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．     

一种基于蚁群算法的模糊C均值聚类

针对模糊C均值(FCM)聚类算法,在选取聚类中心点时采用随机选取易使得迭代过程陷入局部最优解,FCM算法自身并不能确定聚类个数需要人为设定,并在数据分类应用时具有了一定误差的问题,提出了一种基于蚁群算法的FCM聚类算法。该算法根据蚁群聚类算法确定模糊聚类个数和FCM算法的初始聚类中心：利用蚁群算法的全局搜索性、并行计算性等特点避免了聚类陷入局部最优解：仿真结果表明了该算法的有效性。     

模糊c-均值算法和万有引力算法求解模糊聚类问题

针对单纯使用模糊c-均值算法(FCM)求解模糊聚类问题的不足,首先,提出一种改进的万有引力搜索算法,通过一定概率按照不同方式对速度进行更新,有效增大了种群的搜索域.其次,提出了模糊万有引力搜索算法(FG-SA).最后,在模糊万有引力搜索算法(FGSA)和模糊c-均值算法(FCM)的基础上,提出了一种新算法(FGSAFCM)来求解模糊聚类问题,有效避免了单纯使用模糊c-均值算法时对初始值敏感且易于陷入局部最优的缺点.采用目标函数和有效性评价函数作为评价标准,选取10个经典数据集作为测试数据,实验结果表明,新算法比单一的模糊c-均值算法有更高的准确性和鲁棒性.     

广义多变量模糊C均值聚类算法

模糊聚类算法为了保证算法的收敛性,要求模糊指标m取值大于1,这限制了算法的普适性。提出广义多变量模糊C均值聚类算法(GMFCM),在多变量模糊C均值聚类算法(MFCM)的基础上,利用粒子群优化算法对分量模糊隶属度进行优化估计,进而将模糊指标拓展到m>0的情况,同时采用梯度法得到算法聚类中心迭代公式。GMFCM理论分析了模糊指标m扩展的原理,研究了模糊指标m在不同取值情况下的性质,解释了模糊指标m的实际意义,讨论了GMFCM算法的收敛性。GMFCM继承了MFCM算法的样本分量区分性能,弥补了MFCM算法聚类中心分量与样本分量重合时的不完备性,突破了模糊聚类算法对参数m的约束,提高了模糊聚类算法的普适性。基于gauss数据集和UCI数据集的仿真测试验证了所提算法的有效性。     

集粒度计算、蚁群算法与模糊思想的聚类算法

模糊C均值聚类算法在开始时采用随机的方式选取初始聚类中心,该方式使得FCM算法对初始聚类中心的选取极为敏感,且在局部范围内较易得到最优解,但是在全局范围内的效果较差;蚁群聚类算法根据先验知识随意设定蚂蚁拾起或放下数据对象的概率,缺乏严密的数学依据。针对FCM算法和蚁群算法的不足,文中将模糊粒度计算的思想推广应用到蚁群聚类算法中,并将改进后的蚁群聚类算法与模糊C均值聚类算法相结合,提出了一种将粒度计算、蚁群算法与模糊C均值算法思想相结合的聚类算法。经过实验验证,改进后的算法较原算法具有更好的聚类效果。     

熵指数约束的模糊聚类新算法

针对基于模糊C均值聚类(fuzzy C-means,FCM)算法框架的竞争聚集聚类(competitive agglomeration,CA)算法中模糊指数m被限定为2的问题,提出了一种更为普适的模糊聚类新算法.该算法首先在FCM算法框架的基础上引入熵指数约束条件,构造了基于熵指数约束的模糊C均值聚类(entropy index constraint FCM,EIC-FCM)算法,成功地将模糊指数m1的约束条件转换为熵指数0r1的约束条件,经分析该算法具备与经典FCM算法等效的聚类性能.其后进一步在EIC-FCM算法的框架下融入竞争学习机制得到基于熵指数约束的竞争聚集聚类(entropy index constraint CA,EICCA)算法,该算法由于使用(0,1)范围的熵指数约束而不再受到模糊指数仅为2的限制,增强了算法的适应性且更具普适性的特征.在模拟数据集以及UCI数据集上的实验结果同样表明,EICCA方法较之经典的CA算法性能更为优越,参数的选择更为灵活.     

