模糊聚类算法研究
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摘要
聚类分析是数据挖掘与知识发现的核心技术之一。模糊C-均值聚类算法（FCM）是一种基于原型的聚类算法，具有简单、高效、数据适应性强等特点，是聚类分析中使用最为频繁的算法和研究热点。其中最受关注的问题为：（1）如何对FCM算法中目标函数恰当定义使该目标函数既能反映类内“距离”和类间“距离”要求的原则，又能体现各个特征以及不同样本的重要性；（2）无论FCM算法中目标函数如何定义，均会有相应的聚类原型与之对应，从而收敛速度甚至聚类效果必然依赖初始划分，如何建立一种基于模糊理论的聚类算法来规避聚类原型的问题，即从根本上解决对初始划分的敏感性；（3）如何恰当的去刻画半监督FCM算法，使监督样本既能体现其典型性，又不失其局限性；（4）如何减少FCM算法的计算量。
针对问题（1）与（4），引入流形学习的相似度度量,从基于判别近邻嵌入流形学习算法、基于几何流形距离和基于统计流形距离三方面对FCM聚类算法展开研究。通过算例，基于几何流形距离的FCM算法能够有效的识别不规则簇；基于判别近邻嵌入流形学习的FCM聚类算法能够有效的进行特征降维并在人脸识别上取得了良好的效果；基于统计流形距离的FCM聚类算法特别适合处理高维且具有统计特性的样本聚类，计算量也较小。
另外，将数据的统计特型与聚类算法相结合，研究了在传统FCM算法的目标函数中引入K-L信息熵来规则化FCM算法，并将距离函数采用高斯混合分布，应用于图像的分割，能将背景与目标充分分割开来。同时研究了任意高斯混合分布间的K-L距离度量，得到了更为紧凑的K-L距离度量公式，将其改造成具有对称性的距离度量，并引入到传统FCM算法和基于K-L信息熵规则化FCM算法中，建立了一种新的基于高斯混合分布间对称K-L距离及KL信息熵规则化的FCM聚类算法（GMM_PSKL-FCM），应用于图像聚类和检索中，不仅可以同时处理多类别的图像分类，而且大大减少了计算量。
针对问题（1）、（2）与（4），首先研究了样本特征对分类的贡献来确定其权重，提出了基于类间分离度和类内紧缩度的特征加权FCM算法；然后采用加权FCM算法将待分数据集分割成多个小类（冗余类），通过每个样本隶属于各冗余类的隶属度值计算冗余类间的贴近度。以冗余类为图的节点，以冗余类间的贴近度为节点间的权重，并采用Zadeh运算下的Floyd算法计算得到具有较强块对称性的冗余类间的标准贴近度矩阵，提取其谱特征，再次采用FCM算法对谱特征进行聚类完成冗余类的合并。算例表明，基于谱分析的冗余模糊聚类算法既减少了样本容量又规避了聚类原型的影响。
针对问题（3），本文将样本的先验知识转化为监督样本的隶属度约束条件加入到传统的FCM算法求解问题中，并根据监督样本的“典型性”赋予其权重，采用HPR(Hestenes-Powell-Rockafellar)乘子法进行求解，建立了一种新的加权半监督FCM算法（SSFCM-HPR）。监督样本的“典型性”取决于离它所隶属的聚类中心的远近，文中取监督样本的最大与次大隶属度值之比作为该监督样本的权重。该算法不仅保留了FCM算法对监督样本的模糊划分性，使其能有效的引导聚类过程，而且能发现其是否为交叉类样本，特别是当监督样本信息有误时，该算法能有效的减少噪声监督样本对整体分类效果的影响。同时本文在理论上还对半监督可能性聚类算法进行了探讨。结合上述流形学习及冗余聚类的FCM算法，可建立相应的半监督聚类算法，相应算法既可减少样本容量又能减低特征维数，从而大大降低算法的复杂度与计算量，从理论上没有难度，本文不再赘述。
Cluster Analysis is one of the core technologies of Knowledge Discovery and Data Mining (KDD).Fuzzy C-Means (FCM) clustering algorithm is the prototype-based clustering algorithm with thecharacteristics of simpleness, robustness, high efficiency, strong data adaptability, which make it oneof the most frequently-used cluster analysis algorithms. The most concerned issues are:(1) To definethe target function in the FCM algorithm in a proper way, and the target function not only reflects theprinciples set by the distance ratio of within/between class but also reflects the various features andthe importance of different samples;(2) No matter how the target function in the FCM algorithm isdefined, there is a corresponding clustering prototype. And the convergence speed, as well asclustering effect, must depend on the initial partition. How to create a algorithm based on fuzzy theoryto avoid the problem of clustering prototype and fundamentally solve the sensitivity of initial divisionis the issue;(3) Another issue is how to depict the semi-supervised algorithm FCM appropriately tomake supervision sample not only embody its typicality but also not lose its limitations;(4) How toreduce the computation of FCM algorithm is also a concern.
For different types of data samples, the traditional FCM algorithm usually requires different typesof distance measure function. To settle this problem, the FCM clustering algorithm based on manifoldlearning is proposed. For high-dimensional feature samples, a more appropriate distance measure ofsimilarity between samples is put forward from the study of a limited number of training samples.And the FCM clustering algorithm of dimensionality reduction is proposed. In addition, theKullback-Leibler (KL) divergence is introduced into the objective function of traditional FCMalgorithm to regularize the FCM. And the distance function uses a Gaussian mixture distribution,which performs well when applied to image segmentation. A new FCM clustering algorithm(FCM_GMM_PSKL) based on symmetric KL distance of the Gaussian mixture distribution and theKL divergence regularization is proposed. The FCM_GMM_PSKL works well in practice.
The FCM is sensitive to the initial prototypes, and it doesn’t work well only if the clusters areglobular. In this paper, the feature-weighted FCM clustering algorithms are proposed. And amulti-prototype FCM algorithm based on similarity and spectral decomposition is proposed. Largeclusters or elongated shaped clusters are first divided into lots of small clusters by using thefeature-weighted FCM algorithm. The memberships, which represent the degrees those samplesbelong to, are used as the features to compute the similarity among small clusters. The Folydalgorithm, developed from the Zadeh’s operations, is used to modify the similarity. The real affinitymatrix, got from former steps, is regarded as the similarity matrix to generate the Laplacian matrixused in spectral decomposition. At last, the FCM algorithm is applied again to cluster the spectralfeatures to merge the small clusters. The eigenvectors corresponding to the second minimaleigenvalues (issues in connection with curve segmentation) and the third minimal eigenvalues (issuesin connection with surface segmentation) are chosen to be the spectral features.
Most variants of Fuzzy C-Means (FCM) clustering algorithms involving prior knowledge aregenerally based on the modification of the objective function or the clustering process. This paperproposes a new weighted semi-supervised FCM algorithm (SSFCM-HPR) that transforms the prior knowledge in the labeled samples into constraint conditions in terms of fuzzy membership degrees,assigns different weights according to the representativeness of the samples, and then uses the HPRmultiplier to solve the clustering problem. The “representativeness” of the labeled samples is decidedby their distances to the cluster centers they belong to. In this paper we take the ratio of the largest tothe second largest fuzzy membership degree from a labeled sample as its weight. This algorithm notonly retains the fuzzy partition of the labeled samples which guarantees the effective guidance on theclustering process, but also can detect whether a sample is an outlier or not. Moreover, when part ofthe supervised information of the labeled samples is wrong, this algorithm can reduce the influence ofthe incorrectly labeled samples on the final clustering results.
针对问题（1）与（4），引入流形学习的相似度度量,从基于判别近邻嵌入流形学习算法、基于几何流形距离和基于统计流形距离三方面对FCM聚类算法展开研究。通过算例，基于几何流形距离的FCM算法能够有效的识别不规则簇；基于判别近邻嵌入流形学习的FCM聚类算法能够有效的进行特征降维并在人脸识别上取得了良好的效果；基于统计流形距离的FCM聚类算法特别适合处理高维且具有统计特性的样本聚类，计算量也较小。
另外，将数据的统计特型与聚类算法相结合，研究了在传统FCM算法的目标函数中引入K-L信息熵来规则化FCM算法，并将距离函数采用高斯混合分布，应用于图像的分割，能将背景与目标充分分割开来。同时研究了任意高斯混合分布间的K-L距离度量，得到了更为紧凑的K-L距离度量公式，将其改造成具有对称性的距离度量，并引入到传统FCM算法和基于K-L信息熵规则化FCM算法中，建立了一种新的基于高斯混合分布间对称K-L距离及KL信息熵规则化的FCM聚类算法（GMM_PSKL-FCM），应用于图像聚类和检索中，不仅可以同时处理多类别的图像分类，而且大大减少了计算量。
针对问题（1）、（2）与（4），首先研究了样本特征对分类的贡献来确定其权重，提出了基于类间分离度和类内紧缩度的特征加权FCM算法；然后采用加权FCM算法将待分数据集分割成多个小类（冗余类），通过每个样本隶属于各冗余类的隶属度值计算冗余类间的贴近度。以冗余类为图的节点，以冗余类间的贴近度为节点间的权重，并采用Zadeh运算下的Floyd算法计算得到具有较强块对称性的冗余类间的标准贴近度矩阵，提取其谱特征，再次采用FCM算法对谱特征进行聚类完成冗余类的合并。算例表明，基于谱分析的冗余模糊聚类算法既减少了样本容量又规避了聚类原型的影响。
针对问题（3），本文将样本的先验知识转化为监督样本的隶属度约束条件加入到传统的FCM算法求解问题中，并根据监督样本的“典型性”赋予其权重，采用HPR(Hestenes-Powell-Rockafellar)乘子法进行求解，建立了一种新的加权半监督FCM算法（SSFCM-HPR）。监督样本的“典型性”取决于离它所隶属的聚类中心的远近，文中取监督样本的最大与次大隶属度值之比作为该监督样本的权重。该算法不仅保留了FCM算法对监督样本的模糊划分性，使其能有效的引导聚类过程，而且能发现其是否为交叉类样本，特别是当监督样本信息有误时，该算法能有效的减少噪声监督样本对整体分类效果的影响。同时本文在理论上还对半监督可能性聚类算法进行了探讨。结合上述流形学习及冗余聚类的FCM算法，可建立相应的半监督聚类算法，相应算法既可减少样本容量又能减低特征维数，从而大大降低算法的复杂度与计算量，从理论上没有难度，本文不再赘述。
Cluster Analysis is one of the core technologies of Knowledge Discovery and Data Mining (KDD).Fuzzy C-Means (FCM) clustering algorithm is the prototype-based clustering algorithm with thecharacteristics of simpleness, robustness, high efficiency, strong data adaptability, which make it oneof the most frequently-used cluster analysis algorithms. The most concerned issues are:(1) To definethe target function in the FCM algorithm in a proper way, and the target function not only reflects theprinciples set by the distance ratio of within/between class but also reflects the various features andthe importance of different samples;(2) No matter how the target function in the FCM algorithm isdefined, there is a corresponding clustering prototype. And the convergence speed, as well asclustering effect, must depend on the initial partition. How to create a algorithm based on fuzzy theoryto avoid the problem of clustering prototype and fundamentally solve the sensitivity of initial divisionis the issue;(3) Another issue is how to depict the semi-supervised algorithm FCM appropriately tomake supervision sample not only embody its typicality but also not lose its limitations;(4) How toreduce the computation of FCM algorithm is also a concern.
