Dynamic Trend Modeling With Joint Interval Fuzzy Information Granulation: A Framework for Long-Term Forecasting of Interval-Valued Time Series

The long-term forecasting of interval-valued time series (ITS) is vital for applications such as financial risk warning and energy planning. Modeling ITS requires capturing not only long-term temporal dependencies but also the coevolution between interval bounds as well as between centers and radii, which together characterize the intrinsic structure of interval data. Recently, trend fuzzy information granulation methods have been introduced to ITS forecasting, enhancing the uncertainty modeling capabilities. However, these methods impose predefined assumptions on the trend-patterns described by Trend Fuzzy Information Granules (TFIGs), limiting their adaptability to depict nonlinear dynamic trends and interval structural interactions within an ITS. To address these issues, we propose a new kind of TFIGs, that is, the Joint Interval Trend-Pattern Unlimited Fuzzy Information Granules (JITPUFIGs), as well as their construction method, which leverages the AutoEncoder neural network based on Gated Recurrent Unit (GRU-AE) to jointly learn the coevolution patterns of interval centers and radii in a data-driven manner, avoiding predefined trend-pattern assumptions and incorporating an interval structure constraint to ensure logical consistency in JITPUFIGs. Building on this, the JITPUFIG-based Long Short-Term Memory (LSTM) neural networks are developed for long-term ITS forecasting. Experimental results on multiple benchmark ITS datasets show that our proposed model achieves superior forecasting accuracy while ensuring interval logical consistency—effectively preventing counterintuitive outputs such as predicted interval lower bounds exceeding upper bounds. This establishes a theoretically rigorous and effective paradigm for long-term ITS forecasting within the fuzzy system framework.