Granular computing has been an intense research area over the past two decades, focusing on acquiring, processing, and interpreting information granules. In this study, we focus on the granulation of time series and discover the overall structure of the original time series by clustering the granular time series. During the granulation process, when time series exhibit some trend (up trend, equal trend, or down trend) or consist of a variety of tendencies, the trend is essential to be involved to construct the granular time series. Following the principle of justifiable granularity, we propose to form a series of trend-based information granules to describe the original time series and effectively reduce its dimensionality. Then, the similarity measure between trend-based information granules is provided, and considering the dynamic feature of time-series data, dynamic time warping (DTW) distance is generalized to measure the distance for granular time series. In sum, we show here a novel way of forming trend-based granular time series and the corresponding similarity measure, then based on this, the hierarchical clustering of granular time series is realized. The proposed approach can capture the main essence of time series and help to reduce the computing overhead. Experimental results show that the designed approach can reveal meaningful trend-based information granules, and provide promising clustering results on UCR and real-world datasets.

In this article, we propose a two-stage time-series clustering approach to cluster time series with different shapes. The first step is to represent the time series by a suite of information granules following the principle of justifiable granularity to perform dimensionality reduction, while the second step is to realize the fuzzy clustering of the time series in the transformed representation space (viz., the space of information granules). In the dimensionality reduction process, the numerical data are granulated using a collection of information granules forming a new sequence that can well describe the original time series. Then, when clustering the time series, dynamic time warping (DTW) is employed to measure the similarity between time series and DTW barycenter averaging (DBA) is generalized to weighted DBA to be involved in the fuzzy C-means (FCMs) algorithm. Finally, the experiments are conducted on the datasets coming from UCR time-series database and Chinese stocks to demonstrate the effectiveness and advantages of the proposed fuzzy clustering approach.

The highly dimensional characteristic of time series brings many challenges on direct mining time series, such as high cost in time and space. Granular computing provides a potential strategy for representing and dealing with time series at a higher level of abstraction. In this study, we propose an information granulation-based weighted fuzzy ${C}$ -means (wFCM) method to realize time-series clustering, which could avoid high dimensionality processing and provide a concise and visible granular prototype for each cluster. In this method, each time series is first transformed into a series of information granules with trend following the principle of justifiable granularity. The formed granular time series can well capture the main features lying in the original time series and help realize dimensionality reduction. Then, the wFCM method is developed to complete time-series clustering in the granular space. Here, the dynamic time warping (DTW) is extended to capture the similarity for trend-based granular time series. Furthermore, the weighted DTW barycenter averaging is introduced to derive prototypes presented in a granular format, capturing the level, the fluctuation, and the changing trend, which are meaningful and understandable clustering results. The experiments conducted on real-world datasets coming from the UCR time-series database and Chinese stocks are presented to illustrate the effectiveness and practicality of the designed time-series clustering model.

The study is concerned with the representation and aggregation of complex uncertainty information. First, the concept of hesitant Fermatean 2‐tuple linguistic sets (HF2TLSs) is introduced for characterizing an individual’s imprecision preferences and assessing information by combining 2‐tuple linguistic terms and Fermatean fuzzy sets. The advantage of hesitant Fermatean 2‐tuple linguistic information is that it can handle higher levels of uncertainty and express the decision‐makers’ hesitancy. Second, we extend Bonferroni mean (BM) operators under the background of HF2TLSs for the sake of their application in information fusion and decision making. The Archimedean t‐norm and s‐norm‐ (ATS‐) based hesitant Fermatean 2‐tuple linguistic weighted Bonferroni mean (A‐HF2TLWBM) operator and the ATS‐based hesitant Fermatean 2‐tuple linguistic weighted geometric Bonferroni mean (A‐HF2TLWGBM) operator are developed by considering the interrelationship between any two variables. The main benefit of the proposed operators is that these operators deliver more complete and flexible results compared to existing methods. Moreover, some fundamental properties and special cases are examined by adjusting parameter values. Finally, an approach is designed as a support for handling decision making problems, and an example regarding investment selection is provided to demonstrate the practicality of the designed method with a detailed discussion of parameter influence and comparisons with the existing methods.