For different types of data samples, the traditional FCM algorithm usually requires different typesof distance measure function. To settle this problem, the FCM clustering algorithm based on manifoldlearning is proposed. For high-dimensional feature samples, a more appropriate distance measure ofsimilarity between samples is put forward from the study of a limited number of training samples.And the FCM clustering algorithm of dimensionality reduction is proposed. In addition, theKullback-Leibler (KL) divergence is introduced into the objective function of traditional FCMalgorithm to regularize the FCM. And the distance function uses a Gaussian mixture distribution,which performs well when applied to image segmentation. A new FCM clustering algorithm(FCM_GMM_PSKL) based on symmetric KL distance of the Gaussian mixture distribution and theKL divergence regularization is proposed. The FCM_GMM_PSKL works well in practice.
The FCM is sensitive to the initial prototypes, and it doesn’t work well only if the clusters areglobular. In this paper, the feature-weighted FCM clustering algorithms are proposed. And amulti-prototype FCM algorithm based on similarity and spectral decomposition is proposed. Largeclusters or elongated shaped clusters are first divided into lots of small clusters by using thefeature-weighted FCM algorithm. The memberships, which represent the degrees those samplesbelong to, are used as the features to compute the similarity among small clusters. The Folydalgorithm, developed from the Zadeh’s operations, is used to modify the similarity. The real affinitymatrix, got from former steps, is regarded as the similarity matrix to generate the Laplacian matrixused in spectral decomposition. At last, the FCM algorithm is applied again to cluster the spectralfeatures to merge the small clusters. The eigenvectors corresponding to the second minimaleigenvalues (issues in connection with curve segmentation) and the third minimal eigenvalues (issuesin connection with surface segmentation) are chosen to be the spectral features.
Most variants of Fuzzy C-Means (FCM) clustering algorithms involving prior knowledge aregenerally based on the modification of the objective function or the clustering process. This paperproposes a new weighted semi-supervised FCM algorithm (SSFCM-HPR) that transforms the prior knowledge in the labeled samples into constraint conditions in terms of fuzzy membership degrees,assigns different weights according to the representativeness of the samples, and then uses the HPRmultiplier to solve the clustering problem. The “representativeness” of the labeled samples is decidedby their distances to the cluster centers they belong to. In this paper we take the ratio of the largest tothe second largest fuzzy membership degree from a labeled sample as its weight. This algorithm notonly retains the fuzzy partition of the labeled samples which guarantees the effective guidance on theclustering process, but also can detect whether a sample is an outlier or not. Moreover, when part ofthe supervised information of the labeled samples is wrong, this algorithm can reduce the influence ofthe incorrectly labeled samples on the final clustering results.
引文
[1] Ruspini HE. A new approach to clustering. Inf Cont.,1969,15:22~28.
[2] Zadeh LA. Similarity relations and fuzzy orderings. Inf Sci.,1971,3:177~85.
[3] Tamra S. Pattern classification based on fuzzy relations. IEEE Trans. SMC,1971,1(1):217~225.
[4]高新波,谢维信.模糊聚类理论发展及应用的研究进展.科学通报,1999,44(21):2241~2251.
[5]谢维信,高新波,裴继红.模糊聚类理论发展及其应用.中国体视觉与图像分析,1999,4(2):113~119.
[6]张敏,于剑.基于划分的模糊聚类算法.软件学报,2004,15(6):858~868.
[7] Dunn J.C. Well-separated clusters and the optimal fuzzy partitions. J Cybernet,1974,4:95~105.
[8] Bezdek J.C. Pattern recognition with fuzzy objective function algorithms. New York: PlenumPress,1981.
[9] Wu KL., Yang MS. Alternative c-means clustering algorithms. Pattern Recognition,2002,35(10):2267~2278.
[10] Ozdemir D., Akarun L. A fuzzy algorithm for color quantization of images. Pattern Recognition,2002,35(8):1785~1791.
[11] Ding Rui, Liu Xiaodong, Chen Yan. The Fuzzy Clustering Algorithm Based on AFS Topology. In:Proceedings of the Third international conference on Fuzzy Systems and Knowledge Discovery,FSKD2006, Berlin, Germany,2006,89~98.
[12] Ceccarelli M., Maratea A. Improving fuzzy clustering of biological data by metric learning withside information. Int’l Journal of Approximate Reasoning,2008,47(1):45~57.
[13] Yang L., Jin R., Sukthankar R, Liu Y. An efficient algorithm for local distance metric learning. In:Proceedings of American Association for Artificial Intelligence. AAAI2006, Boston, UnitedStates,2006,543~548.
[14] Ye JP, Zhao Z, Liu H. Adaptive distance metric learning for clustering. In: Proceedings of the2007IEEE Computer Society Conference on Computer Vision and Pattern Recognition, CVPR2007, Washington: IEEE Computer Society Press,2007,1~7.
[15] Wang X.Z., Wang Y.D., Wang L.J. Improving fuzzy c-means clustering based on feature-weightlearning. Pattern Recognition Letters,2004,25(10):1123~1132.
[16]王骏,王士同.基于混合距离学习的双指数模糊C均值算法.软件学报,2010,21(8):1878~1888.
[17]王骏,王士同,王晓明.基于特征加权距离的双指数模糊子空间聚类算法.控制与决策,2010,25(8):1207~1210.
[18] LU Ping, ZHAO An-Xin. Fuzzy Clustering with Obstructed Distance Based onQuantum-behaved Particle Swarm Optimization. Second WRI Global Congress on IntelligentSystems,2010,302~305.
[19] Wang Y J., Ji N H., Yao H P. Fuzzy clustering algorithm using fuzzy weighted distance. In:Proceedings of the2010Asia-Pacific Conference on Information Theory, APCIT2010, Xi’an,China,2010,327~329.
[20] Tsai Du-Ming, Lin Chung-Chan. Fuzzy C-means based clustering for linearly and nonlinearlyseparable data. Pattern Recognition,2011,44:1750~1760.
[21] Hamasuna.Y., Endo.Y., Miyamoto.S. On Mahalanobis Distance Based Fuzzy c-Means Clusteringfor Uncertain Data Using Penalty Vector Regularization. In: Proceedings of the2011IEEEInternational Conference on Fuzzy Systems, Taipei, Taiwan,2011,810~815.
[22] Srivastava V., Tripathi B K., Pathak V K. An evolutionary fuzzy clustering with minkowskidistances. In: Proceedings of the2011International Conference on Neural InformationProcessing, ICONIP2011, Shanghai, China,2011,753~760.
[23] Gustafson DE, Kessel WC. Fuzzy clustering with a fuzzy covariance matrix. In: Proceedings ofthe18th IEEE Conference on Decision and Control, CDC1979, San Diego, CA,1979,761~766.
[24] Yang MS, Wu KL, Yu J. A novel fuzzy clustering algorithm. In: Proceedings of the2003IEEEInt’l Symp. On Computational Intelligence in Robotics and Automation. Kobe: IEEE,2003.647~652.
[25] Wu K L., Yu J., Yang M S. A novel fuzzy clustering algorithm based on a fuzzy scatter matrixwith optimality test [J]. Pattern Recognition Letters,2005,26(5):639~652.
[26] Gao J., Wang S T. Clustering algorithm based on fuzzy maximum scatter difference discriminantcriterion [J]. Journal of Software,2009,20(11):2939~2949.
[27] Girolami M. Mercer kernel based clustering in feature space. IEEE Trans. Neural Network,2002,13(3):780~784.
[28]张莉,周伟达,焦李成.核聚类算法.计算机学报,2002,25(6):587~590.
[29]沈红斌,王士同,吴小俊.离群模糊核聚类算法[J].软件学报,2004,15(7):1021~1029.
[30] Shen H B., Yang J., Wang S T., et al. Attribute Weighted Mercer Kernel based Fuzzy ClusteringAlgorithm for General Non-Spherical Datasets[J]. Soft Computing Journal,2006,10(11):1061~1073.
[31] Cai WL., Chen SC., Zhang DQ. Fast and robust fuzzy c-means clustering algorithmsincorporating local information for image segmentation. Pattern Recognition,2007,40(3):825~833.
[32] Daniel Graves, Witold Pedrycz. Kernel-based fuzzy clustering and fuzzy clustering: Acomparative experimental study. Fuzzy Sets and Systems,2010,161:522~543.
[33] Miyamoto S., Umayehara K. Fuzzy clustering by quadratic regularization[C]. In: Proceedings ofthe1998IEEE International Conference on Computational Intelligence, Anchorage, AK, USA:IEEE Computer Society,1998,1394~1399.
[34] Hafinane A., Zavidovique B., Chaudhuri S. A modified FCM with optimal peano scans for imagesegmentation[C]. In: Proceedings of the IEEE International Conference in Image Processing,Genoa, Italy, IEEE Computer Society,2005,3:840~843.
[35] Hou Z., Qian W., Huang S et al. Regularized fuzzy c-means method for brain tissue clustering[J].Pattern Recognition Letters,2007,28(13):1789~1794.
[36] Wang YuPing, Dandpat Ashok Kumar. A hybrid approach of using wavelets and fuzzy clusteringfor classifying multispectral florescence in situ hybridization image[J]. International J.Biomedical Imaging,2006,1~11.
[37] Xie Zhengping, Wang Shitong, Zang Dian You et al. Image segmentation using the enhancedpossibilistic clustering method[J]. Information Technology Journal,2007,6(4):541~546.
[38] Li RP., Mukaidon M. A maximum entropy approach to fuzzy clustering. In: Proceedings of the4th IEEE Int’l Conf. on Fuzzy System, Yokohama: IEEE,1995,2227~2232.
[39] Ichihashi H., Miyagishi K., Honda K. Fuzzy C-means clustering with regularization by K-Linformation. In: Proceedings of the10th IEEE Int’l Conf. on Fuzzy Systems, Vol2. Melbourne:IEEE,2001,924~927.
[40] Miyamoto S., Mukaidono M. Fuzzy c-means as a regulation and maximum entropy approach[C].In: Proceedings of the7th International Fuzzy Systems Association World Congress, Prague,Czech Republic,1997,2:86~92.
[41] Nabila Ferahta, Abdelouahab Moussaoui, Khier B. et al. New fuzzy clustering algorithm appliedto RMN image segnentation[J]. International of Soft Computing,2006,1(2):137~142.
