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].     

Complex fuzzy c-means algorithm

In this paper a new clustering algorithm is presented: A complex-based Fuzzy c-means (CFCM) algorithm. While the Fuzzy c-means uses a real vector as a prototype characterizing a cluster, the CFCM??s prototype is generalized to be a complex vector (complex center). CFCM uses a new real distance measure which is derived from a complex one. CFCM??s formulas for the fuzzy membership are derived. These formulas are extended to derive the complex Gustafson?CKessel algorithm (CGK). Cluster validity measures are used to assess the goodness of the partitions obtained by the complex centers compared those obtained by the real centers. The validity measures used in this paper are the Partition Coefficient, Classification Entropy, Partition Index, Separation Index, Xie and Beni??s Index, Dunn??s Index. It is shown in this paper that the CFCM give better partitions of the data than the FCM and the GK algorithms. It is also shown that the CGK algorithm outperforms the CFCM but at the expense of much higher computational complexity.     

Analysis and efficient implementation of a linguistic fuzzy c-means

The paper is concerned with a linguistic fuzzy c-means (FCM) algorithm with vectors of fuzzy numbers as inputs. This algorithm is based on the extension principle and the decomposition theorem. It turns out that using the extension principle to extend the capability of the standard membership update equation to deal with a linguistic vector has a huge computational complexity. In order to cope with this problem, an efficient method based on fuzzy arithmetic and optimization has been developed and analyzed. We also carefully examine and prove that the algorithm behaves in a way similar to the FCM in the degenerate linguistic case. Synthetic data sets and the iris data set have been used to illustrate the behavior of this linguistic version of the FCM.     

具有数据容错能力的模糊C均值聚类算法

提出基于模糊c均值聚类算法的两个新算法.设置每个数据隶属度的误差阈值,规定每个数据的隶属度误差不能超过给出的误差阈值.使用该类算法可以对有误差的数据进行模糊聚类.先利用隶属度矩阵的误差范围建立新的拉格朗日函数,再使用Kuhn-Tucker条件计算该函数,并通过一组实验来证明这类算法的正确性和有效性.     

Suppressed fuzzy c-means clustering algorithm

Based on the defect of rival checked fuzzy c-means clustering algorithm, a new algorithm: suppressed fuzzy c-means clustering algorithm is proposed. The new algorithm overcomes the shortcomings of the original algorithm, establishes more natural and more reasonable relationships between hard c-means clustering algorithm and fuzzy c-means clustering algorithm.     

A multivariate fuzzy c-means method

Fuzzy c-means (FCMs) is an important and popular unsupervised partitioning algorithm used in several application domains such as pattern recognition, machine learning and data mining. Although the FCM has shown good performance in detecting clusters, the membership values for each individual computed to each of the clusters cannot indicate how well the individuals are classified. In this paper, a new approach to handle the memberships based on the inherent information in each feature is presented. The algorithm produces a membership matrix for each individual, the membership values are between zero and one and measure the similarity of this individual to the center of each cluster according to each feature. These values can change at each iteration of the algorithm and they are different from one feature to another and from one cluster to another in order to increase the performance of the fuzzy c-means clustering algorithm. To obtain a fuzzy partition by class of the input data set, a way to compute the class membership values is also proposed in this work. Experiments with synthetic and real data sets show that the proposed approach produces good quality of clustering.     

On cluster validity for the fuzzy c-means model

Many functionals have been proposed for validation of partitions of object data produced by the fuzzy c-means (FCM) clustering algorithm. We examine the role a subtle but important parameter-the weighting exponent m of the FCM model-plays in determining the validity of FCM partitions. The functionals considered are the partition coefficient and entropy indexes of Bezdek, the Xie-Beni (1991), and extended Xie-Beni indexes, and the Fukuyama-Sugeno index (1989). Limit analysis indicates, and numerical experiments confirm, that the Fukuyama-Sugeno index is sensitive to both high and low values of m and may be unreliable because of this. Of the indexes tested, the Xie-Beni index provided the best response over a wide range of choices for the number of clusters, (2-10), and for m from 1.01-7. Finally, our calculations suggest that the best choice for m is probably in the interval [1.5, 2.5], whose mean and midpoint, m=2, have often been the preferred choice for many users of FCM     

An automatic fuzzy c-means algorithm for image segmentation

Fuzzy c-means (FCM) algorithm is one of the most popular methods for image segmentation. However, the standard FCM algorithm must be estimated by expertise users to determine the cluster number. So, we propose an automatic fuzzy clustering algorithm (AFCM) for automatically grouping the pixels of an image into different homogeneous regions when the number of clusters is not known beforehand. In order to get better segmentation quality, this paper presents an algorithm based on AFCM algorithm, called automatic modified fuzzy c-means cluster segmentation algorithm (AMFCM). AMFCM algorithm incorporates spatial information into the membership function for clustering. The spatial function is the weighted summation of the membership function in the neighborhood of each pixel under consideration. Experimental results show that AMFCM algorithm not only can spontaneously estimate the appropriate number of clusters but also can get better segmentation quality.     

