基于概率典型性和聚类排斥的无噪声模糊聚类方法

提出了建立在概率典型性和聚类排斥基础上的一个新型无噪声模糊聚类方法RTCM，给出了它的迭代算法过程，并验证了它的收敛性．首先引述了一般的聚类方法，它们主要分为两种：噪声聚类，如模糊c均值（FCM）、可能模糊c均值（FPCM）；无噪声聚类，如NC、PCM等，然后给出了RTCM算法模型和过程，并验证了它的局部收敛性．该算法解决噪声环境下的数据聚类问题，避免了重叠聚类．对比试验表明，该算法改善了噪声环境下FCM，NC、PCM、FPCM的聚类中心质量，有效地解决了PCM在近邻聚类数据中的聚类重叠问题．     

A new Kernelized hybrid c-mean clustering model with optimized parameters

A possibilistic approach was initially proposed for c-means clustering. Although the possibilistic approach is sound, this algorithm tends to find identical clusters. To overcome this shortcoming, a possibilistic Fuzzy c-means algorithm (PFCM) was proposed which produced memberships and possibilities simultaneously, along with the cluster centers. PFCM addresses the noise sensitivity defect of Fuzzy c-means (FCM) and overcomes the coincident cluster problem of possibilistic c-means (PCM). Here we propose a new model called Kernel-based hybrid c-means clustering (KPFCM) where PFCM is extended by adopting a Kernel induced metric in the data space to replace the original Euclidean norm metric. Use of Kernel function makes it possible to cluster data that is linearly non-separable in the original space into homogeneous groups in the transformed high dimensional space. From our experiments, we found that different Kernels with different Kernel widths lead to different clustering results. Thus a key point is to choose an appropriate Kernel width. We have also proposed a simple approach to determine the appropriate values for the Kernel width. The performance of the proposed method has been extensively compared with a few state of the art clustering techniques over a test suit of several artificial and real life data sets. Based on computer simulations, we have shown that our model gives better results than the previous models.     

Generalized Possibilistic Fuzzy C-Means with novel cluster validity indices for clustering noisy data

A generalized form of Possibilistic Fuzzy C-Means (PFCM) algorithm (GPFCM) is presented for clustering noisy data. A function of distance is used instead of the distance itself to damp noise contributions. It is shown that when the data are highly noisy, GPFCM finds accurate cluster centers but FCM (Fuzzy C-Means), PCM (Possibilistic C-Means), and PFCM algorithms fail. FCM, PCM, and PFCM yield inaccurate cluster centers when clusters are not of the same size or covariance norm is used, whereas GPFCM performs well for both of the cases even when the data are noisy. It is shown that generalized forms of FCM and PCM (GFCM and GPCM) are also more accurate than FCM and PCM. A measure is defined to evaluate performance of the clustering algorithms. It shows that average error of GPFCM and its simplified forms are about 80% smaller than those of FCM, PCM, and PFCM. However, GPFCM demands higher computational costs due to nonlinear updating equations. Three cluster validity indices are introduced to determine number of clusters in clean and noisy datasets. One of them considers compactness of the clusters; the other considers separation of the clusters, and the third one considers both separation and compactness. Performance of these indices is confirmed to be satisfactory using various examples of noisy datasets.     

Extended Gaussian kernel version of fuzzy c-means in the problem of data analyzing