[42] Li K., Wang Y. Fuzzy clustering based on generalized entropy and its application to imagesegmentation. In: Proceedings of the Third international conference on Artificial intelligence andcomputational intelligence, AICI2011, Taiyuan, China,2011,640~647.
[43] Krishnapuram R., Keller.J. A possibilistic approach to clustering[J]. IEEE Trans. Fuzzy Systems,1993,1(2):98~110.
[44] Krishnapuram R., Keller J M. The possibilistic c-means algorithm: Insights andrecommendations [J]. IEEE Trans. Fuzzy System,1996,4(3):385~393.
[45] Pezdrcy W. Condition fuzzy C-means. Pattern Recognition Letters,1996,17(6):625~631.
[46] Karayiannis NB. Generalized fuzzy c-means algorithms. In: Proceedings of the5th IEEE Int’lConf. on Fuzzy Systems, Vol2. New Orleans: IEEE,1996,1036~1042.
[47] Pal RN, Pal K, Bezedek JC. A mixed c-means clustering model. In: Proceedings of the6th IEEEInt’l Conf. on Fuzzy Systems, Vol1. Barcelona：IEEE,1997,11~21.
[48] Cherkassky V., Mulier F. Learning From Data Concepts, Theory, and Methods. John Wiley&Sons, Inc.,1998,413~417.
[49] Schneider A. Weighted possibilistic clustering algorithms. In: Proceedings of the9th IEEE Int’lConf. on Fuzzy Systems, Texas：IEEE,2000,176~180.
[50] Timm H, Kruse R. A modification to improve possibilistic fuzzy cluster analysis. In: Proceedingsof the2002IEEE Int’l Conf. on Fuzzy Systems, Vol2. Honululu: IEEE,2002,1460~1465.
[51] Zhang J.S., Leung Y.W. Improved possibilistic c-means clustering algorithms. IEEE Trans. FuzzySystems,2004,12(2):209~217.
[52] Pal N. R., Pal K., Bezdek J. C. A possibilistic fuzzy c-means clustering algorithm [J]. IEEETrans. Fuzzy Systems,2005,13(4):517~530.
[53] Yang M.S., Wu K.L. Unsupervised possibilistic clustering. Pattern Recognition,2006,39(1):5~21.
[54]武小红,周建江.可能性模糊C-均值聚类新算法[J].电子学报,2008,36(10):1996~2000.
[55] Zhu L., Chung F. L., Wang S. T. Generalized fuzzy c-means clustering algorithm with improvedfuzzy partitions. IEEE Trans. Systems,2009,39(3):578~591.
[56] Yanling Li, Gang Li. Fuzzy C-Means Cluster Segmentation Algorithm Based on ModifiedMembership. In: Proceedings of the6th International Symposium on Neural Networks:Advances in Neural Networks, ISNN2009, Wuhan, China,2009,135~144.
[57] Qu Fuheng, Ma Siliang, Yating Hu. Generalized possibilistic c-means clustering based ondifferential evolution algorithm. In: Proceedings of2009International IEEE Workshop onIntelligent System and Application, ISA2009, Wuhan, China,2009,1:1~4
[58] Zhang chen, Liu bing. Possibilistic fuzzy clustering algorithm based on sample weighted. In:Proceedings of3rd International Workshop on Intelligent Systems and Applications, ISA2011,Wuhan, China,2011,1~4.
[59]齐淼,张华祥.改进的模糊C-均值聚类算法研究.计算机工程与应用,2009,45(20)：133~135.
[60]张辰,夏士雄,刘兵.一种改进的可能模糊聚类算法.计算机应用研究,2011,28(8)：2848~2851.
[61] Huang Z. Extensions to the k-means algorithm for clustering large data sets with categoricalvalues. Data Mining and Knowledge Discovery,1998,2(3):283~304.
[62] Huang Z. Clustering large data sets with mixed numeric and categorical values. In: Proceedingsof the1st Pacific-Asia conference on Knowledge Discovery and Data Mining, PACDD1997,1997,21~34.
[63] He Z.Y., Xu X.F., Deng S.C. Clustering mixed numeric and categorical data: A cluster ensembleapproach. Artificial Intelligence,2005,1~14.
[64]吴孟书,吴喜之.一种改进的K-Prototypes聚类算法.统计与决策,2008,5：24~26.
[65] Mei Jianping, Chen Lihui. Fuzzy clustering with weighted medoids for relational data. PatternRecognition,2010,43:1964~1974.
[66] Sotirios P., Chatzis. A fuzzy c-means-type algorithm for clustering of data with mixed numericand categorical attributes employing a probabilistic dissimilarity functional. Expert Systems withApplications,2011,38(7):8684~8689.
[67] Grira N., Crucianu M., Boujemaa N. Fuzzy clustering with pairwise constraints forknowledge-driven image categorization[J]. IEEE Conf. on Vision, Image and Signal Processing,2006,153(3):299~304.
[68] Grira N., Crucianu M., Boujemaa N. Active semi-supervised fuzzy clustering[J]. PatternRecognition,2008,41(5):1834~1844.
[69]高翠芳,吴小俊,张松顺.改进的半监督模糊聚类算法.控制与决策,2010,25(1):115~120.
[70] ENDO Y., HAMASUNA Y., YAMASHIRO M. et al. On Semi-Supervised Fuzzy c-MeansClustering. IEEE,2009,1119~1124.
[71] Sayed Mouchaweh M. Semi-supervised classification method for dynamic applications. Fuzzysets and systems,2010,161:544~563,
[72] Willem W., Jan V., Bran S. et al. Supervised learning algorithms for multi-class classificationproblems with partial class memberships. Fuzzy Sets and Systems,2011,106~125.
[73]王亮,王士同.基于成对约束的动态加权半监督模糊核聚类.计算机工程,2012,38(1):148~150.
[74] Zhang Huizhe, Wang Jian. Improved fuzzy c-means clustering algorithm based on selectinginitial clustering centers [J]. Computer Science,2009,36(6):206~209.
[75] Wang Y., Zhu Y. SH. A short-time multifractal approach for arrhythmia detection based on fuzzyneural network[J]. IEEE Trans. Biomedical Engineering,2001,8(9):989~995.
[76] Yang. An Efficient Fuzzy Kohonen Clustering Network Algorithm. In: Proceedings of the5thInternational Conference on Fuzzy Systems and Knowledge Discovery, FSKD2008, Jinan,China,2008,510~513.
[77] Price K., Storn R., Lampinen J. Differential evolution: A practical approach to globaloptimization. Germany, Berlin: Springer,2005.
[78] Kanade P. M., Hall L. O. Fuzzy ants and clustering [J]. IEEE Trans. Systems, Man andCybernetics, Part A: Systems and Humans,2007,37(5):758~769.
[79]王浩,王秀友,陈蕴.新的混合模糊C-均值聚类算法.计算机工程与设计,2008,29(4):917~919.
[80]依玉峰,高立群,郭丽.和声搜索算法在聚类分析中的应用.东北大学学报(自然科学版),2012,33(1):47~51.
[81] Bezdek JC., Hathaway R. et al. Convergence and theory for fuzzy C-means clustering: counterexamples and repairs. IEEE Trans. PAMI,1987,17(5):873~888.
[82] Silva H.B., Brito P., Costa J.P.D. A partitional clustering algorithm validated by a clusteringtendency index based on graph theory. Pattern Recognition,2006,39:776~788.
[83] Saha S. A New Cluster Validity Index Based on Fuzzy Granulation-degranulation Criterion. In:Proceedings of the15th International Conference on Advanced Computing and Communications,ADCOM2007, Indian Institute of Technology Guwahati, INDIA,2007,353~358.
[84] Zhang Yunjie, Wang Weina, Zhang Xiaona. A cluster validity index for fuzzy clustering.Information Sciences: an International J.,2008,178(4):1205~1218.
[85] Lee M. Fuzzy Cluster Validity Index Based on Object Proximities Defined over Fuzzy PartitionMatrices. In: Proceedings of the2008IEEE International Conference on Fuzzy Systems, HongKong, China,2008,336~340.
[86] Saha I., Maulik U., Bandyopadhyay S. A new Differential Evolution based Fuzzy Clustering forAutomatic Cluster Evolution. In: Proceedings of2009IEEE International Advance ComputingConference, Patiala, India,2009,706~711.
[87] Wu K. L., Yang M. S. A cluster validity index for fuzzy clustering. Pattern Recognition Letters,2005,26:1275~1291.
[88] Zhang X B, Li J. A new validity index of fuzzy c-means clustering. In: Proceedings ofInternational Conference on Intelligent Human-Machine Systems and Cybernetics, IHMSC2009,Hangzhou, China, Los Alamitos, Calif.: IEEE Computer Society,2009,218~221.
[89] Li Y, Yu F S. A New Validity Function for Fuzzy Clustering. In: Proceedings of InternationalConference on Computational Intelligence and Natural Computation, CINC2009, Park City,Utah,2009,462~465.
[90] Krista Rizman alik. Cluster validity index for estimation of fuzzy clusters of different sizes anddensities. Pattern Recognition,2010,43(10):3374~3390.
[91] Hu Y. T., Zuo C. C., Yang Y. et al. A cluster validity index for fuzzy c-means clustering.International Conference on System Science, Engineering Design and ManufacturingInformatization,2011,263~266.
[92] Halkidi M., Batistakis Y., Vazirgiannis M. Cluster Validity Methods: Part Ⅰ. SIGMOD Record,2002,31(2):40~45.
[93] Halkidi M., Batistakis Y., Vazirgiannis M. Cluster Validity Methods: Part Ⅱ. SIGMOD Record,2002,31(3):19~27.
[94] L Groll, Jakel G. A New Convergence Proof of Fuzzy c-Means. IEEE trans. Fuzzy systems,2005,13(5):717~720.
[95] Cai W.L., Chen S.C., Zhang, D.Q. Fast and Robust Fuzzy C-means Clustering AlgorithmsIncorporating Local Information for Image Segmentation. Pattern Recognition40,2007,825~838.
[96] Gan G., Wu J. A convergence theorem for the fuzzy subspace clustering (FSC) algorithm. PatternRecognition,2008,41:1939~1947.
[97] Yang Yanqing, Jia Zhenghong, Chang Chun et al. An Efficient Fuzzy Kohonen ClusteringNetwork Algorithm. In: Proceedings of the5th International Conference on Fuzzy Systems andKnowledge Discovery, FSKD2008, Jinan, China,2008,510~513.
[98] Qu F.H., Hu Y.T., Yang Y. et al. A Convergence Theorem for Improved Kernel Based FuzzyC-Means Clustering Algorithm.2011.
[99]王娜,李霞.基于监督信息特性的主动半监督谱聚类算法[J].电子学报,2010,38(1)：172~176.
[100]张巍.基于k邻近分类准则的特征变换算法研究[博士学位论文].复旦大学,2007.