创意FCM算法

针对现有模糊聚类方法仅仅是对已有数据点的聚类的不足,提出了在已有数据集的基础上找到新的一类集群的聚类方法 CFCM。该算法在FCM算法的基础上,通过引入观测点P作为聚类的先验知识,来大致确定未知集群的聚类中心,定义了权重系数λ来限定观测点对新的一类聚类中心形成的影响程度。人造数据集和UCI真实数据集的实验结果表明,该算法不仅对已知数据点有较好的聚类效果,并且可以在观测点P的作用下在特定区域创造出新的一类无已知数据点的集群中心点的大致位置,因而在实际中有潜在应用价值。     

Effective fuzzy c-means clustering algorithms for data clustering problems

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

A size-insensitive integrity-based fuzzy c-means method for data clustering

Fuzzy c-means (FCM) is one of the most popular techniques for data clustering. Since FCM tends to balance the number of data points in each cluster, centers of smaller clusters are forced to drift to larger adjacent clusters. For datasets with unbalanced clusters, the partition results of FCM are usually unsatisfactory. Cluster size insensitive FCM (csiFCM) dealt with “cluster-size sensitivity” problem by dynamically adjusting the condition value for the membership of each data point based on cluster size after the defuzzification step in each iterative cycle. However, the performance of csiFCM is sensitive to both the initial positions of cluster centers and the “distance” between adjacent clusters. In this paper, we present a cluster size insensitive integrity-based FCM method called siibFCM to improve the deficiency of csiFCM. The siibFCM method can determine the membership contribution of every data point to each individual cluster by considering cluster's integrity, which is a combination of compactness and purity. “Compactness” represents the distribution of data points within a cluster while “purity” represents how far a cluster is away from its adjacent cluster. We tested our siibFCM method and compared with the traditional FCM and csiFCM methods extensively by using artificially generated datasets with different shapes and data distributions, synthetic images, real images, and Escherichia coli dataset. Experimental results showed that the performance of siibFCM is superior to both traditional FCM and csiFCM in terms of the tolerance for “distance” between adjacent clusters and the flexibility of selecting initial cluster centers when dealing with datasets with unbalanced clusters.     

Measuring the congruence of fuzzy partitions in fuzzy c-means clustering

We propose an internal cluster validity index for a fuzzy c-means algorithm which combines a mathematical model for the fuzzy c-partition and a heuristic search for the number of clusters in the data. Our index resorts to information theoretic principles, and aims to assess the congruence between such a model and the data that have been observed. The optimal cluster solution represents a trade-off between discrepancy and the complexity of the underlying fuzzy c-partition. We begin by testing the effectiveness of the proposed index using two sets of synthetic data, one comprising a well-defined cluster structure and the other containing only noise. Then we use datasets arising from real life problems. Our results are compared to those provided by several available indices and their goodness is judged by an external measure of similarity. We find substantial evidence supporting our index as a credible alternative to the cluster validation problem, especially when it concerns structureless data.     

Soil clustering by fuzzy c-means algorithm

In this study, hard k-means and fuzzy c-means algorithms are utilized for the classification of fine grained soils in terms of shear strength and plasticity index parameters. In order to collect data, several laboratory tests are performed on 120 undisturbed soil samples, which are obtained from Antalya region. Additionally, for the evaluation of the generalization ability of clustering analysis, 20 fine grained soil samples collected from the other regions of Turkey are also classified using the same clustering algorithms. Fuzzy c-means algorithm exhibited better clustering performance over hard k-means classifier. As expected, clustering analysis produced worse outcomes for soils collected from different regions than those of obtained from a specific region. In addition to its precise classification ability, fuzzy c-means approach is also capable of handling the uncertainty existing in soil parameters. As a result, fuzzy c-means clustering can be successfully applied to classify regional fine grained soils on the basis of shear strength and plasticity index parameters.     

Analytically tractable case of fuzzy c-means clustering

In this paper, we offer a simple and accurate clustering algorithm which was derived as a closed-form analytical solution to a cluster fit function minimization problem. As a result, the algorithm finds the global minimum of the fit function, and combines exceptional efficiency with optimal clustering results.     

