Fuzzy C-means based clustering for linearly and nonlinearly separable data

In this paper we present a new distance metric that incorporates the distance variation in a cluster to regularize the distance between a data point and the cluster centroid. It is then applied to the conventional fuzzy C-means (FCM) clustering in data space and the kernel fuzzy C-means (KFCM) clustering in a high-dimensional feature space. Experiments on two-dimensional artificial data sets, real data sets from public data libraries and color image segmentation have shown that the proposed FCM and KFCM with the new distance metric generally have better performance on non-spherically distributed data with uneven density for linear and nonlinear separation.     

Fuzzy and possibilistic shell clustering algorithms and theirapplication to boundary detection and surface approximation. I

Traditionally, prototype-based fuzzy clustering algorithms such as the Fuzzy C Means (FCM) algorithm have been used to find “compact” or “filled” clusters. Recently, there have been attempts to generalize such algorithms to the case of hollow or “shell-like” clusters, i.e., clusters that lie in subspaces of feature space. The shell clustering approach provides a powerful means to solve the hitherto unsolved problem of simultaneously fitting multiple curves/surfaces to unsegmented, scattered and sparse data. In this paper, we present several fuzzy and possibilistic algorithms to detect linear and quadric shell clusters. We also introduce generalizations of these algorithms in which the prototypes represent sets of higher-order polynomial functions. The suggested algorithms provide a good trade-off between computational complexity and performance, since the objective function used in these algorithms is the sum of squared distances, and the clustering is sensitive to noise and outliers. We show that by using a possibilistic approach to clustering, one can make the proposed algorithms robust     

基于PSO_KFCM的医学图像分割

在核模糊聚类算法(KFCM)的基础上,提出了一种新的PSO KFCM聚类算法.新算法利用高斯核函数,把输入空间的样本映射到高维特征空间,利用微粒群算法的全局搜索、快速收敛的特点,代替KFCM算法逐次迭代的过程,在特征空间中进行聚类,克服了KFCM对初始值和噪声数据敏感、易陷入局部最优的缺点.通过对医学图像进行分割,仿真实验结果表明,新算法在性能上比KFCM聚类算法有较大改进,具有更好的聚类效果,且算法能够很快地收敛.     

核空间中智模糊聚类及图像分割应用

目的 为了更有效地提高中智模糊C-均值聚类对非凸不规则数据的聚类性能和噪声污染图像的分割效果,提出了核空间中智模糊均值聚类算法。方法 引入核函数概念。利用满足Mercer条件的非线性问题,用非线性变换把低维空间线性不可分的输入模式空间映射到一个先行可分的高维特征空间进行中智模糊聚类分割。结果 通过对大量图像添加不同的加性和乘性噪声进行分割测试获得的核空间中智模糊聚类算法提高了现有算法的对含噪声聚类的鲁棒性和分类性能。峰值信噪比至少提高0.8 dB。结论 本文算法具有显著的分割效果和良好的鲁棒性,并适应于医学,遥感图像处理需要。     

Kernel approach to possibilistic C‐means clustering

Kernel approaches can improve the performance of conventional clustering or classification algorithms for complex distributed data. This is achieved by using a kernel function, which is defined as the inner product of two values obtained by a transformation function. In doing so, this allows algorithms to operate in a higher dimensional space (i.e., more degrees of freedom for data to be meaningfully partitioned) without having to compute the transformation. As a result, the fuzzy kernel C‐means (FKCM) algorithm, which uses a distance measure between patterns and cluster prototypes based on a kernel function, can obtain more desirable clustering results than fuzzy C‐means (FCM) for not only spherical data but also nonspherical data. However, it can still be sensitive to noise as in the FCM algorithm. In this paper, to improve the drawback of FKCM, we propose a kernel possibilistic C‐means (KPCM) algorithm that applies the kernel approach to the possibilistic C‐means (PCM) algorithm. The method includes a variance updating method for Gaussian kernels for each clustering iteration. Several experimental results show that the proposed algorithm can outperform other algorithms for general data with additive noise. © 2009 Wiley Periodicals, Inc.     