[101]公茂果,王爽,马萌,等.复杂分布数据的二阶段聚类算法[J].软件学报,2011,22(1):2760~2772.
[102] Ryo Tnoluchi, S Miyamoto. c-Means Clustering on the Multinomial Manifold. In: Proceedingsof the4th international conference on Modeling Decisions for Artificial Intelligence, MDAI2007, Kitakyushu, Japan,2007,261~268.
[103] Balazs Feil, Janos Abonyi. Geodesic Distance based Fuzzy clustering. www.fmt.vein.hu/softcomp.
[104] http://www.itl.nist.gov/iad/humanid/feret/.
[105] Maulik U., Bandyopadhyay S. Genetic algorithm-based clustering technique. PatternRecognition,2000,33(9):1455~1465.
[106] The case western Reserve university Bearing Data Center Website. Bearing data center seededfault test data [EB/OL].[2007-11-27]. http://www/eecs/cwru/edu/laboratory/bearing/.
[107] Li Bo, Chow MoYuen. Neural—network—based motor rolling bearing fault diagnosis. IEEEtrans. industrial electronics,2000,47(5):1060~1069.
[108] Liu X.F., Ma L, Zhang S. et al. Using Fuzzy C-Means and Fuzzy Intergals for Machinery FaultDiagnosis. In: Proceedings of he International Conference on Condition Monitoring, Cambridge,England,2005,1~9.
[109] Wei C., Fahn C. The multisynapse neural network and its application to fuzzy clustering. IEEETrans. Neural Networks,2002,13(3):600~618.
[110] Yu Jian, Hao Pengwei. Comments on ‘The Multisynapse Neural Network and its Application toFuzzy Clustering’. IEEE Trans. Neural Networks,2005,16(3):777~778.
[111] Kannan S.R., Ramathilagam S., Chung P.C. Effective fuzzy c-means clustering algorithms fordata clustering problems. Expert Systems with Applications,2012,39:6292~6300.
[112] Ichihashi H., Miyagishi K., Honda K. Fuzzy c-means clustering with regularization by K-Linformation. In: Proceedings of the10th IEEE International Conference on Fuzzy Systems,Melbourne, Australia,2006,3(2):924~927.
[113] Jun,Zhaojie, Liu Honghai. Fuzzy Gaussian Mixture Models. Pattern Recognition,2012,45:1146~1158.
[114] Sotirios Chatzis. A method for training finite mixture models under a fuzzy clustering principle.Fuzzy sets and systems,2010,161:3000~3013.
[115] Tipping M., Bishop C. Mixtures of probabilistic principal component analyzers. NeuralComputation,1999,11(2):443~482.
[116] Sotirios Chatzis, Theodora Varvarigou. Robust fuzzy clustering using mixtures of Student's-tdistributions. Pattern Recognition Letters,2008,29(13):1901~1905.
[117] Qing Wang, Sanjeev R., Kulkarni et al. Divergence Estimation for Multidimensional DensitiesVia k-Nearest-Neighbor Distances. IEEE Trans. Information Theory,2009,55(5):2392~2405.
[118] Beecks C., Ivanescu A., Kirchhoff S. et al. Modeling image similarity by Gaussian mixturemodels and the Signature Quadratic Form Distance, In: Proceedings of the2011IEEEInternational Conference on Computer Vision, ICCV2011, Barcelona, Spain,2011,1754~1761.
[119] Durrieu J.L., Thiran J.P., Kelly F. Lower and upper bounds for approximation of theKullback-Leibler divergence between Gaussian Mixture Models. In: Proceedings of the IEEEInternational Conference on Acoustics, Speech and Signal Processing, Kyoto, Japan,2012,1~4.
[120]王欢良,韩纪庆,郑铁然.高斯混合分布之间K-L散度的近似计算.自动化学报,2008,34(5):529~534.
[121] Goldberger, J., Gordon, S., Greenspan, H. An Efficient Image Similarity Measure Based onApproximations of KL-Divergence Between Two Gaussian Mixtures. In: Proceedings of theNinth IEEE International Conference on Computer Vision, ICCV2003, Nice, France,2003,1:487~493
[122] Hershey J. R., Olsen P. A.“Approximating the Kullback Leibler divergence between GaussianMixture Models,”. In: Proceedings of the International Conference on Audio, Speech andSignal Processing, Honolulu, Hawai, USA,2007,4:4~317
[123] Johnson D.H., Sinan S. Symmetrizing the Kullback-Leibler distance. IEEE Trans. InformationTheory,2002,1~8.
[124] Xiao-Jun Tong, Shan Zeng, Kang Zhou, et al. Hand-Written Numeral Recognition Based onZernike Moment. In: Proceedings of2008International Conference on Wavelet Analysis andPattern Recognition, ICWAPR2008, Hong Kong, China,2008,368~372.
[125] Li J., Wang J Z. Automatic Linguistic Indexing of Pictures by a Statistical Modeling Approach[J]. IEEE Trans. Pattern Analysis and Machine Intelligence,2003,25(9):1075~1088.
[126] Robotka Z., Zempleni A. IMAGE RETRIEVAL USING GAUSSIAN MIXTURE MODELS.Annales Univ. Sci. Budapest., Sect. Comp.2009,31:93~105.
[127] Carson C., Belongie S., Greenspan H. et al. Blobworld: Image segmentation usingexpectation-maximization and its application to image querying. IEEE Trans. Pattern Analysisand Machine Intelligence,2002,24(8):1026~1038.
[128] Goldberger J.,Gordon S.,Greenspan H. An efficient image similarity measure based onapproximations of KL-divergence between two Gaussian mix. In: Proceedings of the9thInternational Conference on Computer Vision, ICCV2003, Nice, France,2003,487~493.
[129] M Luszczkiewicz-Piatek, Bogdan Smolka. Effective Color Image Retrieval Based on theGaussian Mixture Model. In: Proceedings of the Third international conference onComputational Color Imaging Workshop, CCIW2011, Milan, Italy,2011,199~213.
[130] Maria Luszczkiewicz, Bogdan Smolka. Gaussian mixture model based retrieval technique forlossy compressed color Images. Computer Science including subseries Lecture Notes inArtificial Intelligence and Lecture Notes in Bioinformatics,2007,46(33):662~673.
[131] Maria Luszczkiewicz, Bogdan Smolka. Gaussian mixture model based approach to color imageretrieval. In: Proceedings of the15th International Conference on Digital Signal Processing,DSP2007, Wales, UK,2007,527~530.
[132] Yuan Hua, Zhang Xiao-ping. Texture Image Retrieval Based on a Gaussian Mixture Model andSimilarity Measure Using a Kullback Divergence, In: Proceedings of the2004IEEEInternational Conference on Multimedia and Expo, ICME2004, Taipei, Taiwan,2004,3:1867~1870.
[133] Xing Xing, Zhang Yi, Gong Bo. Mixture model based contextual image retrieval. In:Proceedings of the ACM International Conference on Image and Video Retrieval, CIVR2010,Xi’an, China,2010,251~258.
[134] Wang Yaonan, Li Chunsheng, Zuo Yi. A selection model for optimal fuzzy clustering algorithmand number of clusters based on competitive comprehensive fuzzy evaluation, IEEE Trans.Fuzzy Systems,2009,17(3):568~577.
[135] Babak Rezaee. A cluster validity index for fuzzy clustering. Fuzzy Sets and Systems,2010,161(23):3014~3025.
[136] Wang Weina, Zhang Yunjie. On fuzzy cluster validity indices.Fuzzy Sets and Systems,2007,158(19):2095~2117.
[137] Wu Kuo-Lung, Yang Miin-Shen, Hsieh June-Nan. Robust cluster validity indexes.PatternRecognition,2009,42(11):2541~2550.
[138] Frank C.H.Rhee., Choi K,S., Choi B.I. Kernel Approach to Possibilistic c-means clustering.International J. Intelligent Systems,2009,24(3):272~292
[139] Gan G., Wu J., Yang Z. A genetic fuzzy k-Modes algorithm for clustering categoricaldata. Expert Systems with Applications,2009,36(2):1615~1620.
[140] Han Yanfang, Shi Pengfei. An improved ant colony algorithm for fuzzy clustering in imagesegmentation. Neuro computing,2007,70(46):665~671.
[141] Zhang H.X., Lu J. Semi-supervised fuzzy clustering: A kernel-based approach.Knowledge-Based Systems,2009,22(6):477~481.
[142] Bouchachia A., Pedrycz W. Data Clustering with Partial Supervision. Data Mining andKnowledge Discovery,2006,(12):47~78.
[143] Pedrycz W., Vukovich G. Fuzzy clustering with supervision. Pattern Recognition,2004,37(7):1339~1349.
[144] Tari L., Baral C., Kim S. Fuzzy c-means clustering with prior biological knowledge. Journal ofBiomedical Informatics,2009,42(1):74~81.
[145] Shi J., Malik J. Normalized cuts and image segmentation. IEEE Trans. Pattern Analysis andMachine Intelligence,2000,22(8):888~905.
[146] Meila M., Liang X. Multi-way cuts and spectral clustering. U.Washington Tech Report.2003.
[147] Hagen L., Kahng A.B. New spectral methods for ratio cut partitioning and clustering. IEEETrans. Computer-Aided Design of Integrated Circuits and Systems,1992,11(9):1074~1085.
[148] Sudeep S., Padmanabhan S. Supervised learning of large perceptual organization: Graphspectral partitioning and learning automata. IEEE Trans. pattern Analysis and MachineIntelligence,2000,22(5):504~525.
[149] Pietro P., William T.F. A factorization approach to grouping. In: proceedings of the5thEuropean Conference on Computer Vision, ECCV1998, London, UK,1998,1:655~670.
[150] Ravi K., Santosh V., Adrian V. On clusterings: Good, bad and spectral. Journal of the ACM,2004,51(3):497~515.
[151] Andrew Y.N., Michael I.J, Yair W. On spectral clustering: Analysis and algorithm. In:Proceedings of Advances in Neural Information Processing Systems, NIPS2002, Cambridge,MA, MIT Press,2002,14:849~856.
[152] Weiss Y. Segmentation using eigenvectors: A unified view. In: Proceedings of the7th IEEEInternational Conference on Computer Vision, ICCV1999, Kerkyra, Greece,1999,2:975~982.
[153] Fischer I., Poland J. Amplifying the block matrix structure for spectral clustering. TechnicalReport No.IDSIA-03, Institute Dalle Molle Institute for Artificial Intelligence (IDSIA),2005,1~10.
[154] Chan P. K., Schlag D. F., Zien J. Spectral k-way ratio-cut partitioning and clustering[J]. IEEETrans. Computer Aided Design of Integrated Circuits and Systems,1994,13(9):1088~1096.
[155] Luxburg U. A Tutorial on Spectral Clustering. Statistics and Computing,2007,14(4):1~32.
