Introduction to Fuzzy Sets and Logic

Time-series clustering algorithms have been used in a variety of areas to extract valuable information from complex and massive data sets. However, these algorithms suffer from two shortcomings. On the one hand, most of them are designed for the equal-length time series, while clustering of unequal-length time series is often encountered in real-world problems. On the other hand, commonly used distance measures of time series cannot fully reveal trend differences. To overcome these two shortcomings, this paper focuses on the trend of time series and employs the area-based shape distance to measure their similarity. In addition, we present a new hierarchical clustering for unequal-length time series based on area-based shape distance measure. A series of experiments illustrates the performance of the proposed clustering algorithm.

Hierarchical clustering of unequal-length time series with area-based shape distance

Multiview spectral clustering aims to separate data into different clusters efficiently by the use of multiview information. Many studies learn the affinity matrix from the original high‐dimensional data, whose noise goes against the clustering results. Besides, some methods based on self‐representation subspace clustering have a high time complexity. In this paper, we propose a simple, yet effective, and efficient method named Kernel‐based Low‐rank Tensorized Multiview Spectral Clustering (KLTMSC) to address these issues. Instead of using the original data to get the affinity matrix, KLTMSC learns the affinity matrix from kernel representation of the high‐dimensional data to reduce the noisy information. Furthermore, to be robust to noise, the low‐rank tensor is learned in the process of exploring the high‐order correlations between data. Experiments on real‐world data sets show that our method not only yields better results but also is quite time‐saving compared with other state‐of‐the‐art models.

Kernel‐based low‐rank tensorized multiview spectral clustering

As a hot research topic, many multi-view clustering approaches are proposed over the past few years. Nevertheless, most existing algorithms merely take the consensus information among different views into consideration for clustering. Actually, it may hinder the multi-view clustering performance in real-life applications, since different views usually contain diverse statistic properties. To address this problem, we propose a novel Tensor-based Intrinsic Subspace Representation Learning (TISRL) for multi-view clustering in this paper. Concretely, the rank preserving decomposition is proposed firstly to effectively deal with the diverse statistic information contained in different views. Then, to achieve the intrinsic subspace representation, the tensor-singular value decomposition based low-rank tensor constraint is also utilized in our method. It can be seen that specific information contained in different views is fully investigated by the rank preserving decomposition, and the high-order correlations of multi-view data are also mined by the low-rank tensor constraint. The objective function can be optimized by an augmented Lagrangian multiplier based alternating direction minimization algorithm. Experimental results on nine common used real-world multi-view datasets illustrate the superiority of TISRL.

Tensor-based Intrinsic Subspace Representation Learning for Multi-view Clustering.

In horizontal collaborative fuzzy clustering, the collaboration is implemented by adapting the difference between the collaborated partition matrix and the collaborating partition matrixes (called knowledge) in the objective function. Given a partition matrix of a dataset, the matrix obtained by exchanging its rows arbitrarily is also a partition matrix of the dataset. These partition matrices are the same except the orders of the rows, thus they represent the same knowledge. Let the original partition matrix be the collaborated one, the other be the collaborating ones, the final partition matrix given by collaborative fuzzy clustering should be the same as the original one. But the experiments result is different. If we change the collaborating matrixes to the same as the collaborated one, the final partition matrix remains the same as the collaborated one. This tells us the matching of the rows of the collaborated matrix and the collaborating matrixes is necessary. Furthermore, similar problems happened in practice. To deal with this problem, this paper proposes a knowledge matching algorithm based on the Kuhn Munkras (KM) algorithm in bipartite graph matching, then gives an improved horizontal collaborative fuzzy clustering algorithm. Experimental results show that the proposed horizontal collaborative clustering algorithm has superior performance than the existing ones.

Knowledge Matching in Horizontal Collaborative Fuzzy Clustering

Clustering a group of large-scale time series with the same length is a frequently met problem in real world. However, the existing clustering methods often show high computational cost and low efficiency when dealing with this problem. In this paper, we propose a granulation-based horizontal collaborative fuzzy clustering method for this problem. In this method, some new subgroups are built from the given large-scale time series group by segmenting all the time series with time alignment. Thus, all the subsequences in each subgroup have the same length, which are much smaller than the length of the original time series. Just because of the smaller length, the clustering of all subsequences in each subgroup can be easily carried out with lower computation cost and higher efficiency. How to aggregate the clustering results of all subgroups to obtain the clustering result of the given group of large-scale time series becomes the main task of this paper. To solve this problem, we carry out the horizontal collaborative fuzzy clustering on the last subgroup by collaborating the clustering information of the previous subgroups. To obtain much better performance, we first perform fuzzy information granulation on the original group of large-scale time series. After that, the original group of time series is transformed into a group of granular time series. Therefore, the collaborative clustering is carried out on the corresponding granular subgroups. Simulation experiments presented here illustrate the good performance of the algorithm of our method.

Large-Scale Time Series Clustering Based on Fuzzy Granulation and Collaboration

The problem of horizontal collaboration clustering which collaboration coefficient is usually given by priori knowledge is carefully researched. This paper provides an approach to obtain the optimal weight by taking the weight factor as a variable in the term of the objective function. Theory analysis and experimental results demonstrate that the new algorithm can get better clustering result.

Implement the horizontal collaborative fuzzy clustering by optimizing the collaboration coefficient

The clustering of a group of large-scale time series with same lengths is a challenging problem. Facing with this problem, the existing clustering algorithms usually show high computation cost and low efficiency. In this paper, a knowledge-guided clustering approach is proposed for this problem. In this new approach, the given group of large-scale time series is first changed into some new groups of subsequences by segmentation. All the new groups have same sizes to that of the given group of large-scale time series. In each new group, all the subsequences have same lengths. Different subsequences are obtained from different large-scale time series and start from the same index. Thus, the clustering of the given group of large-scale time series is changed into the clustering of the last new group of subsequences which is implemented by the guidance of the cluster knowledge obtained from the previous new groups. In order to obtain better performance, we perform Haar wavelet transformation on the given group of large-scale time series, and the clustering is carried on the transformed time series. The simulation experiments are given and the results show that the new clustering approach exhibits with high efficiency in revealing the cluster property of the original group of large-scale time series.

Knowledge-Guided clustering of large-scale time series under wavelet transformation

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Large-Scale Time Series Clustering Based on Fuzzy Granulation and Collaboration

Implement the horizontal collaborative fuzzy clustering by optimizing the collaboration coefficient

Knowledge-Guided clustering of large-scale time series under wavelet transformation