ECM: An evidential version of the fuzzy c-means algorithm

A new clustering method for object data, called ECM (evidential c-means) is introduced, in the theoretical framework of belief functions. It is based on the concept of credal partition, extending those of hard, fuzzy, and possibilistic ones. To derive such a structure, a suitable objective function is minimized using an FCM-like algorithm. A validity index allowing the determination of the proper number of clusters is also proposed. Experiments with synthetic and real data sets show that the proposed algorithm can be considered as a promising tool in the field of exploratory statistics.     

基于模糊核c-means 算法的位置指纹聚类

提出一种针对位置指纹的模糊核c-means聚类算法.将位置指纹归结为一种服从正态分布的区间值数据以反映接入点信号强度采样值的不确定性,通过区间中值和大小确定的正态分布函数将位置指纹映射为特征空间中的一点,并在该特征空间中采用基于核方法的模糊c-means算法对其进行聚类.通过ZigBee定位实验表明,该方法对于位置指纹的分类效果明显好于基于信号强度平均值的c-means聚类,可在保证定位精度的前提下有效降低定位的计算量.     

Strong fuzzy c-means in medical image data analysis

This paper presents a robust fuzzy c-means (FCM) for an automatic effective segmentation of breast and brain magnetic resonance images (MRI). This paper obtains novel objective functions for proposed robust fuzzy c-means by replacing original Euclidean distance with properties of kernel function on feature space and using Tsallis entropy. By minimizing the proposed effective objective functions, this paper gets membership partition matrices and equations for successive prototypes. In order to reduce the computational complexity and running time, center initialization algorithm is introduced for initializing the initial cluster center. The initial experimental works have done on synthetic image and benchmark dataset to investigate the effectiveness of proposed, and then the proposed method has been implemented to differentiate the different region of real breast and brain magnetic resonance images. In order to identify the validity of proposed fuzzy c-means methods, segmentation accuracy is computed by using silhouette method. The experimental results show that the proposed method is more capable in segmentation of medical images than existed methods.     

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.     

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.     

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.     

Clustering incomplete relational data using the non-Euclidean relational fuzzy c-means algorithm

An approach for clustering on the basis of incomplete dissimilarity data is given. The data is first completed using simple triangle inequality-based approximation schemes and then clustered using the non-Euclidean relational fuzzy c-means algorithm. Results of numerical tests are included.     

多路径高斯核模糊C均值聚类算法

聚类算法单一迭代路径限制了参数优值的搜索。提出一种多路径高斯核模糊C均值聚类算法（MGKFCMs）,MGKFCMs算法首先取核目标函数及模糊隶属度函数中的核函数为高斯核函数;然后利用梯度法得到聚类中心迭代公式,并基于该迭代公式和粒子群算法作聚类中心的并行参数迭代,在每一次聚类迭代时,选择聚类目标函数值小的路径作为参数迭代最终路径。对比分析了MGKFCMs算法的相关性质,通过仿真实验验证了所提算法的有效性。     

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.     

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.     

Parameterized attribute reduction with Gaussian kernel based fuzzy rough sets

Fuzzy rough sets are considered as an effective tool to deal with uncertainty in data analysis, and fuzzy similarity relations are used in fuzzy rough sets to calculate similarity between objects. On the other hand in kernel tricks, a kernel maps data into a higher dimensional feature space where the resulting structure of the learning task is linearly separable, while the kernel is the inner product of this feature space and can also be viewed as a similarity function. It has been reported there is an overlap between family of kernels and collection of fuzzy similarity relations. This fact motivates the idea in this paper to use some kernels as fuzzy similarity relations and develop kernel based fuzzy rough sets. First, we consider Gaussian kernel and propose Gaussian kernel based fuzzy rough sets. Second we introduce parameterized attribute reduction with the derived model of fuzzy rough sets. Structures of attribute reduction are investigated and an algorithm with discernibility matrix to find all reducts is developed. Finally, a heuristic algorithm is designed to compute reducts with Gaussian kernel fuzzy rough sets. Several experiments are provided to demonstrate the effectiveness of the idea.     

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.     

A multi-objective genetic algorithm with fuzzy c-means for automatic data clustering

This article presents a multi-objective genetic algorithm which considers the problem of data clustering. A given dataset is automatically assigned into a number of groups in appropriate fuzzy partitions through the fuzzy c-means method. This work has tried to exploit the advantage of fuzzy properties which provide capability to handle overlapping clusters. However, most fuzzy methods are based on compactness and/or separation measures which use only centroid information. The calculation from centroid information only may not be sufficient to differentiate the geometric structures of clusters. The overlap-separation measure using an aggregation operation of fuzzy membership degrees is better equipped to handle this drawback. For another key consideration, we need a mechanism to identify appropriate fuzzy clusters without prior knowledge on the number of clusters. From this requirement, an optimization with single criterion may not be feasible for different cluster shapes. A multi-objective genetic algorithm is therefore appropriate to search for fuzzy partitions in this situation. Apart from the overlap-separation measure, the well-known fuzzy J_m index is also optimized through genetic operations. The algorithm simultaneously optimizes the two criteria to search for optimal clustering solutions. A string of real-coded values is encoded to represent cluster centers. A number of strings with different lengths varied over a range correspond to variable numbers of clusters. These real-coded values are optimized and the Pareto solutions corresponding to a tradeoff between the two objectives are finally produced. As shown in the experiments, the approach provides promising solutions in well-separated, hyperspherical and overlapping clusters from synthetic and real-life data sets. This is demonstrated by the comparison with existing single-objective and multi-objective clustering techniques.     

