Optimality test for generalized FCM and its application to parameter selection

In cluster analysis, the fuzzy c-means (FCM) clustering algorithm is the best known and most widely used method. It was proven to converge to either a local minimum or saddle points by Bezdek et al. Wei and Mendel produced efficient optimality tests for FCM fixed points. Recently, a weighting exponent selection for FCM was proposed by Yu et al. Inspired by these results, we unify several alternative FCM algorithms into one model, called the generalized fuzzy c-means (GFCM). This GFCM model presents a wide variation of FCM algorithms and can easily lead to new and interesting clustering algorithms. Moreover, we construct a general optimality test for GFCM fixed points. This is applied to theoretically choose the parameters in the GFCM model. The experimental results demonstrate the precision of the theoretical analysis.     

A contribution to convergence theory of fuzzy c-means and derivatives

In this paper, we revisit the convergence and optimization properties of fuzzy clustering algorithms, in general, and the fuzzy c-means (FCM) algorithm, in particular. Our investigation includes probabilistic and (a slightly modified implementation of) possibilistic memberships, which will be discussed under a unified view. We give a convergence proof for the axis-parallel variant of the algorithm by Gustafson and Kessel, that can be generalized to other algorithms more easily than in the usual approach. Using reformulated fuzzy clustering algorithms, we apply Banach's classical contraction principle and establish a relationship between saddle points and attractive fixed points. For the special case of FCM we derive a sufficient condition for fixed points to be attractive, allowing identification of them as (local) minima of the objective function (excluding the possibility of a saddle point).     

基于PSO的模糊聚类算法

提出了一种基于模糊C-均值算法和粒子群算法的混合聚类算法。该算法结合PSO的全局搜索和FCM局部搜索的特点，将PSO优化聚类结果作为后续FCM算法的初始值，有效地克服了FCM对初始值敏感、易陷入局部最优和PSO算法局部搜索较弱的问题，同时增强了跳出局部最优的能力。实验表明，新算法得到的目标函数值更小，并能减小分类错误率，聚类效果优于单一使用FCM或PSO。     

一种改进的模糊C均值聚类算法

针对数据集中若存在孤立点或者是噪声数据会影响模糊C均值聚类算法(FCM)的聚类性能问题,本文将离群点的辨认方法与FCM算法相结合,提出一种改进的FCM聚类算法。该算法有效地降低了孤立点或噪声数据对正常数据的影响,提高了FCM算法的聚类精度。将该算法在入侵检测系统中进行实验验证,通过与FCM算法进行对比分析,证明了该算法的有效性和可行性。     

一种基于萤火虫算法的模糊聚类方法

针对模糊C-均值聚类对初始值敏感、容易陷入局部最优的缺陷,提出了一种基于萤火虫算法的模糊聚类方法。该方法结合萤火虫算法良好的全局寻优能力和模糊C-均值算法的较强的局部搜索特性,用萤火虫算法优化搜索FCM的聚类中心,利用FCM进行聚类,有效地克服了模糊C-均值聚类的不足,同时增强了萤火虫算法的局部搜索能力。实验结果表明,该算法具有很好的全局寻优能力和较快的收敛速度,能有效地收敛于全局最优解,具有较好的聚类效果。     

Visualization and Genetic Algorithms in Minimax Theory for Nonlinear Functionals

In this paper, evolution and visualization of the existence of saddle points of nonlinear functionals or multi-variable functions in finite dimensional spaces are presented. New algorithms are developed based on the mountain pass lemma and link thery in nonlinear analysis. Further more, a simple comparison of the steepest descent algorithm and the genetic algorithm is given. The process of the saddle point finding is visualised in an inteactive graphical interface.     

色彩量化的模糊粒子群优化技术

把粒子群算法应用到色彩量化中,结合已有的模糊C均值聚类量化方法,提出了一种基于粒子群优化的色彩量化算法。模糊C均值聚类量化算法是一种局部搜索算法,对初始值较为敏感,容易陷入局部极小值而不能得到全局最优解;PSO算法是一种基于群体的具有全局寻优能力的优化方法。将模糊C均值聚类量化算法和PSO算法结合起来,把模糊C均值聚类量化算法的聚类准则函数作为PSO算法中的粒子适应度函数。仿真实验表明,新算法在均方根误差和峰值信噪比评判准则下能够得到最优的量化结果。     