Fuzzy c-means clustering with spatial constraints is considered as suitable algorithm for data clustering or data analyzing. But FCM has still lacks enough robustness to employ with noise data, because of its Euclidean distance measure objective function for finding the relationship between the objects. It can only be effective in clustering ‘spherical’ clusters, and it may not give reasonable clustering results for “non-compactly filled” spherical data such as “annular-shaped” data. This paper realized the drawbacks of the general fuzzy c-mean algorithm and it tries to introduce an extended Gaussian version of fuzzy C-means by replacing the Euclidean distance in the original object function of FCM. Firstly, this paper proposes initial kernel version of fuzzy c-means to aim at simplifying its computation and then extended it to extended Gaussian kernel version of fuzzy c-means. It derives an effective method to construct the membership matrix for objects, and it derives a robust method for updating centers from extended Gaussian version of fuzzy C-means. Furthermore, this paper proposes a new prototypes learning method and it obtains initial cluster centers using new mathematical initialization centers for the new effective objective function of fuzzy c-means, so that this paper tries to minimize the iteration of algorithms to obtain more accurate result. Initial experiment will be done with an artificially generated data to show how effectively the new proposed Gaussian version of fuzzy C-means works in obtaining clusters, and then the proposed methods can be implemented to cluster the Wisconsin breast cancer database into two clusters for the classes benign and malignant. To show the effective performance of proposed fuzzy c-means with new initialization of centers of clusters, this work compares the results with results of recent fuzzy c-means algorithm; in addition, it uses Silhouette method to validate the obtained clusters from breast cancer datasets.     

Suppressed possibilistic c-means clustering algorithm

The possibilistic c-means (PCM) clustering algorithm always suffers from a coincident clustering problem since it relaxes the probabilistic constraint in the fuzzy c-means (FCM) clustering algorithm. In this paper, to overcome the shortcoming of the PCM, a novel suppressed possibilistic c-means (S-PCM) clustering algorithm by introducing a suppressed competitive learning strategy into the PCM so as to improve the between-cluster relationships is proposed. Specifically, in the updating process the new algorithm searches for the biggest typicality which is regarded as winner by a competitive mechanism. Then it suppresses the non-winner typicalities with a suppressed rate which is used to control the learning strength. Moreover, the parameter setting problems of the suppressed rate and the penalty parameter in the S-PCM are also discussed in detail. In addition, the suppressed competitive learning strategy is still introduced into the possibilistic Gustafson–Kessel (PGK) clustering algorithm and a novel suppressed possibilistic Gustafson–Kessel (S-PGK) clustering model is proposed, which is more applicable to the ellipsoidal data clustering. Finally, experiments on several synthetic and real datasets with noise injection demonstrate the effectiveness of the proposed algorithms.     

Rough set based generalized fuzzy c-means algorithm and quantitative indices.

A generalized hybrid unsupervised learning algorithm, which is termed as rough-fuzzy possibilistic c-means (RFPCM), is proposed in this paper. It comprises a judicious integration of the principles of rough and fuzzy sets. While the concept of lower and upper approximations of rough sets deals with uncertainty, vagueness, and incompleteness in class definition, the membership function of fuzzy sets enables efficient handling of overlapping partitions. It incorporates both probabilistic and possibilistic memberships simultaneously to avoid the problems of noise sensitivity of fuzzy c-means and the coincident clusters of PCM. The concept of crisp lower bound and fuzzy boundary of a class, which is introduced in the RFPCM, enables efficient selection of cluster prototypes. The algorithm is generalized in the sense that all existing variants of c-means algorithms can be derived from the proposed algorithm as a special case. Several quantitative indices are introduced based on rough sets for the evaluation of performance of the proposed c-means algorithm. The effectiveness of the algorithm, along with a comparison with other algorithms, has been demonstrated both qualitatively and quantitatively on a set of real-life data sets.     

Fuzzy clustering with volume prototypes and adaptive cluster merging

Two extensions to objective function-based fuzzy clustering are proposed. First, the (point) prototypes are extended to hypervolumes, whose size can be fixed or can be determined automatically from the data being clustered. It is shown that clustering with hypervolume prototypes can be formulated as the minimization of an objective function. Second, a heuristic cluster merging step is introduced where the similarity among the clusters is assessed during optimization. Starting with an overestimation of the number of clusters in the data, similar clusters are merged in order to obtain a suitable partitioning. An adaptive threshold for merging is proposed. The extensions proposed are applied to Gustafson-Kessel and fuzzy c-means algorithms, and the resulting extended algorithm is given. The properties of the new algorithm are illustrated by various examples.     