基于模糊相关度的模糊C均值聚类加权指数研究

在极小化模糊C均值（FCM）聚类目标函数的过程中,针对目前模糊加权指数m的确定缺乏理论依据和有效评价方法的问题,提出了一种基于模糊相关度的模糊加权指数计算方法。首先定义模糊相关度的聚类有效性函数,然后通过Gauss迭代计算FCM聚类有效性并将其反馈到模糊加权指数的变化中,从而使m收敛到一个稳定的最优解。理论分析和实验结果表明,该算法是有效的,所得到加权指数m符合预期的结果。     

一种新的模糊聚类有效性指标

针对模糊C均值(FCM)算法聚类数需要预先设定的问题,提出了一种新的模糊聚类有效性指标。首先,计算簇中每个属性的方差,给方差较小的属性赋予较大的权值,给方差较大的属性赋予较小的权值,得到一种基于属性加权的FCM算法;然后,根据FCM改进算法得到的隶属度矩阵计算类内紧致性和类间分离性;最后,利用类内紧致性和类间分离性定义一个新的聚类有效性指标。实验结果表明,该指标可以找到符合数据自然分布的类的数目。基于属性加权的FCM算法可以识别不同属性的重要程度,增加聚类结果的准确率,使用FCM改进算法得到的隶属度矩阵定义的有效性指标,能够发现正确的聚类个数,实现聚类无监督的学习过程。     

基于密度函数加权的模糊C均值聚类算法研究

模糊聚类算法具有较强的实用性,但传统模糊C均值算法（FCM）具有对样本集进行等划分趋势的缺陷,没有考虑不同样本的实际分布对聚类效果的影响,当数据集中各样本密集程度相差较大时,聚类结果不是很理想。因此,提出一种基于密度函数加权的模糊C均值聚类算法（DFCM算法）,该算法利用数据对象的密度函数作为每个数据点权值。实验结果表明,与传统的模糊C均值算法相比,DFCM算法具有较好的聚类效果。     

一种基于决策粗糙集的模糊C均值聚类数的确定方法

Fuzzy C-Means(FCM)是模糊聚类中聚类效果较好且应用较为广泛的聚类算法,但是其对初始聚类数的敏感性导致如何选择一个较好的C值 变得十分重要。因此,确定FCM的聚类数是使用FCM进行聚类分析时的一个至关重要的步骤。通过扩展决策粗糙集模型进行聚类的有效性分析,并进一步确定FCM的聚类数,从而避免了使用FCM时不好的初始化所带来的影响。文中提出了一种基于扩展粗糙集模型的模糊C均值聚类数的确定方法,并通过图像分割实验来验证聚类的效果。实验通过比对不同聚类数下分类结果的代价获得了一个较好的分割结果,并将结果与Z.Yu等人于2015年提出的蚁群模糊C均值混合算法(AFHA)以及提高的AFHA算法(IAFHA)进行对比,结果表明所提方法的聚类结果较好,图像分割效果较明显,Bezdek分割系数比AFHA和IAFHA算法的更高,且在Xie-Beni系数上也有较大优势。     

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.     

基于模糊层次法的改进型网络安全态势评估方法

为最大限度降低网络安全问题带来的损失,提出一种基于模糊层次分析法(FAHP)的改进型网络安全态势评估模型。鉴于未来的大规模网络环境,首先建立一套符合实际环境的,由指标层、准则层、决策层三层组成的态势指标体系;针对态势评估中的数据分布不确定性、模糊性对评估结果的影响,利用模糊C-均值(FCM)聚类和最佳聚类准则进行数据预处理,得到最佳聚类数和聚类中心;最终建立多因素二级评估模型得到态势评估向量。仿真结果表明,与目前的基于模糊层次法的态势评估方法相比,更好地考虑到某些权重小的因素,因而标准偏差更小,评估结果更加客观、准确。     

Fuzziness parameter selection in fuzzy c-means: The perspective of cluster validation

Fuzzy c-means （FCM） algorithm is an important clustering method in pattern recognition, while the fuzziness parameter, m, in FCM algorithm is a key parameter that can significantly affect the result of clustering. Cluster validity index （CVI） is a kind of criterion function to validate the clustering results, thereby determining the optimal cluster number of a data set. From the perspective of cluster validation, we propose a novel method to select the optimal value of m in FCM, and four well-known CVIs, namely XB, VK, VT, and SC, for fuzzy clustering are used. In this method, the optimal value of m is determined when CVIs reach their minimum values. Experimental results on four synthetic data sets and four real data sets have demonstrated that the range of m is [2, 3.5] and the optimal interval is [2.5, 3].     

Double indices-induced FCM clustering and its integration with fuzzy subspace clustering

As one of the most popular algorithms for cluster analysis, fuzzy c-means (FCM) and its variants have been widely studied. In this paper, a novel generalized version called double indices-induced FCM (DI-FCM) is developed from another perspective. DI-FCM introduces a power exponent r into the constraints of the objective function such that the fuzziness index m is generalized and a new criterion of selecting an appropriate fuzziness index m is defined. Furthermore, it can be explained from the viewpoint of entropy concept that the power exponent r facilitates the introduction of entropy-based constraints into fuzzy clustering algorithms. As an attractive and judicious application, DI-FCM is integrated with a fuzzy subspace clustering (FSC) algorithm so that a new fuzzy subspace clustering algorithm called double indices-induced fuzzy subspace clustering (DI-FSC) algorithm is proposed for high-dimensional data. DI-FSC replaces the commonly used Euclidean distance with the feature-weighted distance, resulting in having two fuzzy matrices in the objective function. A convergence proof of DI-FSC is also established by applying Zangwill’s convergence theorem. Several experiments on both artificial data and real data were conducted and the experimental results show the effectiveness of the proposed algorithm.     