[156] Brand M., Huang K. A Unifying Theorem for Spectral Embedding and Clustering. In:Proceeding of the9th International Conference on Artificial Intelligence and Statistics, KeyWest, Florida,2003,1~12.
[157] Hathaway R.J., Hu Y. Density-weighted fuzzy C-means clustering. IEEE Trans. Fuzzy Systems,2009,17(1):243~252.
[158] Tong Xiaojun, Zhang Shemin. Similarity and nearness of fuzzy sets. In: Proceedings of2005International Conference on Machine Learning and Cybernetics, ICMLC2005, Guangzhou,China,2005,8:2668~2670.
[159] Chaudhuri D.,Chaudhuri.B.B. A Novel Multi-seed Nonhierarchical Data ClusteringTechnique[J]. IEEE Trans. systems, man, and cybernetics—part b: cybernetics,1997,27(5):871~877.
[160] Ester, Martin, Hans Peter Krieg el. et al. A Density Based Algorithm for Discovering Clusters inLarge Spatial Databases with Noise. In: Proceedings of the2nd International Conference onKnowledge Discovery and Data Mining, KDD1996, Ortland, Oregon,1996.
[161]于剑,程乾生.模糊聚类方法中的最佳聚类数的搜索范围[J].中国科学（E辑）,2002,32（2）：274~280.
[162] Alp Erilli N., Ufuk Yolcu, Erol E rio lu et al. Determining the most proper number of cluster infuzzy clustering by using artificial neural networks. Expert Systems with Applications: AnInternational J.,2011,38(3):2248~2252.
[163] Wang X.Z., Wang Y.D.,Wang L.J. Improving fuzzy c-means clustering based onfeature-weighted learning. Pattern Recognition Letters,2004,25(10):1123~1132.
[164] Huang Pengfei, Zhang Daoqiang. Locality sensitive C-means clustering algorithms. Neurocomputing,2010,71:1~9.
[165] Wang Fei, Zhang Changshui. Robust self-tuning semi-supervised learning. Neuro computing,2007,70:2931~2939.
[166] Wagstaff K., Cardie C., Rogers S. et al. Constrained K-means clustering with backgroundknowledge. In: Proceedings of the18th international conference on Machine Learning, ICML2001, MA, USA,2001,577~584.
[167] Basu S., Banerjee A., Mooney RJ. Semi-Supervised clustering by seeding. In: Proceedings ofthe19th international conference on Machine Learning, ICML2002, Sydney, Australia,2002,19~26.
[168] Basu S., Banerjee A., Mooney RJ. Active semi-supervision for pairwise constrained clustering.In: Proceedings of the SIAM international conference on Data Mining, CDM2004, Lake,Florida,2004,333~344.
[169] Xing E.P., Ng A.Y., Jordan M.I. et al. Distance metric learning with application to clusteringwith side-information. In: Proceedings of the16th Annual conference on Neural InformationProcessing System, NIPS2002, Vancouver, British Columbia,2002,505~512.
[170] Yeung D.Y., Chang H. Extending the relevant component analysis algorithm for metric learningusing both positive and negative equivalence constraints. Pattern Recognition,2006,39(5):1007~1010.
[171] Basu S., Banerjee A., Mooney RJ. A probabilistic framework for semi-supervised clustering. In:Proceedings of the10th ACM international conference on Knowledge Discovery and DataMining, SIGKDD2004, Seattle, WA,2004,59~68.
[172] Bilenko M., Basu S., Mooney RJ. Integrating constraints and metric teaming is semi-supervisedclustering. In: Proceedings of the21th international conference on Machine Learning, ICML2004, Alberta, Canada,2004,81~88.
[173] Tang W., Xiong H., Zhong S. et al. Enhancing semi-supervised clustering: a feature projectionperspective. In: Proceedings of the13th ACM international conference on KnowledgeDiscovery and Data Mining, SIGKDD2007, San Jose, California,2007,707~716.
[174] Zhang D.Q., Zhou Z.H., Chen S.C. Semi-Supervised dimensionality reduction. In: Proceedingsof the7th SIAM international conference on Data Mining, SIAM2007, Minneapolis,Minnesota,2007,629~634.
[175] Pedrycz W. Algorithms of Fuzzy Clustering with Partial Supervision. Pattern RecognitionLetters,1985,3(1):13~20.
[176] Pedrycz W., Waletzky J. Fuzzy clustering with partial supervision. IEEE Trans. Systems Manand Cybernetics B27,1997,(5):787~795.
[177] Pedrycz W., Waletzky J. Neural-network front ends in unsupervised learning. IEEE Trans.Neural Networks,1997,8(2):390~401.
[178] Pedrycz W. Knowledge-Based Clustering: From Data to Information Granules. Wiley, NewYork,2005.
[179] Stutz C., Runkler T.A.Classification and prediction of road traffic using application-specificfuzzy clustering. IEEE Trans. Fuzzy Systems,2002,10(3):297~308.
[180] Bouchachia A., Pedrycz W. Enhancement of Fuzzy Clustering by Mechanisms of PartialSupervision. Fuzzy Sets and Systems,2006,157(13):1773~1759.
[181] Pedrycz W. Collaborative and knowledge-based fuzzy clustering. International Journal ofInnovative Computing, Information and Control,2007,3(1):1~12.
[182]Pedrycz W., Amato A. Lecce VD Fuzzy clustering with partial supervision in organization andclassification of digital images. IEEE Trans. Fuzzy Systems,2008,16(4):1008~1026.
[183] Kanzawa Y., Endo Y., Miyamoto S. Some Pairwise Constrained Semi-supervised Fuzzyc-Means Clustering Algorithms. In: Proceedings of the7th International Conference onModeling Decisions for Artificial Intelligence, MDAI2010, Perpignan, France,2010,268~281
[184] Bensaid A., Hall L.O., Bezdek J.C. et al. Partially Supervised Clustering for ImageSegmentation. Pattern Recognition,1996,29(5):859~871.
[185] Benkhalifa M., Bensaid A., Mouradi A. Text Categorization using the Semi-Supervised FuzzyC-Means Algorithm. In: Proceedings of the18th International Conference of the NorthAmerican Fuzzy Information Processing Society, NAFIPS1999, New York, USA,1999,561~565.
[186] Kang J.Y., Min L.Q., Luan Q.X. et al. Novel modified fuzzy c-means algorithm withapplications. Digital Signal Processing,2009,19(2):309~319.
[187] http://www.cl.cam.ac.uk/research/dtg/attarchive/facedatabase.html.
[188] http://cvc.yale.edu/projects/yalefaces/yalefaces.html.
[189] Barni M., Cappellini V., Mecocci A. Comments on A possibilistic approach to clustering [J].IEEE Trans. Fuzzy Systems,1996,4(3):393~396.
[190] Zhang D.Q., Chen S.C. Kernel-based fuzzy and possibilistic c-means clustering. In:Proceedings of the International Conference on Artificial Neural Networks, ICANN2003,Istanbul, Turkey,2003,122~125.
[191] Wu Xiaohong. A Possibilistic C-Means Clustering Algorithm Based on Kernel Methods. In:Proceedings of the International Conference on Communications, Circuits and SystemsProceedings,2006,2062~2066.
[2] Zadeh LA. Similarity relations and fuzzy orderings. Inf Sci.,1971,3:177~85.
[3] Tamra S. Pattern classification based on fuzzy relations. IEEE Trans. SMC,1971,1(1):217~225.
[4]高新波,谢维信.模糊聚类理论发展及应用的研究进展.科学通报,1999,44(21):2241~2251.
[5]谢维信,高新波,裴继红.模糊聚类理论发展及其应用.中国体视觉与图像分析,1999,4(2):113~119.
[6]张敏,于剑.基于划分的模糊聚类算法.软件学报,2004,15(6):858~868.
[7] Dunn J.C. Well-separated clusters and the optimal fuzzy partitions. J Cybernet,1974,4:95~105.
[8] Bezdek J.C. Pattern recognition with fuzzy objective function algorithms. New York: PlenumPress,1981.
[9] Wu KL., Yang MS. Alternative c-means clustering algorithms. Pattern Recognition,2002,35(10):2267~2278.
[10] Ozdemir D., Akarun L. A fuzzy algorithm for color quantization of images. Pattern Recognition,2002,35(8):1785~1791.
[11] Ding Rui, Liu Xiaodong, Chen Yan. The Fuzzy Clustering Algorithm Based on AFS Topology. In:Proceedings of the Third international conference on Fuzzy Systems and Knowledge Discovery,FSKD2006, Berlin, Germany,2006,89~98.
[12] Ceccarelli M., Maratea A. Improving fuzzy clustering of biological data by metric learning withside information. Int’l Journal of Approximate Reasoning,2008,47(1):45~57.
[13] Yang L., Jin R., Sukthankar R, Liu Y. An efficient algorithm for local distance metric learning. In:Proceedings of American Association for Artificial Intelligence. AAAI2006, Boston, UnitedStates,2006,543~548.
[14] Ye JP, Zhao Z, Liu H. Adaptive distance metric learning for clustering. In: Proceedings of the2007IEEE Computer Society Conference on Computer Vision and Pattern Recognition, CVPR2007, Washington: IEEE Computer Society Press,2007,1~7.
[15] Wang X.Z., Wang Y.D., Wang L.J. Improving fuzzy c-means clustering based on feature-weightlearning. Pattern Recognition Letters,2004,25(10):1123~1132.
[16]王骏,王士同.基于混合距离学习的双指数模糊C均值算法.软件学报,2010,21(8):1878~1888.
[17]王骏,王士同,王晓明.基于特征加权距离的双指数模糊子空间聚类算法.控制与决策,2010,25(8):1207~1210.
[18] LU Ping, ZHAO An-Xin. Fuzzy Clustering with Obstructed Distance Based onQuantum-behaved Particle Swarm Optimization. Second WRI Global Congress on IntelligentSystems,2010,302~305.
[19] Wang Y J., Ji N H., Yao H P. Fuzzy clustering algorithm using fuzzy weighted distance. In:Proceedings of the2010Asia-Pacific Conference on Information Theory, APCIT2010, Xi’an,China,2010,327~329.
[20] Tsai Du-Ming, Lin Chung-Chan. Fuzzy C-means based clustering for linearly and nonlinearlyseparable data. Pattern Recognition,2011,44:1750~1760.
[21] Hamasuna.Y., Endo.Y., Miyamoto.S. On Mahalanobis Distance Based Fuzzy c-Means Clusteringfor Uncertain Data Using Penalty Vector Regularization. In: Proceedings of the2011IEEEInternational Conference on Fuzzy Systems, Taipei, Taiwan,2011,810~815.
[22] Srivastava V., Tripathi B K., Pathak V K. An evolutionary fuzzy clustering with minkowskidistances. In: Proceedings of the2011International Conference on Neural InformationProcessing, ICONIP2011, Shanghai, China,2011,753~760.