动态权值混合C-均值模糊核聚类算法*

PCM算法存在聚类重叠的缺陷,PFCM算法同时利用隶属度与典型值把数据样本划分到不同的类中,提高了算法的抗噪能力,但PFCM算法对样本分布不均衡的聚类效果并不十分理想。针对此不足,可以通过Mercer核把原来的数据空间映射到特征空间,并为特征空间的每个向量分配一个动态权值,从而得到特征空间内的目标函数。理论分析和实验结果表明,相对于其他经典模糊聚类算法,新算法具有更好的健壮性和聚类效果。     

Novel Cluster Validity Index for FCM Algorithm

How to determine an appropriate number of clusters is very important when implementing a specific clustering algorithm, like c-means, fuzzy c-means （FCM）. In the literature, most cluster validity indices are originated from partition or geometrical property of the data set. In this paper, the authors developed a novel cluster validity index for FCM, based on the optimality test of FCM. Unlike the previous cluster validity indices, this novel cluster validity index is inherent in FCM itself. Comparison experiments show that the stability index can be used as cluster validity index for the fuzzy c-means.     

一种协同的可能性模糊聚类算法

模糊C-均值聚类（FCM）对噪声数据敏感和可能性C-均值聚类（PCM）对初始中心非常敏感易导致一致性聚类。协同聚类算法利用不同特征子集之间的协同关系并与其他算法相结合,可提高原有的聚类性能。对此,在可能性C-均值聚类算法（PCM）基础上将其与协同聚类算法相结合,提出一种协同的可能性C-均值模糊聚类算法（C-FCM）。该算法在改进的PCM的基础上,提高了对数据集的聚类效果。在对数据集Wine和Iris进行测试的结果表明,该方法优于PCM算法,说明该算法的有效性。     

Strong fuzzy c-means in medical image data analysis

This paper presents a robust fuzzy c-means (FCM) for an automatic effective segmentation of breast and brain magnetic resonance images (MRI). This paper obtains novel objective functions for proposed robust fuzzy c-means by replacing original Euclidean distance with properties of kernel function on feature space and using Tsallis entropy. By minimizing the proposed effective objective functions, this paper gets membership partition matrices and equations for successive prototypes. In order to reduce the computational complexity and running time, center initialization algorithm is introduced for initializing the initial cluster center. The initial experimental works have done on synthetic image and benchmark dataset to investigate the effectiveness of proposed, and then the proposed method has been implemented to differentiate the different region of real breast and brain magnetic resonance images. In order to identify the validity of proposed fuzzy c-means methods, segmentation accuracy is computed by using silhouette method. The experimental results show that the proposed method is more capable in segmentation of medical images than existed methods.     

A Novel Similarity-Based Fuzzy Clustering Algorithm by Integrating PCM and Mountain Method

The fuzzy c-means (FCM) and possibilistic c-means (PCM) algorithms have been utilized in a wide variety of fields and applications. Although many methods are derived from the FCM and PCM for clustering various types of spatial data, relational clustering has received much less attention. Most fuzzy clustering methods can only process the spatial data (e.g., in Euclidean space) instead of the nonspatial data (e.g., where the Pearson's correlation coefficient is used as similarity measure). In this paper, we propose a novel clustering method, similarity-based PCM (SPCM), which is fitted for clustering nonspatial data without requesting users to specify the cluster number. The main idea behind the SPCM is to extend the PCM for similarity-based clustering applications by integration with the mountain method. The SPCM has the merit that it can automatically generate clustering results without requesting users to specify the cluster number. Through performance evaluation on real and synthetic data sets, the SPCM method is shown to perform excellently for similarity-based clustering in clustering quality, even in a noisy environment with outliers. This complements the deficiency of other fuzzy clustering methods when applied to similarity-based clustering applications.     

