ABSTRACT Time series clustering is an important research problem in machine learning and data mining. With the rapid increase in the amount of time series data, many traditional clustering algorithms cannot directly deal with large‐scale time series due to some limitations in the memory capacity and the execution time. In this study, we suggest a distributed information granulation method for large‐scale time clustering problem. First, a distributed time series partitioning method is designed to randomly divide the original time series dataset into some data blocks. Then, the distributed time series granulation method is developed in the map‐reduce framework by the principle of reasonable granularity, where each time series can be described by some representative data points to show the trend state information. Finally, we introduce the large‐scale time series clustering method in terms of the fuzzy C‐means clustering algorithm. The experimental studies demonstrate the feasibility and the effectiveness on several UCR publicly benchmark time series datasets. Compared with the classical clustering methods, the proposed method can achieve a 4.86–9.65% improvement in average clustering accuracy. Meanwhile, the proposed method exhibits more advantages in both unequal length time series clustering and execution time.

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.

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.

Financial time series are generally high-dimensional, nonstationary, and exhibit heteroscedasticity. To derive a suitable way to cluster financial time series, these characteristics have to be taken into consideration. With this aim, in this article, the financial time series is firstly modeled using generalized autoregressive conditional heteroscedasticity (GARCH) models, where the parameters of GARCH models can represent the dynamic feature of the volatility in each time series. Therefore, the following clustering is realized based on the GARCH model parameters, which can help reduce the dimensionality of the original time series at the same time. Then, to produce semantically sound clustering results, we granulate the parameters based on the axiomatic fuzzy set (AFS) theory and structure them into a collection of meaningful and semantically sound entities, i.e., AFS information granules. Furthermore, the hierarchical structure of AFS information granules is built to realize time series clustering under the framework of granular computing. In the proposed approach, the characteristics of financial time series is fully considered to proceed dimensionality reduction, and the semantic clustering results obtained for different numbers of clusters are guaranteed to be the most informative. In the experiments, an application for clustering the time series coming from Chinese Yuan exchange rates against international currencies is presented to demonstrate the performance of the proposed clustering method. The results of clustering of the proposed method are the same as those of the fuzzy C-means algorithm and the hierarchical clustering with ward linkage, where the clustering results produced by the AFS hierarchical clustering exhibit well-articulated semantics at each level of the hierarchy.

Information granulation is an effective vehicle to explore data structure and has become a timely research topic. In this article, a top-down granulation model (T-GrM) adhering to the two fundamental requirements (coverage and specificity) in the principle of justifiable granularity is designed. Here, this principle is employed as a fundamental method for constructing information granules in each layer, and as an indicator to determine whether the obtained partitions need to be further divided in the next layer. For capturing the correlation among features and reflecting the data structure, an improved version of a T-GrM is offered by incorporating the principal component analysis. The proposed granulation models are evaluated on synthetic datasets, where the obtained experimental results offer some insights into the feasibility of the designed models and describe the effect of parameter values on the constructed information granules. The improved version of the T-GrM has the advantage of being transparent by generating information granules on several principal components, and thereby, the description of information granules is simpler and descriptive. As application examples, two real-world datasets are analyzed to exhibit the advantage of the constructed information granules.

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.

In time series forecasting, a challenging and important task is to realize long-term forecasting that is both accurate and transparent. In this study, we propose a long-term prediction approach by transforming the original numerical data into some meaningful and interpretable entities following the principle of justifiable granularity. The obtained sequences exhibiting sound semantics may have different lengths, which bring some difficulties when carrying out predictions. To equalize these temporal sequences, we propose to adjust their lengths by involving the dynamic time warping (DTW) distance. Two theorems are included to ensure the correctness of the proposed equalization approach. Finally, we exploit hidden Markov models (HMM) to derive the relations existing in the granular time series. A series of experiments using publicly available data are conducted to assess the performance of the proposed prediction method. The comparative analysis demonstrates the performance of the prediction delivered by the proposed model.