A SSM is Polymerized from Multivariate Time Series

Abstract: For multivariate time series (MTS) tasks, previous state space models (SSMs) followed the modeling paradigm of Transformer-based methods. However, none of them explicitly model the complex dependencies of MTS: the Channel Dependency variations with Time (CDT). In view of this, we delve into the derivation of SSM, which involves approximating continuously updated functions by orthogonal function basis. We then develop Poly-Mamba, a novel method for MTS forecasting. Its core concept is to expand the original orthogonal function basis space into a multivariate orthogonal function space containing variable mixing terms, and make a projection on this space so as to explicitly describe the CDT by weighted coefficients. In Poly-Mamba, we propose the Multivariate Orthogonal Polynomial Approximation (MOPA) as a simplified implementation of this concept. For the simple linear relationship between channels, we propose Linear Channel Mixing (LCM) and generate CDT patterns adaptively for different channels through a proposed Order Combining method. Experiments on six real-world datasets demonstrate that Poly-Mamba outperforms the SOTA methods, especially when dealing with datasets having a large number of channels and complex correlations. The codes and log files will be released at: https://github.com/Joeland4/Poly-Mamba.