Fuzzy c-means clustering with non local spatial information for noisy image segmentation

As an effective image segmentation method, the standard fuzzy c-means (FCM) clustering algorithm is very sensitive to noise in images. Several modified FCM algorithms, using local spatial information, can overcome this problem to some degree. However, when the noise level in the image is high, these algorithms still cannot obtain satisfactory segmentation performance. In this paper, we introduce a non local spatial constraint term into the objective function of FCM and propose a fuzzy cmeans clustering algorithm with non local spatial information (FCM_NLS). FCM_NLS can deal more effectively with the image noise and preserve geometrical edges in the image. Performance evaluation experiments on synthetic and real images, especially magnetic resonance (MR) images, show that FCM_NLS is more robust than both the standard FCM and the modified FCM algorithms using local spatial information for noisy image segmentation.     

New fuzzy C-means clustering method based on feature-weight and cluster-weight learning

Among fuzzy clustering methods, fuzzy c-means (FCM) is the most recognized algorithm. In this algorithm, it is assumed that all the features are of equal importance. In real applications, however, the importance of the features are different and there exist some features that are more important than the others. These important features should basically have more effects than the other features in the forming of optimal clusters. The basic FCM algorithm does not support this idea. Also, the FCM algorithm suffers from another problem; the algorithm is very sensitive to initialization, whereas a bad initialization leads to a poor local optima. Some improved versions of FCM have been proposed in the literature, each of which has somehow mitigated the first problem or the second one. In this paper, motivated by these weaknesses of the FCM, the goal is to solve the two problems at the same time. In doing so, an automatic local feature weighting scheme is proposed to properly weight the features of each clusters. And, a cluster weighting process is performed to mitigate the initialization sensitivity of the FCM. Feature weighting and cluster weighting are performed simultaneously and automatically during the clustering process resulting in high quality clusters, regardless of the initial centers. Extensive experiments conducted on a synthetic dataset and 16 real world datasets indicate that the proposed algorithm outperforms the state-of-the-arts algorithms. The convergence proof of the proposed algorithm is also provided.     

基于全局优化搜索算法的图像分割研究

基于聚类的图像分割算法中,由于模糊C-均值算法需要初始化,并且目标函数存在许多局部极小点,如果初始化落在目标函数的局部极小点附近,就会造成算法收敛到局部极小.为了解决此问题,采用全局优化搜索算法,提出了将全局优化搜索技术引入进来对模糊C-均值算法加以改进,分析了在不同初始条件下,对许多样本的聚类分析时,全局优化搜索算法比传统的模糊C-均值聚类算法更加有效,通过仿真实验验证并对算法性能进行理论分析.     

一种多空间FCM算法

FCM是经典的聚类算法,广泛地应用于模式识别、数据挖掘等领域。FCM算法是一种梯度下降优化算法,对初始解敏感并且容易获得局部最优解。空间平滑能够避免启发式局部搜索算法掉入局部最优解。采用空间平滑策略构造一系列光滑程度不同的搜索空间,在不同的搜索空间中执行FCM算法,并利用前层搜索空间的聚类结果来引导本层搜索空间的聚类。FCMS（FCM based on multi-Space）能够跳过局部最优解的“陷阱”,增大获得全局最优解的概率,达到提高聚类质量的目的。给出了等距法空间平滑策略,并通过实验对比了FCMS算法与FCM算法的聚类质量。实验结果表明,空间平滑对FCM算法非常有效。     

Research on 3D medial axis transform via the saddle point programming method

The present paper investigates the 3D medial axis transform of objects bounded by freeform surfaces via the saddle point programming method, a mathematical programming approach used to identify the saddle points of a function. After exploring the local geometry and saddle point property of 3D medial axis transform, the mathematical programming method is employed to construct the saddle point programming models. Based on the optimality conditions that the optimal solutions should satisfy, a generic algorithm for computing various medial axis points is developed. In order to identify the junction points and localize the problem, the boundary and the skeletal curves are divided into skeletal segments, and it is proved to be efficient and accurate by numerical examples.     