一种快速鲁棒核空间图形模糊聚类分割算法

针对现有鲁棒图形模糊聚类算法难以满足强噪声干扰下大幅面图像快速分割的需要，提出一种快速鲁棒核空间图形模糊聚类分割算法。该算法将欧氏空间样本通过核函数映射至高维空间；采用待分割图像中像素邻域的灰度和空间等信息构建线性加权滤波图像，对其进行鲁棒核空间图形模糊聚类；并引入当前聚类像素与其邻域像素均值所对应的二维直方图信息，获得鲁棒核空间图形模糊聚类快速迭代表达式。对大幅面图像添加高斯和椒盐噪声进行分割测试，实验结果表明:本文算法相比基于图形模糊聚类等分割算法的分割性能、抗噪鲁棒性和实时性有了显著提高。     

核空间广义均衡模糊C-均值聚类算法

目的 针对现有广义均衡模糊C-均值聚类不收敛问题,提出一种改进广义均衡模糊聚类新算法,并将其推广至再生希尔伯特核空间以便提高该类算法的普适性。方法 在现有广义均衡模糊C-均值聚类目标函数的基础上,利用Schweizer T范数极限表达式的性质构造了新的广义均衡模糊C-均值聚类最优化目标函数,然后采用拉格朗日乘子法获取其迭代求解所对应的隶属度和聚类中心表达式,同时对其聚类中心迭代表达式进行修改并得到一类聚类性能显著改善的修正聚类算法;最后利用非线性函数将数据样本映射至高维特征空间获得核空间广义均衡模糊聚类算法。结果 对Iris标准文本数据聚类和灰度图像分割测试表明,提出的改进广义均衡模模糊聚类新算法及其修正算法具有良好的分类性能,核空间广义均衡模糊聚类算法对比现有融入类间距离的改进模糊C-均值聚类(FCS)算法和改进再生核空间的模糊局部C-均值聚类(KFLICM)算法能将图像分割的误分率降低10%30%。结论 本文算法克服了现有广义均衡模糊C-均值聚类算法的缺陷,同时改善了聚类性能,适合复杂数据聚类分析的需要。     

一种基于改进混合蛙跳的KFCM算法

针对核模糊C-均值（KFCM）聚类算法存在易陷入局部极小值,对初始值敏感的缺点。将混合蛙跳算法（shuffled frog leaping algorithm,SFLA）用于KFCM中,但在聚类数较大和维数较高时,聚类效果不理想,为此提出将自适应惯性权重引入混合蛙跳算法的更新策略中,再用改进后的混合蛙跳算法求得最优解作为KFCM算法的初始聚类中心,利用KFCM算法优化初始聚类中心,求得全局最优解,从而有效克服了KFCM算法的缺点。人造数据和经典数据集的实验结果表明,新算法与KFCM和FCM聚类算法相比,寻优能力更强,迭代次数更少,聚类效果更好。     

Attribute weighted mercer kernel based fuzzy clustering algorithm for general non-spherical datasets

Clustering analysis is an important topic in artificial intelligence, data mining and pattern recognition research. Conventional clustering algorithms, for instance, the famous Fuzzy C-means clustering algorithm (FCM), assume that all the attributes are equally relevant to all the clusters. However in most domains, especially for high-dimensional dataset, some attributes are irrelevant, and some relevant ones are less important than others with respect to a specific class. In this paper, such imbalances between the attributes are considered and a new weighted fuzzy kernel-clustering algorithm (WFKCA) is presented. WFKCA performs clustering in a kernel feature space mapped by mercer kernels. Compared with the conventional hard kernel-clustering algorithm, WFKCA can yield the meaningful prototypes (cluster centers) of the clusters. Numerical convergence properties of WFKCA are also discussed. For in-depth studies, WFKCA is extended to WFKCA2, which has been demonstrated as a useful tool for clustering incomplete data. Numerical examples demonstrate the effectiveness of the new WFKCA algorithm     

密度敏感模糊核最大熵聚类算法EI北大核心CSCD

提出一种密度敏感模糊核最大熵聚类算法.该算法首先通过核函数将原始非线性非高斯的数据集转化为核空间数据集,然后利用核函数的相似性抵消不属于该聚类的样本数据在聚类过程中对聚类中心求解的干扰,消除正则化系数对聚类结果的影响,进而抑制传统最大熵聚类算法的趋同性.最后通过引入相对密度项,解决因样本数据在特征空间的分布差异而导致的聚类中心求解偏差问题,从而提高聚类结果的准确性.实验部分,本文讨论了算法参数间的关系以及对聚类结果的影响.通过与传统模糊C均值聚类算法、核模糊C均值聚类算法、最大熵聚类算法、最大熵规范化权重核模糊C均值聚类算法以及其他两种改进最大熵聚类算法的聚类结果进行对比分析,结果表明本文提出的密度敏感模糊核最大熵聚类算法的聚类性能明显优于其他算法.     