Fast automatic medical image segmentation based on spatial kernel fuzzy c-means on level set method

Fast two-cycle (FTC) model is an efficient and the fastest Level set image segmentation. But, its performance is highly dependent on appropriate manual initialization. This paper proposes a new algorithm by combining a spatially constrained kernel-based fuzzy c-means (SKFCM) algorithm and an FTC model to overcome the mentioned problem. The approach consists of two successive stages. First, the SKFCM makes a rough segmentation to select the initial contour automatically. Then, a fuzzy membership matrix of the region of interest, which is generated by the SKFCM, is used in the next stage to produce an initial contour. Eventually, the FTC scheme segments the image by a curve evolution based on the level set. Moreover, the fuzzy membership degree from the SKFCM is incorporated into the fidelity term of the Chan–Vese model to improve the robustness and accuracy, and it is utilized for the data-dependent speed term of the FTC. A performance evaluation of the proposed algorithm is carried out on the synthetic and real images. The experimental results show that the proposed algorithm has advantages in accuracy, computational time and robustness against noise in comparison with the KFCM, the SKFCM, the hybrid model of the KFCM and the FTC, and five different level set methods on medical image segmentation.     

Integrated use of fuzzy c-means and convex programming for capacitated multi-facility location problem

In this study a fuzzy c-means clustering algorithm based method is proposed for solving a capacitated multi-facility location problem of known demand points which are served from capacitated supply centres. It involves the integrated use of fuzzy c-means and convex programming. In fuzzy c-means, data points are allowed to belong to several clusters with different degrees of membership. This feature is used here to split demands between supply centers. The cluster number is determined by an incremental method that starts with two and designated when capacity of each cluster is sufficient for its demand. Finally, each group of cluster and each model are solved as a single facility location problem. Then each single facility location problem given by fuzzy c-means is solved by convex programming which optimizes transportation cost is used to fine-tune the facility location. Proposed method is applied to several facility location problems from OR library (Osman & Christofides, 1994) and compared with centre of gravity and particle swarm optimization based algorithms. Numerical results of an asphalt producer’s real-world data in Turkey are reported. Numerical results show that the proposed approach performs better than using original fuzzy c-means, integrated use of fuzzy c-means and center of gravity methods in terms of transportation costs.     

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.     

Gaussian kernel optimization: Complex problem and a simple solution

The Gaussian kernel function implicitly defines the feature space of an algorithm and plays an essential role in the application of kernel methods. The parameter of Gaussian kernel function is a scalar that has significant influences on final results. However, until now, it is still unclear how to choose an optimal kernel parameter. In this paper, we propose a novel data-driven method to optimize the Gaussian kernel parameter, which only depends on the original dataset distribution and yields a simple solution to this complex problem. The proposed method is task irrelevant and can be used in any Gaussian kernel-based approach, including supervised and unsupervised machine learning. Simulation experiments demonstrate the efficacy of the obtained results. A user-friendly online calculator is implemented at: www.csbio.sjtu.edu.cn/bioinf/kernel/ for public use.     

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。     

An edge-cloud-aided incremental tensor-based fuzzy c-means approach with big data fusion for exploring smart data

Recently, smart data has attracted great attention in the smart city community since it can provide valuable information to support intelligent services such as planning, monitoring, and decision making. However, it imposes a big challenge to explore smart data from big data gathered from smart city with various advanced fusion and analysis approaches. This paper proposes an incremental tensor-based fuzzy c-means approach (IT-FCM) for obtaining smart data from continuously generated big data. Specifically, a weighted version of the tensor-based fuzzy c-means approach (T-FCM) is firstly proposed to cluster the dataset that combines the previous cluster centroids and the new generated data. Aiming to improve the clustering efficiency, the old data objects are represented by the centroids to avoid repeat clustering. Furthermore, this paper presents an edge-cloud-aided clustering scheme to fuse big data from different sources and perspectives and further to implement co-clustering on the fused datasets for exploring smart data. Finally, the proposed IT-FCM approach is evaluated by comparing with T-FCM regarding clustering accuracy and efficiency on two different datasets in the experiments. The results state that IT-FCM outperforms T-FCM in clustering streaming big data in terms of accuracy and efficiency for obtaining smart data.     

On semi-supervised fuzzy c-means clustering for data with clusterwise tolerance by opposite criteria

This paper presents a new semi-supervised fuzzy c-means clustering for data with clusterwise tolerance by opposite criteria. In semi-supervised clustering, pairwise constraints, that is, must-link and cannot-link, are frequently used in order to improve clustering performances. From the viewpoint of handling pairwise constraints, a new semi-supervised fuzzy c-means clustering is proposed by introducing clusterwise tolerance-based pairwise constraints. First, a concept of clusterwise tolerance-based pairwise constraints is introduced. Second, the optimization problems of the proposed method are formulated. Especially, must-link and cannot-link are handled by opposite criteria in our proposed method. Third, a new clustering algorithm is constructed based on the above discussions. Finally, the effectiveness of the proposed algorithm is verified through numerical examples.