Effective fuzzy c-means clustering algorithms for data clustering problems

Clustering is a well known technique in identifying intrinsic structures and find out useful information from large amount of data. One of the most extensively used clustering techniques is the fuzzy c-means algorithm. However, computational task becomes a problem in standard objective function of fuzzy c-means due to large amount of data, measurement uncertainty in data objects. Further, the fuzzy c-means suffer to set the optimal parameters for the clustering method. Hence the goal of this paper is to produce an alternative generalization of FCM clustering techniques in order to deal with the more complicated data; called quadratic entropy based fuzzy c-means. This paper is dealing with the effective quadratic entropy fuzzy c-means using the combination of regularization function, quadratic terms, mean distance functions, and kernel distance functions. It gives a complete framework of quadratic entropy approaching for constructing effective quadratic entropy based fuzzy clustering algorithms. This paper establishes an effective way of estimating memberships and updating centers by minimizing the proposed objective functions. In order to reduce the number iterations of proposed techniques this article proposes a new algorithm to initialize the cluster centers.In order to obtain the cluster validity and choosing the number of clusters in using proposed techniques, we use silhouette method. First time, this paper segments the synthetic control chart time series directly using our proposed methods for examining the performance of methods and it shows that the proposed clustering techniques have advantages over the existing standard FCM and very recent ClusterM-k-NN in segmenting synthetic control chart time series.     

Vector quantization in DCT domain using fuzzy possibilistic c-means based on penalized and compensated constraints

In this paper, fuzzy possibilistic c-means (FPCM) approach based on penalized and compensated constraints are proposed to vector quantization (VQ) in discrete cosine transform (DCT) for image compression. These approaches are named penalized fuzzy possibilistic c-means (PFPCM) and compensated fuzzy possibilistic c-means (CFPCM). The main purpose is to modify the FPCM strategy with penalized or compensated constraints so that the cluster centroids can be updated with penalized or compensated terms iteratively in order to find near-global solution in optimal problem. The information transformed by DCT was separated into DC and AC coefficients. Then, the AC coefficients are trained by using the proposed methods to generate better codebook based on VQ. The compression performances using the proposed approaches are compared with FPCM and conventional VQ method. From the experimental results, the promising performances can be obtained using the proposed approaches.     

Parallel fuzzy clustering on fixed size hypercube SIMD computers

This article presents PFCM, a parallel algorithm for fuzzy clustering of large data sets. Being a generalization of FCM, the algorithm enables arbitrary numbers of data points, features and clusters to be handled cost-optimally by hypercube SIMD computers of arbitrary cube dimension, the only limitation being the size of the local memories of the processors. Speedup responds optimally to enlarging the hypercube. PFCM owes its flexibility to the technique employed in its derivation from the sequential fuzzy C-means algorithm FCM: the association of each of the three dimensions of the problem (numbers of data points, features and clusters) with a distinct subset of hypercube dimensions.     

Fuzzy c-means improvement using relaxed constraints support vector machines

Fuzzy clustering is a widely applied method for extracting the underlying models within data. It has been applied successfully in many real-world applications. Fuzzy c-means is one of the most popular fuzzy clustering methods because it produces reasonable results and its implementation is straightforward. One problem with all fuzzy clustering algorithms such as fuzzy c-means is that some data points which are assigned to some clusters have low membership values. It is possible that many samples may be assigned to a cluster with low-confidence. In this paper, an efficient and noise-aware implementation of support vector machines, namely relaxed constraints support vector machines, is used to solve the mentioned problem and improve the performance of fuzzy c-means algorithm. First, fuzzy c-means partitions data into appropriate clusters. Then, the samples with high membership values in each cluster are selected for training a multi-class relaxed constraints support vector machine classifier. Finally, the class labels of the remaining data points are predicted by the latter classifier. The performance of the proposed clustering method is evaluated by quantitative measures such as cluster entropy and Minkowski scores. Experimental results on real-life data sets show the superiority of the proposed method.     