基于成对约束的动态加权半监督模糊核聚类

针对样本间的不均衡性,提出一种基于成对约束的动态加权半监督模糊核聚类算法。在传统模糊聚类算法中加入半监督学习机制,通过Mercer核将原数据空间映射到特征空间,为特征空间中的每个向量分配一个动态权值,由此得到新的目标函数,并结合一种简单的核参数选择方法实现数据分类。理论分析和实验结果表明,与模糊核聚类算法及成对约束的竞争群算法相比,该算法具有更好的聚类效果。     

Fast global kernel fuzzy c-means clustering algorithm for consonant/vowel segmentation of speech signal

We propose a novel clustering algorithm using fast global kernel fuzzy c-means-F（FGKFCM-F）, where F refers to kernelized feature space. This algorithm proceeds in an incremental way to derive the near-optimal solution by solving all intermediate problems using kernel-based fuzzy c-means-F（KFCM-F） as a local search procedure. Due to the incremental nature and the nonlinear properties inherited from KFCM-F, this algorithm overcomes the two shortcomings of fuzzy c-means（FCM）： sen- sitivity to initialization and inability to use nonlinear separable data. An accelerating scheme is developed to reduce the compu-tational complexity without significantly affecting the solution quality. Experiments are carried out to test the proposed algorithm on a nonlinear artificial dataset and a real-world dataset of speech signals for consonant/vowel segmentation. Simulation results demonstrate the effectiveness of the proposed algorithm in improving clustering performance on both types of datasets.     

融合块对角约束的鲁棒低秩多核聚类

针对现有的多核学习(multiple kernel learning, MKL)子空间聚类方法忽略噪声和特征空间中数据的低秩结构问题,提出一种新的鲁棒多核子空间聚类方法(low-rank robust multiple kernel clustering, LRMKC),该方法结合块对角表示(block diagonal representation, BDR)与低秩共识核(low-rank consensus kernel, LRCK)学习,可以更好地挖掘数据的潜在结构.为了学习最优共识核,设计一种基于混合相关熵度量(mixture correntropy induced metric, MCIM)的自动加权策略,其不仅为每个核设置最优权重,而且通过抑制噪声提高模型的鲁棒性;为了探索特征空间数据的低秩结构,提出一种非凸低秩共识核学习方法;考虑到亲和度矩阵的块对角性质,对系数矩阵应用块对角约束.LRMKC将MKL、LRCK与BDR巧妙融合,以迭代提高各种方法的效率,最终形成一个处理非线性结构数据的全局优化方法.与最先进的MKL子空间聚类方法相比,通过在图像和文本数据集上的大量实验验证了LRMKC的优越性.     

Semi-supervised fuzzy clustering: A kernel-based approach

Semi-supervised clustering algorithms aim to improve the clustering accuracy under the supervisions of a limited amount of labeled data. Since kernel-based approaches, such as kernel-based fuzzy c-means algorithm (KFCM), have been successfully used in classification and clustering problems, in this paper, we propose a novel semi-supervised clustering approach using the kernel-based method based on KFCM and denote it the semi-supervised kernel fuzzy c-mean algorithm (SSKFCM). The objective function of SSKFCM is defined by adding classification errors of both the labeled and the unlabeled data, and its global optimum has been obtained through repeatedly updating the fuzzy memberships and the optimized kernel parameter. The objective function may have more than one local optimum, so we employ a function transformation technique to reformulate the objective function after a local minimum has been obtained, and select the best optimum as the solution to the objective function. Experimental results on both the artificial and several real data sets show SSKFCM performs better than its conventional counterparts and it achieves the best accurate clustering results when the parameter is optimized.     