基于改进遗传算法的加权模糊C均值聚类算法

针对模糊C均值(FCM)聚类算法具有初始聚类中心敏感和容易陷入局部最优的问题,提出了一种基于改进遗传算法(GA)的加权模糊c均值聚类算法,采用高斯变异算子,提高了遗传算法在每个峰值附近的局部搜索能力,用基于复相关系数的加权欧式距离代替欧式距离,改进了FCM算法的聚类目标函数.用改进的算法对国际标准测试数据Iris进行测试,实验结果表明改进后的算法具有更好的稳定性和健壮性,提高了聚类的效果.     

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。     

改进的粒子群优化模糊C均值聚类算法

针对传统模糊C均值聚类算法（FCM）存在对初值敏感和易陷入局部收敛的缺陷,利用改进的粒子群算法对FCM进行优化,提出一种新的模糊C均值聚类算法Improved PSOFCM,并建立基于熵的聚类有效性函数,对聚类算法的性能进行客观评价。数据集实验表明,Improved PSOFCM算法不仅能克服传统FCM算法的不足,而且在聚类正确率和有效性上也优于基于粒子群与基于遗传优化的FCM算法。     

Local convergence of the fuzzy c-Means algorithms

Much understanding has recently been gained concerning global convergence properties of the fuzzy c-Means (FCM) family of clustering algorithms. These global convergence properties, which hold for all iteration sequences, guarantee that every FCM iteration sequence converges, at least along a subsequence, to a stationary point of an FCM objective function. In this paper we prove a local convergence property, that is, a property pertaining to iteration sequences started near a solution. Specifically, a simple result is proved which shows that whenever an FCM algorithm is started sufficiently near a minimizer of the corresponding objective function, then the iteration sequence must converge to that particular minimizer. The result guarantees that once captured by the local neighborhood of a minimizer, the succeeding iterate sequence will not escape—thus, infinite oscillation of such a sequence cannot occur. The rate of convergence of the sequence to such a point is also discussed.     

基于粒子群优化的模糊C-均值聚类改进算法

针对模糊C-均值聚类算法(FCM)存在易陷入局部优化的问题,将粒子群优化算法(PSO)和模糊C-均值聚类算法FCM相结合,提出了一种新的模糊聚类算法PSO-FCM.该算法使用PSO算法来代替FCM的迭代过程以实现模糊聚类,具有了很强的全局搜索能力,从而不用再为得到好的聚类效果而反复选择初值.仿真实验结果表明,提出的模糊聚类算法提高了FCM的搜索能力,具有更好的稳定性和健壮性,优化能力增强,提高了聚类的效率和效果.     

基于简化随机场模型的高分辨率遥感影像分割方法

提出了一种灰度分割的基础上添加辅助的纹理分割的基于简化随机场模型的遥感影像目标分割方法,即用常用的描述局部图像特点的特征代替MRF中定义的特征,将这些特征组合成特征向量进行模糊C均值聚类完成分割。给出了算法流程和实验结果,并将该结果与基于高斯马尔可夫随机场模型法分割的结果进行比较,实验结果表明简化随机场模型法在保证一定的分割精度的情况下,分割速度明显快于高斯马尔可夫随机场模型法。     

基于PSO的模糊C-均值聚类算法的图像分割

根据粒子群优化算法（PSO）强大的全局搜索能力,提出了用PSO算法优化模糊C均值聚类（FCM）的聚类中心的方法,有效地避免了传统的FCM对初始值及噪声数据敏感,容易陷入局部最优的缺点,同时图像分割的效果也得到了提高,性能也比传统的FCM方法更加稳定。实验结果反映了该方法的有效性。     

A Generalized Fuzzy Clustering Regularization Model With Optimality Tests and Model Complexity Analysis

In this paper, we propose a generalized fuzzy clustering regularization (GFCR) model and then study its theoretical properties. GFCR unifies several fuzzy clustering algorithms, such as fuzzy c-means (FCM), maximum entropy clustering (MEC), fuzzy clustering based on Fermi-Dirac entropy, and fuzzy bidirectional associative clustering network, etc. The proposed GFCR becomes an alternative model of the generalized FCM (GFCM) that was recently proposed by Yu and Yang. To advance theoretical study, we have the following three considerations. 1) We give an optimality test to monitor if GFCR converges to a local minimum. 2) We relate the GFCR optimality tests to Occam's razor principle, and then analyze the model complexity for fuzzy clustering algorithms. 3) We offer a general theoretical method to evaluate the performance of fuzzy clustering algorithms. Finally, some numerical experiments are used to demonstrate the validity of our theoretical results and complexity analysis.