[23] Gustafson DE, Kessel WC. Fuzzy clustering with a fuzzy covariance matrix. In: Proceedings ofthe18th IEEE Conference on Decision and Control, CDC1979, San Diego, CA,1979,761~766.
[24] Yang MS, Wu KL, Yu J. A novel fuzzy clustering algorithm. In: Proceedings of the2003IEEEInt’l Symp. On Computational Intelligence in Robotics and Automation. Kobe: IEEE,2003.647~652.
[25] Wu K L., Yu J., Yang M S. A novel fuzzy clustering algorithm based on a fuzzy scatter matrixwith optimality test [J]. Pattern Recognition Letters,2005,26(5):639~652.
[26] Gao J., Wang S T. Clustering algorithm based on fuzzy maximum scatter difference discriminantcriterion [J]. Journal of Software,2009,20(11):2939~2949.
[27] Girolami M. Mercer kernel based clustering in feature space. IEEE Trans. Neural Network,2002,13(3):780~784.
[28]张莉,周伟达,焦李成.核聚类算法.计算机学报,2002,25(6):587~590.
[29]沈红斌,王士同,吴小俊.离群模糊核聚类算法[J].软件学报,2004,15(7):1021~1029.
[30] Shen H B., Yang J., Wang S T., et al. Attribute Weighted Mercer Kernel based Fuzzy ClusteringAlgorithm for General Non-Spherical Datasets[J]. Soft Computing Journal,2006,10(11):1061~1073.
[31] Cai WL., Chen SC., Zhang DQ. Fast and robust fuzzy c-means clustering algorithmsincorporating local information for image segmentation. Pattern Recognition,2007,40(3):825~833.
[32] Daniel Graves, Witold Pedrycz. Kernel-based fuzzy clustering and fuzzy clustering: Acomparative experimental study. Fuzzy Sets and Systems,2010,161:522~543.
[33] Miyamoto S., Umayehara K. Fuzzy clustering by quadratic regularization[C]. In: Proceedings ofthe1998IEEE International Conference on Computational Intelligence, Anchorage, AK, USA:IEEE Computer Society,1998,1394~1399.
[34] Hafinane A., Zavidovique B., Chaudhuri S. A modified FCM with optimal peano scans for imagesegmentation[C]. In: Proceedings of the IEEE International Conference in Image Processing,Genoa, Italy, IEEE Computer Society,2005,3:840~843.
[35] Hou Z., Qian W., Huang S et al. Regularized fuzzy c-means method for brain tissue clustering[J].Pattern Recognition Letters,2007,28(13):1789~1794.
[36] Wang YuPing, Dandpat Ashok Kumar. A hybrid approach of using wavelets and fuzzy clusteringfor classifying multispectral florescence in situ hybridization image[J]. International J.Biomedical Imaging,2006,1~11.
[37] Xie Zhengping, Wang Shitong, Zang Dian You et al. Image segmentation using the enhancedpossibilistic clustering method[J]. Information Technology Journal,2007,6(4):541~546.
[38] Li RP., Mukaidon M. A maximum entropy approach to fuzzy clustering. In: Proceedings of the4th IEEE Int’l Conf. on Fuzzy System, Yokohama: IEEE,1995,2227~2232.
[39] Ichihashi H., Miyagishi K., Honda K. Fuzzy C-means clustering with regularization by K-Linformation. In: Proceedings of the10th IEEE Int’l Conf. on Fuzzy Systems, Vol2. Melbourne:IEEE,2001,924~927.
[40] Miyamoto S., Mukaidono M. Fuzzy c-means as a regulation and maximum entropy approach[C].In: Proceedings of the7th International Fuzzy Systems Association World Congress, Prague,Czech Republic,1997,2:86~92.
[41] Nabila Ferahta, Abdelouahab Moussaoui, Khier B. et al. New fuzzy clustering algorithm appliedto RMN image segnentation[J]. International of Soft Computing,2006,1(2):137~142.
[42] Li K., Wang Y. Fuzzy clustering based on generalized entropy and its application to imagesegmentation. In: Proceedings of the Third international conference on Artificial intelligence andcomputational intelligence, AICI2011, Taiyuan, China,2011,640~647.
[43] Krishnapuram R., Keller.J. A possibilistic approach to clustering[J]. IEEE Trans. Fuzzy Systems,1993,1(2):98~110.
[44] Krishnapuram R., Keller J M. The possibilistic c-means algorithm: Insights andrecommendations [J]. IEEE Trans. Fuzzy System,1996,4(3):385~393.
[45] Pezdrcy W. Condition fuzzy C-means. Pattern Recognition Letters,1996,17(6):625~631.
[46] Karayiannis NB. Generalized fuzzy c-means algorithms. In: Proceedings of the5th IEEE Int’lConf. on Fuzzy Systems, Vol2. New Orleans: IEEE,1996,1036~1042.
[47] Pal RN, Pal K, Bezedek JC. A mixed c-means clustering model. In: Proceedings of the6th IEEEInt’l Conf. on Fuzzy Systems, Vol1. Barcelona：IEEE,1997,11~21.
[48] Cherkassky V., Mulier F. Learning From Data Concepts, Theory, and Methods. John Wiley&Sons, Inc.,1998,413~417.
[49] Schneider A. Weighted possibilistic clustering algorithms. In: Proceedings of the9th IEEE Int’lConf. on Fuzzy Systems, Texas：IEEE,2000,176~180.
[50] Timm H, Kruse R. A modification to improve possibilistic fuzzy cluster analysis. In: Proceedingsof the2002IEEE Int’l Conf. on Fuzzy Systems, Vol2. Honululu: IEEE,2002,1460~1465.
[51] Zhang J.S., Leung Y.W. Improved possibilistic c-means clustering algorithms. IEEE Trans. FuzzySystems,2004,12(2):209~217.
[52] Pal N. R., Pal K., Bezdek J. C. A possibilistic fuzzy c-means clustering algorithm [J]. IEEETrans. Fuzzy Systems,2005,13(4):517~530.
[53] Yang M.S., Wu K.L. Unsupervised possibilistic clustering. Pattern Recognition,2006,39(1):5~21.
[54]武小红,周建江.可能性模糊C-均值聚类新算法[J].电子学报,2008,36(10):1996~2000.
[55] Zhu L., Chung F. L., Wang S. T. Generalized fuzzy c-means clustering algorithm with improvedfuzzy partitions. IEEE Trans. Systems,2009,39(3):578~591.
[56] Yanling Li, Gang Li. Fuzzy C-Means Cluster Segmentation Algorithm Based on ModifiedMembership. In: Proceedings of the6th International Symposium on Neural Networks:Advances in Neural Networks, ISNN2009, Wuhan, China,2009,135~144.
[57] Qu Fuheng, Ma Siliang, Yating Hu. Generalized possibilistic c-means clustering based ondifferential evolution algorithm. In: Proceedings of2009International IEEE Workshop onIntelligent System and Application, ISA2009, Wuhan, China,2009,1:1~4
[58] Zhang chen, Liu bing. Possibilistic fuzzy clustering algorithm based on sample weighted. In:Proceedings of3rd International Workshop on Intelligent Systems and Applications, ISA2011,Wuhan, China,2011,1~4.
[59]齐淼,张华祥.改进的模糊C-均值聚类算法研究.计算机工程与应用,2009,45(20)：133~135.
[60]张辰,夏士雄,刘兵.一种改进的可能模糊聚类算法.计算机应用研究,2011,28(8)：2848~2851.
[61] Huang Z. Extensions to the k-means algorithm for clustering large data sets with categoricalvalues. Data Mining and Knowledge Discovery,1998,2(3):283~304.
[62] Huang Z. Clustering large data sets with mixed numeric and categorical values. In: Proceedingsof the1st Pacific-Asia conference on Knowledge Discovery and Data Mining, PACDD1997,1997,21~34.
[63] He Z.Y., Xu X.F., Deng S.C. Clustering mixed numeric and categorical data: A cluster ensembleapproach. Artificial Intelligence,2005,1~14.
[64]吴孟书,吴喜之.一种改进的K-Prototypes聚类算法.统计与决策,2008,5：24~26.
[65] Mei Jianping, Chen Lihui. Fuzzy clustering with weighted medoids for relational data. PatternRecognition,2010,43:1964~1974.
[66] Sotirios P., Chatzis. A fuzzy c-means-type algorithm for clustering of data with mixed numericand categorical attributes employing a probabilistic dissimilarity functional. Expert Systems withApplications,2011,38(7):8684~8689.
[67] Grira N., Crucianu M., Boujemaa N. Fuzzy clustering with pairwise constraints forknowledge-driven image categorization[J]. IEEE Conf. on Vision, Image and Signal Processing,2006,153(3):299~304.
[68] Grira N., Crucianu M., Boujemaa N. Active semi-supervised fuzzy clustering[J]. PatternRecognition,2008,41(5):1834~1844.
[69]高翠芳,吴小俊,张松顺.改进的半监督模糊聚类算法.控制与决策,2010,25(1):115~120.
[70] ENDO Y., HAMASUNA Y., YAMASHIRO M. et al. On Semi-Supervised Fuzzy c-MeansClustering. IEEE,2009,1119~1124.
[71] Sayed Mouchaweh M. Semi-supervised classification method for dynamic applications. Fuzzysets and systems,2010,161:544~563,
[72] Willem W., Jan V., Bran S. et al. Supervised learning algorithms for multi-class classificationproblems with partial class memberships. Fuzzy Sets and Systems,2011,106~125.
[73]王亮,王士同.基于成对约束的动态加权半监督模糊核聚类.计算机工程,2012,38(1):148~150.
[74] Zhang Huizhe, Wang Jian. Improved fuzzy c-means clustering algorithm based on selectinginitial clustering centers [J]. Computer Science,2009,36(6):206~209.
[75] Wang Y., Zhu Y. SH. A short-time multifractal approach for arrhythmia detection based on fuzzyneural network[J]. IEEE Trans. Biomedical Engineering,2001,8(9):989~995.
[76] Yang. An Efficient Fuzzy Kohonen Clustering Network Algorithm. In: Proceedings of the5thInternational Conference on Fuzzy Systems and Knowledge Discovery, FSKD2008, Jinan,China,2008,510~513.
[77] Price K., Storn R., Lampinen J. Differential evolution: A practical approach to globaloptimization. Germany, Berlin: Springer,2005.
[78] Kanade P. M., Hall L. O. Fuzzy ants and clustering [J]. IEEE Trans. Systems, Man andCybernetics, Part A: Systems and Humans,2007,37(5):758~769.
[79]王浩,王秀友,陈蕴.新的混合模糊C-均值聚类算法.计算机工程与设计,2008,29(4):917~919.
[80]依玉峰,高立群,郭丽.和声搜索算法在聚类分析中的应用.东北大学学报(自然科学版),2012,33(1):47~51.
[81] Bezdek JC., Hathaway R. et al. Convergence and theory for fuzzy C-means clustering: counterexamples and repairs. IEEE Trans. PAMI,1987,17(5):873~888.