自抗扰控制器的阶次与参数的选取

本文就选择不同阶次的自抗扰控制器时,对系统的控制参数选取进行了研究.结果表明:线性时不变系统的线性自抗扰控制,可等效为一个复合控制系统,其等效反馈补偿器为一超前校正单元串联一积分器;其等效前置滤波器为一滞后校正单元串联一微分器.观测器带宽和控制器带宽的比值,决定着反馈补偿器的最大相位超前角,而频带则决定着最大相位超前角的发生位置.同时,随着自抗扰控制器阶次的增加,补偿器的最大超前校正角也增加.通过对开环系统的频域分析,本文给出了利用该补偿器的频域特性进行自抗扰控制器参数设计的一般步骤,可大幅度减少工程师的反复试验过程,方便工程师应用.     

On the use of the weighted fuzzy c-means in fuzzy modeling

This paper proposes a fuzzy clustering-based algorithm for fuzzy modeling. The algorithm incorporates unsupervised learning with an iterative process into a framework, which is based on the use of the weighted fuzzy c-means. In the first step, the learning vector quantization (LVQ) algorithm is exploited as a data pre-processor unit to group the training data into a number of clusters. Since different clusters may contain different number of objects, the centers of these clusters are assigned weight factors, the values of which are calculated by the respective cluster cardinalities. These centers accompanied with their weights are considered to be a new data set, which is further elaborated by an iterative process. This process consists of applying in sequence the weighted fuzzy c-means and the back-propagation algorithm. The application of the weighted fuzzy c-means ensures that the contribution of each cluster center to the final fuzzy partition is determined by its cardinality, meaning that the real data structure can be easier discovered. The algorithm is successfully applied to three test cases, where the produced fuzzy models prove to be very accurate as well as compact in size.     

Hyperkernel-based intuitionistic fuzzy c-means for denoising color archival document images

International Journal on Document Analysis and Recognition (IJDAR) - In this article, we have addressed the problem of denoising and enhancement of color archival handwritten document images by...     

基于模糊C均值的图像分割

图像分割的质量直接影响后期的图像分析、识别和解释的质量。本文主要研究了基于模糊c均值算法的图像分割,它通过优化目标函数得到每个样本点对所有类中心的隶属度,从而决定样本点的类属以达到自动对样本数据进行分类的目的。实验结果表明文中用到的图像分割算法对图像分割的效果均优于对比算法的分割效果。     

Novel modified fuzzy c-means algorithm with applications

Fuzzy c-means (FCM) clustering algorithm has been widely used in many medical image segmentations. However, the conventionally standard FCM algorithm is noise sensitive because of not taking into account the spatial information. To overcome the above problem, a novel modified FCM algorithm (called FCM–AWA later) for image segmentation is presented in this paper. The algorithm is realized by modifying the objective function in the conventional FCM algorithm, i.e., by incorporating the spatial neighborhood information into the standard FCM algorithm. An adaptive weighted averaging (AWA) filter is given to indicate the spatial influence of the neighboring pixels on the central pixel. The parameters (weighting coefficients) of control template (neighboring widow) are automatically determined in the implementation of the weighted averaging image by a predefined nonlinear function. The presented algorithm is applied to both artificial synthesized image and real image. Furthermore, the quantifications of dental plaque using proposed algorithm-based segmentation were conducted. Experimental results show that the presented algorithm performs more robust to noise than the standard FCM algorithm and another FCM algorithm (proposed by Ahmed) do. Furthermore, the results of dental plaque quantification using proposed method indicate the FCM–AWA provides a quantitative, objective and efficient analysis of dental plaque, and possesses great promise.     

An enhanced fuzzy c-means algorithm for audio segmentation and classification

Automated audio segmentation and classification play important roles in multimedia content analysis. In this paper, we propose an enhanced approach, called the correlation intensive fuzzy c-means (CIFCM) algorithm, to audio segmentation and classification that is based on audio content analysis. While conventional methods work by considering the attributes of only the current frame or segment, the proposed CIFCM algorithm efficiently incorporates the influence of neighboring frames or segments in the audio stream. With this method, audio-cuts can be detected efficiently even when the signal contains audio effects such as fade-in, fade-out, and cross-fade. A number of audio features are analyzed in this paper to explore the differences between various types of audio data. The proposed CIFCM algorithm works by detecting the boundaries between different kinds of sounds and classifying them into clusters such as silence, speech, music, speech with music, and speech with noise. Our experimental results indicate that the proposed method outperforms the state-of-the-art FCM approach in terms of audio segmentation and classification.