Uncertain Fuzzy Clustering: Interval Type-2 Fuzzy Approach to C-Means

In many pattern recognition applications, it may be impossible in most cases to obtain perfect knowledge or information for a given pattern set. Uncertain information can create imperfect expressions for pattern sets in various pattern recognition algorithms. Therefore, various types of uncertainty may be taken into account when performing several pattern recognition methods. When one performs clustering with fuzzy sets, fuzzy membership values express assignment availability of patterns for clusters. However, when one assigns fuzzy memberships to a pattern set, imperfect information for a pattern set involves uncertainty which exist in the various parameters that are used in fuzzy membership assignment. When one encounters fuzzy clustering, fuzzy membership design includes various uncertainties (e.g., distance measure, fuzzifier, prototypes, etc.). In this paper, we focus on the uncertainty associated with the fuzzifier parameter m that controls the amount of fuzziness of the final C-partition in the fuzzy C-means (FCM) algorithm. To design and manage uncertainty for fuzzifier m, we extend a pattern set to interval type-2 fuzzy sets using two fuzzifiers m₁ and m₂ which creates a footprint of uncertainty (FOU) for the fuzzifier m. Then, we incorporate this interval type-2 fuzzy set into FCM to observe the effect of managing uncertainty from the two fuzzifiers. We also provide some solutions to type-reduction and defuzzification (i.e., cluster center updating and hard-partitioning) in FCM. Several experimental results are given to show the validity of our method     

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.     

Upper and lower values for the level of fuzziness in FCM

The level of fuzziness is a parameter in fuzzy system modeling which is a source of uncertainty. In order to explore the effect of this uncertainty, one needs to investigate and identify effective upper and lower boundaries of the level of fuzziness. For this purpose, Fuzzy c-means (FCM) clustering methodology is investigated to determine the effective upper and lower boundaries of the level of fuzziness in order to capture the uncertainty generated by this parameter. In this regard, we propose to expand the membership function around important information points of FCM. These important information points are, cluster centers and the mass center. At these points, it is known that, the level of fuzziness has no effect on the membership values. In this way, we identify the counter-intuitive behavior of membership function near these particular information points. It will be shown that the upper and lower values of the level of fuzziness can be identified. Hence the uncertainty generated by this parameter can be encapsulated.     

MiniMax ε-stable cluster validity index for Type-2 fuzziness

In this paper, we concentrate on the usage of uncertainty associated with the level of fuzziness in determination of the number of clusters in FCM for any data set. We propose a MiniMax ε-stable cluster validity index based on the uncertainty associated with the level of fuzziness within the framework of interval valued Type 2 fuzziness. If the data have a clustered structure, the optimum number of clusters may be assumed to have minimum uncertainty under upper and lower levels of fuzziness. Upper and lower values of the level of fuzziness for Fuzzy C-Mean (FCM) clustering methodology have been found as m = 2.6 and 1.4, respectively, in our previous studies. Our investigation shows that the stability of cluster centers with respect to the level of fuzziness is sufficient for the determination of the number of clusters.     

融合KNN优化的密度峰值和FCM聚类算法

针对模糊C均值（Fuzzy C-Means,FCM）聚类算法对初始聚类中心和噪声敏感、对边界样本聚类不够准确且易收敛于局部极小值等问题,提出了一种K邻近（KNN）优化的密度峰值（DPC）算法和FCM相结合的融合聚类算法（KDPC-FCM）。算法利用样本的K近邻信息定义样本局部密度,快速准确搜索样本的密度峰值点样本作为初始类簇中心,改善FCM聚类算法存在的不足,从而达到优化FCM聚类算法效果的目的。在多个UCI数据集、单个人造数据集、多种基准数据集和Geolife项目中的6个较大规模数据集上的实验结果表明,改进后的新算法与传统FCM算法、DSFCM算法对比,有着更好的抗噪性、聚类效果和更快的全局收敛速度,证明了新算法的可行性和有效性。     

融合空间信息的改进FCM图像分割算法

针对FCM(Fuzzy C-Means)算法在图像分割时存在选取初始聚类中心不佳与算法抗噪性差的问题,提出一种融合空间信息的改进FCM图像分割算法；首先采用了直方图算法和LOF(Local Outlier Factor)算法自适应地选取初始聚类中心,之后使用马尔科夫随机场得到先验概率改进目标函数,使用修正隶属度矩阵的方法改进算法流程,最后使用改进算法进行图像分割；为验证该算法性能,使用Berkeley图像数据集作为实验数据,选取Dice系数、JS系数、SA系数、PSNR指数、运行时间及迭代次数作为评价标准；实验结果表明,该算法能够获取更优初始聚类中心,在处理不同噪声图像上有更好的鲁棒性。