[82] Silva H.B., Brito P., Costa J.P.D. A partitional clustering algorithm validated by a clusteringtendency index based on graph theory. Pattern Recognition,2006,39:776~788.
[83] Saha S. A New Cluster Validity Index Based on Fuzzy Granulation-degranulation Criterion. In:Proceedings of the15th International Conference on Advanced Computing and Communications,ADCOM2007, Indian Institute of Technology Guwahati, INDIA,2007,353~358.
[84] Zhang Yunjie, Wang Weina, Zhang Xiaona. A cluster validity index for fuzzy clustering.Information Sciences: an International J.,2008,178(4):1205~1218.
[85] Lee M. Fuzzy Cluster Validity Index Based on Object Proximities Defined over Fuzzy PartitionMatrices. In: Proceedings of the2008IEEE International Conference on Fuzzy Systems, HongKong, China,2008,336~340.
[86] Saha I., Maulik U., Bandyopadhyay S. A new Differential Evolution based Fuzzy Clustering forAutomatic Cluster Evolution. In: Proceedings of2009IEEE International Advance ComputingConference, Patiala, India,2009,706~711.
[87] Wu K. L., Yang M. S. A cluster validity index for fuzzy clustering. Pattern Recognition Letters,2005,26:1275~1291.
[88] Zhang X B, Li J. A new validity index of fuzzy c-means clustering. In: Proceedings ofInternational Conference on Intelligent Human-Machine Systems and Cybernetics, IHMSC2009,Hangzhou, China, Los Alamitos, Calif.: IEEE Computer Society,2009,218~221.
[89] Li Y, Yu F S. A New Validity Function for Fuzzy Clustering. In: Proceedings of InternationalConference on Computational Intelligence and Natural Computation, CINC2009, Park City,Utah,2009,462~465.
[90] Krista Rizman alik. Cluster validity index for estimation of fuzzy clusters of different sizes anddensities. Pattern Recognition,2010,43(10):3374~3390.
[91] Hu Y. T., Zuo C. C., Yang Y. et al. A cluster validity index for fuzzy c-means clustering.International Conference on System Science, Engineering Design and ManufacturingInformatization,2011,263~266.
[92] Halkidi M., Batistakis Y., Vazirgiannis M. Cluster Validity Methods: Part Ⅰ. SIGMOD Record,2002,31(2):40~45.
[93] Halkidi M., Batistakis Y., Vazirgiannis M. Cluster Validity Methods: Part Ⅱ. SIGMOD Record,2002,31(3):19~27.
[94] L Groll, Jakel G. A New Convergence Proof of Fuzzy c-Means. IEEE trans. Fuzzy systems,2005,13(5):717~720.
[95] Cai W.L., Chen S.C., Zhang, D.Q. Fast and Robust Fuzzy C-means Clustering AlgorithmsIncorporating Local Information for Image Segmentation. Pattern Recognition40,2007,825~838.
[96] Gan G., Wu J. A convergence theorem for the fuzzy subspace clustering (FSC) algorithm. PatternRecognition,2008,41:1939~1947.
[97] Yang Yanqing, Jia Zhenghong, Chang Chun et al. An Efficient Fuzzy Kohonen ClusteringNetwork Algorithm. In: Proceedings of the5th International Conference on Fuzzy Systems andKnowledge Discovery, FSKD2008, Jinan, China,2008,510~513.
[98] Qu F.H., Hu Y.T., Yang Y. et al. A Convergence Theorem for Improved Kernel Based FuzzyC-Means Clustering Algorithm.2011.
[99]王娜,李霞.基于监督信息特性的主动半监督谱聚类算法[J].电子学报,2010,38(1)：172~176.
[100]张巍.基于k邻近分类准则的特征变换算法研究[博士学位论文].复旦大学,2007.
[101]公茂果,王爽,马萌,等.复杂分布数据的二阶段聚类算法[J].软件学报,2011,22(1):2760~2772.
[102] Ryo Tnoluchi, S Miyamoto. c-Means Clustering on the Multinomial Manifold. In: Proceedingsof the4th international conference on Modeling Decisions for Artificial Intelligence, MDAI2007, Kitakyushu, Japan,2007,261~268.
[103] Balazs Feil, Janos Abonyi. Geodesic Distance based Fuzzy clustering. www.fmt.vein.hu/softcomp.
[104] http://www.itl.nist.gov/iad/humanid/feret/.
[105] Maulik U., Bandyopadhyay S. Genetic algorithm-based clustering technique. PatternRecognition,2000,33(9):1455~1465.
[106] The case western Reserve university Bearing Data Center Website. Bearing data center seededfault test data [EB/OL].[2007-11-27]. http://www/eecs/cwru/edu/laboratory/bearing/.
[107] Li Bo, Chow MoYuen. Neural—network—based motor rolling bearing fault diagnosis. IEEEtrans. industrial electronics,2000,47(5):1060~1069.
[108] Liu X.F., Ma L, Zhang S. et al. Using Fuzzy C-Means and Fuzzy Intergals for Machinery FaultDiagnosis. In: Proceedings of he International Conference on Condition Monitoring, Cambridge,England,2005,1~9.
[109] Wei C., Fahn C. The multisynapse neural network and its application to fuzzy clustering. IEEETrans. Neural Networks,2002,13(3):600~618.
[110] Yu Jian, Hao Pengwei. Comments on ‘The Multisynapse Neural Network and its Application toFuzzy Clustering’. IEEE Trans. Neural Networks,2005,16(3):777~778.
[111] Kannan S.R., Ramathilagam S., Chung P.C. Effective fuzzy c-means clustering algorithms fordata clustering problems. Expert Systems with Applications,2012,39:6292~6300.
[112] Ichihashi H., Miyagishi K., Honda K. Fuzzy c-means clustering with regularization by K-Linformation. In: Proceedings of the10th IEEE International Conference on Fuzzy Systems,Melbourne, Australia,2006,3(2):924~927.
[113] Jun,Zhaojie, Liu Honghai. Fuzzy Gaussian Mixture Models. Pattern Recognition,2012,45:1146~1158.
[114] Sotirios Chatzis. A method for training finite mixture models under a fuzzy clustering principle.Fuzzy sets and systems,2010,161:3000~3013.
[115] Tipping M., Bishop C. Mixtures of probabilistic principal component analyzers. NeuralComputation,1999,11(2):443~482.
[116] Sotirios Chatzis, Theodora Varvarigou. Robust fuzzy clustering using mixtures of Student's-tdistributions. Pattern Recognition Letters,2008,29(13):1901~1905.
[117] Qing Wang, Sanjeev R., Kulkarni et al. Divergence Estimation for Multidimensional DensitiesVia k-Nearest-Neighbor Distances. IEEE Trans. Information Theory,2009,55(5):2392~2405.
[118] Beecks C., Ivanescu A., Kirchhoff S. et al. Modeling image similarity by Gaussian mixturemodels and the Signature Quadratic Form Distance, In: Proceedings of the2011IEEEInternational Conference on Computer Vision, ICCV2011, Barcelona, Spain,2011,1754~1761.
[119] Durrieu J.L., Thiran J.P., Kelly F. Lower and upper bounds for approximation of theKullback-Leibler divergence between Gaussian Mixture Models. In: Proceedings of the IEEEInternational Conference on Acoustics, Speech and Signal Processing, Kyoto, Japan,2012,1~4.
[120]王欢良,韩纪庆,郑铁然.高斯混合分布之间K-L散度的近似计算.自动化学报,2008,34(5):529~534.
[121] Goldberger, J., Gordon, S., Greenspan, H. An Efficient Image Similarity Measure Based onApproximations of KL-Divergence Between Two Gaussian Mixtures. In: Proceedings of theNinth IEEE International Conference on Computer Vision, ICCV2003, Nice, France,2003,1:487~493
[122] Hershey J. R., Olsen P. A.“Approximating the Kullback Leibler divergence between GaussianMixture Models,”. In: Proceedings of the International Conference on Audio, Speech andSignal Processing, Honolulu, Hawai, USA,2007,4:4~317
[123] Johnson D.H., Sinan S. Symmetrizing the Kullback-Leibler distance. IEEE Trans. InformationTheory,2002,1~8.
[124] Xiao-Jun Tong, Shan Zeng, Kang Zhou, et al. Hand-Written Numeral Recognition Based onZernike Moment. In: Proceedings of2008International Conference on Wavelet Analysis andPattern Recognition, ICWAPR2008, Hong Kong, China,2008,368~372.
[125] Li J., Wang J Z. Automatic Linguistic Indexing of Pictures by a Statistical Modeling Approach[J]. IEEE Trans. Pattern Analysis and Machine Intelligence,2003,25(9):1075~1088.
[126] Robotka Z., Zempleni A. IMAGE RETRIEVAL USING GAUSSIAN MIXTURE MODELS.Annales Univ. Sci. Budapest., Sect. Comp.2009,31:93~105.
[127] Carson C., Belongie S., Greenspan H. et al. Blobworld: Image segmentation usingexpectation-maximization and its application to image querying. IEEE Trans. Pattern Analysisand Machine Intelligence,2002,24(8):1026~1038.
[128] Goldberger J.,Gordon S.,Greenspan H. An efficient image similarity measure based onapproximations of KL-divergence between two Gaussian mix. In: Proceedings of the9thInternational Conference on Computer Vision, ICCV2003, Nice, France,2003,487~493.
[129] M Luszczkiewicz-Piatek, Bogdan Smolka. Effective Color Image Retrieval Based on theGaussian Mixture Model. In: Proceedings of the Third international conference onComputational Color Imaging Workshop, CCIW2011, Milan, Italy,2011,199~213.
[130] Maria Luszczkiewicz, Bogdan Smolka. Gaussian mixture model based retrieval technique forlossy compressed color Images. Computer Science including subseries Lecture Notes inArtificial Intelligence and Lecture Notes in Bioinformatics,2007,46(33):662~673.
[131] Maria Luszczkiewicz, Bogdan Smolka. Gaussian mixture model based approach to color imageretrieval. In: Proceedings of the15th International Conference on Digital Signal Processing,DSP2007, Wales, UK,2007,527~530.
[132] Yuan Hua, Zhang Xiao-ping. Texture Image Retrieval Based on a Gaussian Mixture Model andSimilarity Measure Using a Kullback Divergence, In: Proceedings of the2004IEEEInternational Conference on Multimedia and Expo, ICME2004, Taipei, Taiwan,2004,3:1867~1870.
[133] Xing Xing, Zhang Yi, Gong Bo. Mixture model based contextual image retrieval. In:Proceedings of the ACM International Conference on Image and Video Retrieval, CIVR2010,Xi’an, China,2010,251~258.
[134] Wang Yaonan, Li Chunsheng, Zuo Yi. A selection model for optimal fuzzy clustering algorithmand number of clusters based on competitive comprehensive fuzzy evaluation, IEEE Trans.Fuzzy Systems,2009,17(3):568~577.
