The existing long-term time-series forecasting methods based on the neural networks suffer from multiple limitations, such as accumulated errors and diminishing temporal correlation, which compromise the prediction quality. To overcome these shortcomings, in this article, we build trend fuzzy granulation-based long short-term memory (LSTM) neural networks to carry out long-term forecasting, where data points with consistent trend characteristics, including trend change, fluctuation range, and trend persistence, are predicted in unison rather than individually. Noticing that these trend characteristics are more urgently needed than just magnitude information, a question on how to granulate a time series into a granular time series consists of meaningful granules containing trend information that comes to be a crucial step. Only if the established granules fit the varying patterns of time series at utmost, such a granulation can make sense for the improvement of the forecasting accuracy; thus, an optimization method of trend-oriented fuzzy granulation is proposed to meliorate the granulation results. With the built trend fuzzy granulation-based LSTM networks, the successive iterations of one-step forecasting are prevented, and the prediction errors of data within a granule will not further increase. This is the first attempt to build trend fuzzy granule-based LSTM to predict the trend characteristics. Experiments on publicly available time series show good performance of the proposed model.

One drawback of using the existing one-step forecasting models for long-term time series prediction is the cumulative errors caused by iterations. In order to overcome this shortcoming, this paper proposes a trend-fuzzy-granulation-based adaptive fuzzy cognitive map (FCM) for long-term time series forecasting. Different from the original FCM-based forecasting models, a class of trend fuzzy information granules is built to represent the trend, fluctuation range and trend persistence of various segments of time series, which are more instrumental and comprehensive than simple magnitude information. Thus, the proposed forecasting model is a granular model according to the form of its inputs and outputs. In an original FCM-based forecasting model, the causal relationships among concepts remain unchanged throughout the training of the whole dataset, however, in reality the causal relationships may change with the state of concepts. Therefore, it is unreasonable to use the invariable causal relationships which often result in poor predictions. In view of this, we construct an adaptive FCM where different causal relationships are built to forecast concepts of different states. This is the first time to forecast trend fuzzy information granules using an adaptive FCM. Compared with the existing classical forecasting models, the proposed forecasting model achieves superior performance which is verified through a series of experimental studies.

Although the idea of information granulation has been shown to be a research craze in short-term time series forecasting, it is still urgent to develop a granular framework so that information granulation can characterize the trend distribution of data to the significant extent under a common concept of time. This article puts forward a novel granulation algorithm involving two-stage partitioning scheme so that information granule established there exhibits well-articulated semantics at the time level, while at the same time, it gives full consideration to the varying patterns of data. On this basis, a new association rule based on this type of information granules is presented. Unlike most fuzzy association rules, the proposed association rules can extract and derive the correlations between two collections of trend features corresponding to the past and future, which is in accord with human's reasoning. Keeping in mind that the prediction process should center on essential rules while freeing from the interference of irrelevant rules, which contributes to a reliable prediction result, thus, a rule selection algorithm is involved so as one makes sure the accuracy and interpretability of the forecasting results. The design of short-term forecasting model based on fuzzy inference system is implemented, where the concept of granulation eliminates the commonly used alternative, i.e., the recursive iterations of one-step prediction. Experimental results have verified the effectiveness of the proposed model.

This article aims to design a trend-oriented-granulation-based fuzzy $C$ -means (FCM) algorithm that can cluster a group of time series at an abstract (granular) level. To achieve a better trend-oriented granulation of a time series, ${{l}}_{{1}}$ trend filtering is firstly carried out to result in segments which are then optimized by the proposed segment merging algorithm. By constructing a linear fuzzy information granule (LFIG) on each segment, a granular time series which well reflects the linear trend characteristic of the original time series is produced. With the novel designed distance that can well measure the trend similarity of two LFIGs, the distance between two granular time series is calculated by the modified dynamic time warping (DTW) algorithm. Based on this distance, the LFIG-based FCM algorithm is developed for clustering time series. In this algorithm, cluster prototypes are iteratively updated by the specifically designed granule splitting and merging algorithm, which allows the lengths of prototypes to change in the process of iteration. This overcomes the serious drawback of the existing approaches, where the lengths of prototypes cannot be changed. Experimental studies demonstrate the superior performance of the proposed algorithm in clustering time series with different shapes or trends.