Forecasting

TL;DR

This work tackles the challenge of forecasting multivariate time series by modeling both intra- and inter-series relationships through multi-order couplings and time-lag effects. It introduces DeepCN, a neural architecture built around a coupling mechanism informed by mutual information, a coupled variable representation module, and a one-forward-step inference module, achieving state-of-the-art results on seven real-world datasets. The approach demonstrates that explicit, hierarchical cross-variable interactions improve predictive accuracy, especially in settings with strong inter-series coupling such as traffic data. The study provides insights into when higher-order couplings are beneficial and lays groundwork for efficient, scalable modeling of complex dependencies in MTS data.

Abstract

Multivariate time series (MTS) forecasting is crucial in many real-world applications. To achieve accurate MTS forecasting, it is essential to simultaneously consider both intra- and inter-series relationships among time series data. However, previous work has typically modeled intra- and inter-series relationships separately and has disregarded multi-order interactions present within and between time series data, which can seriously degrade forecasting accuracy. In this paper, we reexamine intra- and inter-series relationships from the perspective of mutual information and accordingly construct a comprehensive relationship learning mechanism tailored to simultaneously capture the intricate multi-order intra- and inter-series couplings. Based on the mechanism, we propose a novel deep coupling network for MTS forecasting, named DeepCN, which consists of a coupling mechanism dedicated to explicitly exploring the multi-order intra- and inter-series relationships among time series data concurrently, a coupled variable representation module aimed at encoding diverse variable patterns, and an inference module facilitating predictions through one forward step. Extensive experiments conducted on seven real-world datasets demonstrate that our proposed DeepCN achieves superior performance compared with the state-of-the-art baselines.

Figures (12)

• Figure 1: Illustration of modeling the intra- and inter-series relationships. (a) RNN-based models connect two values at consecutive time steps while ignoring the inter-series relationships. (b) Attention-based models directly link variable values across different time steps but do not consider inter-series relationships. (c) GNN-based models construct a graph to model inter-series relationships at each timestamp and then connect values of adjacent time steps for each variable. (d) Our model DeepCN proposes a coupling mechanism to comprehensively capture intra- and inter-series relationships by leveraging the multi-order couplings of various time lags.
• Figure 2: The overview framework of DeepCN, comprising a coupling mechanism, a coupled variable representation module, and an inference module. Coupling mechanism: it comprehensively explores the complicated multi-order intra- and inter-series relationships simultaneously among time series data $X$ by a recursive multiplication (see Equation (\ref{['crossnet']})), and output couplings $\mathcal{C}$. Coupled variable representation: it first generates a dense representation $h$ by a fully connected network, and then captures the corresponding variable representations by conducting multiplications between ${h}$ and the weights $W_{var}$ (see Equation (\ref{['equation_10']})), and outputs $\mathcal{X}$. Inference: it makes predictions by a FFN network according to $\mathcal{X}$ and outputs $\hat{X}$.
• Figure 3: Multi-order couplings diagram. In the figure, we take four variables, five time lags, and second-order as an example. From the figure, there are various combinations of cross-variable and cross-time, and several of them are marked with dotted lines. Moreover, the combinations include both intra-series (e.g., $\{X_{1}^{t-2}X_{1}^{t-3}\}, \{X_{1}^{t-1}X_{1}^{t-3}\}$) and inter-series (e.g., $\{X_{1}^{t}X_{3}^{t}\}, \{X_{2}^{t-2}X_{4}^{t-1}$}) information.
• Figure 4: Coupling-based model for relationships between variables. First, we transform the input matrix $X \in \mathbb{R}^{N \times T}$ to an input vector ${x}=\mathbb{R}^{NT\times 1}$ by a reshape operation where N is the number of variables and $T$ is the input length. Then we calculate the different order couplings respectively. $x$ is the first order.
• Figure 5: Multi-step forecasting error result analysis (MAE and RMSE) on the ECG dataset. We compare the changing curve of DeepCN with the other five baseline models on the ECG dataset under different prediction lengths (3, 6, 9, 12), respectively.