[135] Babak Rezaee. A cluster validity index for fuzzy clustering. Fuzzy Sets and Systems,2010,161(23):3014~3025.
[136] Wang Weina, Zhang Yunjie. On fuzzy cluster validity indices.Fuzzy Sets and Systems,2007,158(19):2095~2117.
[137] Wu Kuo-Lung, Yang Miin-Shen, Hsieh June-Nan. Robust cluster validity indexes.PatternRecognition,2009,42(11):2541~2550.
[138] Frank C.H.Rhee., Choi K,S., Choi B.I. Kernel Approach to Possibilistic c-means clustering.International J. Intelligent Systems,2009,24(3):272~292
[139] Gan G., Wu J., Yang Z. A genetic fuzzy k-Modes algorithm for clustering categoricaldata. Expert Systems with Applications,2009,36(2):1615~1620.
[140] Han Yanfang, Shi Pengfei. An improved ant colony algorithm for fuzzy clustering in imagesegmentation. Neuro computing,2007,70(46):665~671.
[141] Zhang H.X., Lu J. Semi-supervised fuzzy clustering: A kernel-based approach.Knowledge-Based Systems,2009,22(6):477~481.
[142] Bouchachia A., Pedrycz W. Data Clustering with Partial Supervision. Data Mining andKnowledge Discovery,2006,(12):47~78.
[143] Pedrycz W., Vukovich G. Fuzzy clustering with supervision. Pattern Recognition,2004,37(7):1339~1349.
[144] Tari L., Baral C., Kim S. Fuzzy c-means clustering with prior biological knowledge. Journal ofBiomedical Informatics,2009,42(1):74~81.
[145] Shi J., Malik J. Normalized cuts and image segmentation. IEEE Trans. Pattern Analysis andMachine Intelligence,2000,22(8):888~905.
[146] Meila M., Liang X. Multi-way cuts and spectral clustering. U.Washington Tech Report.2003.
[147] Hagen L., Kahng A.B. New spectral methods for ratio cut partitioning and clustering. IEEETrans. Computer-Aided Design of Integrated Circuits and Systems,1992,11(9):1074~1085.
[148] Sudeep S., Padmanabhan S. Supervised learning of large perceptual organization: Graphspectral partitioning and learning automata. IEEE Trans. pattern Analysis and MachineIntelligence,2000,22(5):504~525.
[149] Pietro P., William T.F. A factorization approach to grouping. In: proceedings of the5thEuropean Conference on Computer Vision, ECCV1998, London, UK,1998,1:655~670.
[150] Ravi K., Santosh V., Adrian V. On clusterings: Good, bad and spectral. Journal of the ACM,2004,51(3):497~515.
[151] Andrew Y.N., Michael I.J, Yair W. On spectral clustering: Analysis and algorithm. In:Proceedings of Advances in Neural Information Processing Systems, NIPS2002, Cambridge,MA, MIT Press,2002,14:849~856.
[152] Weiss Y. Segmentation using eigenvectors: A unified view. In: Proceedings of the7th IEEEInternational Conference on Computer Vision, ICCV1999, Kerkyra, Greece,1999,2:975~982.
[153] Fischer I., Poland J. Amplifying the block matrix structure for spectral clustering. TechnicalReport No.IDSIA-03, Institute Dalle Molle Institute for Artificial Intelligence (IDSIA),2005,1~10.
[154] Chan P. K., Schlag D. F., Zien J. Spectral k-way ratio-cut partitioning and clustering[J]. IEEETrans. Computer Aided Design of Integrated Circuits and Systems,1994,13(9):1088~1096.
[155] Luxburg U. A Tutorial on Spectral Clustering. Statistics and Computing,2007,14(4):1~32.
[156] Brand M., Huang K. A Unifying Theorem for Spectral Embedding and Clustering. In:Proceeding of the9th International Conference on Artificial Intelligence and Statistics, KeyWest, Florida,2003,1~12.
[157] Hathaway R.J., Hu Y. Density-weighted fuzzy C-means clustering. IEEE Trans. Fuzzy Systems,2009,17(1):243~252.
[158] Tong Xiaojun, Zhang Shemin. Similarity and nearness of fuzzy sets. In: Proceedings of2005International Conference on Machine Learning and Cybernetics, ICMLC2005, Guangzhou,China,2005,8:2668~2670.
[159] Chaudhuri D.,Chaudhuri.B.B. A Novel Multi-seed Nonhierarchical Data ClusteringTechnique[J]. IEEE Trans. systems, man, and cybernetics—part b: cybernetics,1997,27(5):871~877.
[160] Ester, Martin, Hans Peter Krieg el. et al. A Density Based Algorithm for Discovering Clusters inLarge Spatial Databases with Noise. In: Proceedings of the2nd International Conference onKnowledge Discovery and Data Mining, KDD1996, Ortland, Oregon,1996.
[161]于剑,程乾生.模糊聚类方法中的最佳聚类数的搜索范围[J].中国科学（E辑）,2002,32（2）：274~280.
[162] Alp Erilli N., Ufuk Yolcu, Erol E rio lu et al. Determining the most proper number of cluster infuzzy clustering by using artificial neural networks. Expert Systems with Applications: AnInternational J.,2011,38(3):2248~2252.
[163] Wang X.Z., Wang Y.D.,Wang L.J. Improving fuzzy c-means clustering based onfeature-weighted learning. Pattern Recognition Letters,2004,25(10):1123~1132.
[164] Huang Pengfei, Zhang Daoqiang. Locality sensitive C-means clustering algorithms. Neurocomputing,2010,71:1~9.
[165] Wang Fei, Zhang Changshui. Robust self-tuning semi-supervised learning. Neuro computing,2007,70:2931~2939.
[166] Wagstaff K., Cardie C., Rogers S. et al. Constrained K-means clustering with backgroundknowledge. In: Proceedings of the18th international conference on Machine Learning, ICML2001, MA, USA,2001,577~584.
[167] Basu S., Banerjee A., Mooney RJ. Semi-Supervised clustering by seeding. In: Proceedings ofthe19th international conference on Machine Learning, ICML2002, Sydney, Australia,2002,19~26.
[168] Basu S., Banerjee A., Mooney RJ. Active semi-supervision for pairwise constrained clustering.In: Proceedings of the SIAM international conference on Data Mining, CDM2004, Lake,Florida,2004,333~344.
[169] Xing E.P., Ng A.Y., Jordan M.I. et al. Distance metric learning with application to clusteringwith side-information. In: Proceedings of the16th Annual conference on Neural InformationProcessing System, NIPS2002, Vancouver, British Columbia,2002,505~512.
[170] Yeung D.Y., Chang H. Extending the relevant component analysis algorithm for metric learningusing both positive and negative equivalence constraints. Pattern Recognition,2006,39(5):1007~1010.
[171] Basu S., Banerjee A., Mooney RJ. A probabilistic framework for semi-supervised clustering. In:Proceedings of the10th ACM international conference on Knowledge Discovery and DataMining, SIGKDD2004, Seattle, WA,2004,59~68.
[172] Bilenko M., Basu S., Mooney RJ. Integrating constraints and metric teaming is semi-supervisedclustering. In: Proceedings of the21th international conference on Machine Learning, ICML2004, Alberta, Canada,2004,81~88.
[173] Tang W., Xiong H., Zhong S. et al. Enhancing semi-supervised clustering: a feature projectionperspective. In: Proceedings of the13th ACM international conference on KnowledgeDiscovery and Data Mining, SIGKDD2007, San Jose, California,2007,707~716.
[174] Zhang D.Q., Zhou Z.H., Chen S.C. Semi-Supervised dimensionality reduction. In: Proceedingsof the7th SIAM international conference on Data Mining, SIAM2007, Minneapolis,Minnesota,2007,629~634.
[175] Pedrycz W. Algorithms of Fuzzy Clustering with Partial Supervision. Pattern RecognitionLetters,1985,3(1):13~20.
[176] Pedrycz W., Waletzky J. Fuzzy clustering with partial supervision. IEEE Trans. Systems Manand Cybernetics B27,1997,(5):787~795.
[177] Pedrycz W., Waletzky J. Neural-network front ends in unsupervised learning. IEEE Trans.Neural Networks,1997,8(2):390~401.
[178] Pedrycz W. Knowledge-Based Clustering: From Data to Information Granules. Wiley, NewYork,2005.
[179] Stutz C., Runkler T.A.Classification and prediction of road traffic using application-specificfuzzy clustering. IEEE Trans. Fuzzy Systems,2002,10(3):297~308.
[180] Bouchachia A., Pedrycz W. Enhancement of Fuzzy Clustering by Mechanisms of PartialSupervision. Fuzzy Sets and Systems,2006,157(13):1773~1759.
[181] Pedrycz W. Collaborative and knowledge-based fuzzy clustering. International Journal ofInnovative Computing, Information and Control,2007,3(1):1~12.
[182]Pedrycz W., Amato A. Lecce VD Fuzzy clustering with partial supervision in organization andclassification of digital images. IEEE Trans. Fuzzy Systems,2008,16(4):1008~1026.
[183] Kanzawa Y., Endo Y., Miyamoto S. Some Pairwise Constrained Semi-supervised Fuzzyc-Means Clustering Algorithms. In: Proceedings of the7th International Conference onModeling Decisions for Artificial Intelligence, MDAI2010, Perpignan, France,2010,268~281
[184] Bensaid A., Hall L.O., Bezdek J.C. et al. Partially Supervised Clustering for ImageSegmentation. Pattern Recognition,1996,29(5):859~871.
[185] Benkhalifa M., Bensaid A., Mouradi A. Text Categorization using the Semi-Supervised FuzzyC-Means Algorithm. In: Proceedings of the18th International Conference of the NorthAmerican Fuzzy Information Processing Society, NAFIPS1999, New York, USA,1999,561~565.
[186] Kang J.Y., Min L.Q., Luan Q.X. et al. Novel modified fuzzy c-means algorithm withapplications. Digital Signal Processing,2009,19(2):309~319.
[187] http://www.cl.cam.ac.uk/research/dtg/attarchive/facedatabase.html.
[188] http://cvc.yale.edu/projects/yalefaces/yalefaces.html.
[189] Barni M., Cappellini V., Mecocci A. Comments on A possibilistic approach to clustering [J].IEEE Trans. Fuzzy Systems,1996,4(3):393~396.
[190] Zhang D.Q., Chen S.C. Kernel-based fuzzy and possibilistic c-means clustering. In:Proceedings of the International Conference on Artificial Neural Networks, ICANN2003,Istanbul, Turkey,2003,122~125.
[191] Wu Xiaohong. A Possibilistic C-Means Clustering Algorithm Based on Kernel Methods. In:Proceedings of the International Conference on Communications, Circuits and SystemsProceedings,2006,2062~2066.