Design Trend Fuzzy Granulation-Based Three-Layer Fuzzy Cognitive Map for Long-Term Forecasting of Multivariate Time Series

Fuzzy Cognitive Maps (FCMs) are directed graphs with multiple nodes, rendering them well-suited for tackling the challenges of multivariate time series (MTS) forecasting. However, the conventional FCMs encounter obstacles in long-term forecasting, primarily due to the cumulated errors arising from iterative one-step forecasting. Drawing inspiration from recent advancements on fuzzy information granulation, this paper introduces a novel trend fuzzy granulation-based three-layer FCM model that operates at a granular level, effectively addressing abovementioned obstacles. This model leverages an optimization algorithm to determine the optimal number of granules for granulating an MTS into a granular time series (GTS), enabling the simultaneous consideration of trend information across various dimensions of the given MTS. Subsequently, viewing the obtained GTS as a complex structured MTS, a novel three-layer FCM architecture is devised. This FCM comprises a layer-3 FCM for extracting spatial relationships among parameters, a layer-2 FCM for extracting spatial relationships among variables, and a layer-1 FCM for capturing temporal relationships. By embedding the layer-3 FCM into the nodes of the layer-2 FCM and further embedding the layer-2 FCM into the nodes of the layer-1 FCM, the three-layer FCM can effectively capture and reflect temporal and spatial relationships while treating each complex element of the obtained GTS as a cohesive entity during forecasting. By constructing the three-layer FCM-based model at a granular level for MTS, the proposed approach mitigates accumulated errors and enhance the ability to forecast future trends with superior accuracy.