Constructing Spatial Relationship and Temporal Relationship Oriented Composite Fuzzy Cognitive Maps for Multivariate Time Series Forecasting

Fuzzy cognitive maps (FCMs) are directed graphs with multiple nodes, making them well-suited for addressing multivariate time series (MTS) forecasting problems. When forecasting MTS, it is crucial to treat each vector of the MTS as a whole, considering both the causalities between different variables of the vector at a timepoint (spatial relationship) and the causalities between multiple historical vectors and future vector (temporal relationship). Existing FCM-based MTS forecasting models often fail to treat the vectors as a whole and do not distinctly reflect the temporal relationship and spatial relationship in MTS. To address these limitations, this paper introduces the concept of composite fuzzy cognitive maps (CFCMs). A CFCM comprises two layers of FCMs: the layer-1 FCM describes the temporal relationship in an MTS, while the layer-2 FCM describes the spatial relationship. By embedding the layer-2 FCMs into the nodes of the layer-1 FCM, the relationships within the MTS can be separately reflected while still treating each vector as a whole. In this structure, the nodes of the layer-1 FCM represent historical vectors used to forecast the future vector, and each node of the layer-1 FCM corresponds to a layer-2 FCM whose nodes represent the variables of the vector at a specific historical timepoint in the MTS. Based on the novel CFCM concept, this paper proposes a new MTS forecasting model that can distinctly reflect the temporal and spatial relationships in an MTS and utilize multiple historical vectors to forecast the future vector. Experimental results demonstrate the effectiveness of the proposed MTS forecasting model.