Possibilistic Exponential Fuzzy Clustering

Generally, abnormal points (noise and outliers) cause cluster analysis to produce low accuracy especially in fuzzy clustering. These data not only stay in clusters but also deviate the centroids from their true positions. Traditional fuzzy clustering like Fuzzy C-Means (FCM) always assigns data to all clusters which is not reasonable in some circumstances. By reformulating objective function in exponential equation, the algorithm aggressively selects data into the clusters. However noisy data and outliers cannot be properly handled by clustering process therefore they are forced to be included in a cluster because of a general probabilistic constraint that the sum of the membership degrees across all clusters is one. In order to improve this weakness, possibilistic approach relaxes this condition to improve membership assignment. Nevertheless, possibilistic clustering algorithms generally suffer from coincident clusters because their membership equations ignore the distance to other clusters. Although there are some possibilistic clustering approaches that do not generate coincident clusters, most of them require the right combination of multiple parameters for the algorithms to work. In this paper, we theoretically study Possibilistic Exponential Fuzzy Clustering (PXFCM) that integrates possibilistic approach with exponential fuzzy clustering. PXFCM has only one parameter and not only partitions the data but also filters noisy data or detects them as outliers. The comprehensive experiments show that PXFCM produces high accuracy in both clustering results and outlier detection without generating coincident problems.     

A novel hybrid method based on fuzzy cognitive maps and fuzzy clustering algorithms for grading celiac disease

This paper presents a new method based on fuzzy cognitive map (FCM) and possibilistic fuzzy c-means (PFCM) clustering algorithm for categorizing celiac disease (CD). CD is a complex disorder whose development is affected by genetics (HLA alleles) and gluten ingestion. The celiac patients who are not treated are at a high risk of cancer, malignant lymphoma, and small bowel neoplasia. Therefore, CD diagnosis and grading are of paramount importance. The proposed FCM models human thinking for the purpose of classifying patients suffering from CD. We used the latest grading method where three grades A, B1, and B2 are used. To improve FCM efficiency and classification capability, a nonlinear Hebbian learning algorithm is applied for adjusting the FCM weights. To this end, 89 cases are studied. Three experts extracted seven main determinant characteristics of CD which were considered as FCM concepts. The mutual effects of these concepts on one another and on the final concept were expressed in the form of fuzzy rules and linguistic variables. Using the center of gravity defuzzifier, we obtained the numerical values of these weights and obtained the total weight matrix. Ultimately, combining the FCM model with PFCM algorithm, we obtained the grades A, B1, and B2 accuracies as 88, 90, and 91%, respectively. The main advantage of the proposed FCM is the good transparency and interpretability in the decision-making procedure, which make it a suitable tool for daily usage in the clinical practice.

     

Alpha-cut implemented fuzzy clustering algorithms and switching regressions.

In the fuzzy c-means (FCM) clustering algorithm, almost none of the data points have a membership value of 1. Moreover, noise and outliers may cause difficulties in obtaining appropriate clustering results from the FCM algorithm. The embedding of FCM into switching regressions, called the fuzzy c-regressions (FCRs), still has the same drawbacks as FCM. In this paper, we propose the alpha-cut implemented fuzzy clustering algorithms, referred to as FCMalpha, which allow the data points being able to completely belong to one cluster. The proposed FCMalpha algorithms can form a cluster core for each cluster, where data points inside a cluster core will have a membership value of 1 so that it can resolve the drawbacks of FCM. On the other hand, the fuzziness index m plays different roles for FCM and FCMalpha. We find that the clustering results obtained by FCMalpha are more robust to noise and outliers than FCM when a larger m is used. Moreover, the cluster cores generated by FCMalpha are workable for various data shape clusters, so that FCMalpha is very suitable for embedding into switching regressions. The embedding of FCMalpha into switching regressions is called FCRalpha. The proposed FCRalpha provides better results than FCR for environments with noise or outliers. Numerical examples show the robustness and the superiority of our proposed methods.     

模糊c均值聚类算法中参数m的优选

本文利用模糊决策理论提出了一种模糊c均值(FCM)聚类算法中加权指数m的优选方法.文中定义了合适的模糊目标和模糊约束,通过模糊决策确定最佳的m值,以保证FCM算法获得好的聚类效果.实验结果显示了该方法的有效性,并得到实际应用中m的最佳取值范围为[1.5,2.5].