Identification of control chart patterns using wavelet filtering and robust fuzzy clustering

This paper proposes a hybrid framework composed of filtering module and clustering module to identify six common types of control chart patterns, including natural pattern, cyclic pattern, upward shift, downward shift, upward trend, and downward trend. In particular, a multi-scale wavelet filter is designed for denoising and its performance is compared to single-scale filters, including mean filter and exponentially weighted moving average (EWMA) filter. Moreover, three fuzzy clustering algorithms, based on fuzzy c means (FCM), entropy fuzzy c means (EFCM) and kernel fuzzy c means (KFCM), are adopted to compare their performance of pattern classification. Experimental results demonstrate that the excellent performance of EFCM and KFCM against outliers, especially in the case of high noise level embedded in the input data. Therefore, a hybrid framework combining wavelet filter with robust fuzzy clustering is suggested and proposed in this paper. Compared to neural network based approaches, the proposed method provides a promising way for the on-line recognition of control chart patterns because of its efficient computation and robustness against outliers.     

基于改进核模糊C均值类间极大化聚类算法

传统的核聚类仅考虑了类内元素的关系而忽略了类间的关系,对边界模糊或边界存在噪声点的数据集进行聚类分析时,会造成边界点的误分问题。为解决上述问题,在核模糊C均值（KFCM）聚类算法的基础上提出了一种基于改进核模糊C均值类间极大化聚类（MKFCM）算法。该算法考虑了类内元素和类间元素的联系,引入了高维特征空间的类间极大惩罚项和调控因子,拉大类中心间的距离,使得边界处的样本得到了较好的划分。在各模拟数据集的实验中,该算法在类中心的偏移距离相对其他算法均有明显降低。在人造高斯数据集的实验中,该算法的精度（ACC）、归一化互信息（NMI）、芮氏指标（RI）指标分别提升至0.9132,0.7575,0.9138。     

An axiomatic approach to soft learning vector quantization andclustering

This paper presents an axiomatic approach to soft learning vector quantization (LVQ) and clustering based on reformulation. The reformulation of the fuzzy c-means (FCM) algorithm provides the basis for reformulating entropy-constrained fuzzy clustering (ECFC) algorithms. According to the proposed approach, the development of specific algorithms reduces to the selection of a generator function. Linear generator functions lead to the FCM and fuzzy learning vector quantization algorithms while exponential generator functions lead to ECFC and entropy-constrained learning vector quantization algorithms. The reformulation of LVQ and clustering algorithms also provides the basis for developing uncertainty measures that can identify feature vectors equidistant from all prototypes. These measures are employed by a procedure developed to make soft LVQ and clustering algorithms capable of identifying outliers in the data set. This procedure is evaluated by testing the algorithms generated by linear and exponential generator functions on speech data.     

核空间直觉模糊局部C-均值聚类分割算法研究

针对现有直觉模糊C-均值聚类仅适合呈团状数据的不足,采用非线性函数将数据样本从欧式空间映射至再生希尔伯特高维特征空间,得到核空间直觉模糊聚类算法;同时考虑相邻像素的相互影响,将邻域像素融入核空间直觉模糊聚类的最优化目标函数中,经数学推导便得到嵌入像素局部信息的核空间直觉模糊聚类分割算法。图像分割测试结果表明,核直觉模糊C-均值聚类分割法相比现有直觉模糊C-均值聚类分割法能获得更满意的分割效果;同时,嵌入局部信息的核直觉模糊C-均值聚类分割法表现出良好的抗噪鲁棒性。     

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

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

A fast and automated segmentation method for detection of masses using folded kernel based fuzzy c-means clustering algorithm

Present study proposes a fast, accurate and automated segmentation approach of mammographic images using kernel based fuzzy c-means (FCM) clustering technique. This approach exploits the significant regional features of mammograms which address the properties of different breast densities. The proposed segmentation approach captures those regional features using appropriate kernel and hence apply fuzzy clustering technique for segmenting the masses. This study also introduces kernel based FCM (KFCM) approach in a folded way to process a combination of significant features simultaneously. Suitable choice of kernel size also assists to collect all possible variations of regional features with minimum blocking effect in the output results. Performances of the proposed methodology are analyzed qualitatively and quantitatively in compare to other clustering-based segmentation techniques. Since the proposed approach is able to resolve uncertain and imprecise characteristics of mammograms, it performs superior to other techniques. Convergence time of the proposed method is also assessed and compared with other conventional clustering techniques. Kernel based approach of the proposed segmentation technique reduces the number of data points for clustering and hence convergence speed improves over the conventional algorithms. This study also shows a variation of convergence speed of the proposed segmentation method with different image sizes.