A cluster validity index for fuzzy clustering

A new cluster validity index is proposed for the validation of partitions of object data produced by the fuzzy c-means algorithm. The proposed validity index uses a variation measure and a separation measure between two fuzzy clusters. A good fuzzy partition is expected to have a low degree of variation and a large separation distance. Testing of the proposed index and nine previously formulated indices on well-known data sets shows the superior effectiveness and reliability of the proposed index in comparison to other indices and the robustness of the proposed index in noisy environments.     

Fast and robust fuzzy c-means clustering algorithms incorporating local information for image segmentation

Fuzzy c-means (FCM) algorithms with spatial constraints (FCM_S) have been proven effective for image segmentation. However, they still have the following disadvantages: (1) although the introduction of local spatial information to the corresponding objective functions enhances their insensitiveness to noise to some extent, they still lack enough robustness to noise and outliers, especially in absence of prior knowledge of the noise; (2) in their objective functions, there exists a crucial parameter α used to balance between robustness to noise and effectiveness of preserving the details of the image, it is selected generally through experience; and (3) the time of segmenting an image is dependent on the image size, and hence the larger the size of the image, the more the segmentation time. In this paper, by incorporating local spatial and gray information together, a novel fast and robust FCM framework for image segmentation, i.e., fast generalized fuzzy c-means (FGFCM) clustering algorithms, is proposed. FGFCM can mitigate the disadvantages of FCM_S and at the same time enhances the clustering performance. Furthermore, FGFCM not only includes many existing algorithms, such as fast FCM and enhanced FCM as its special cases, but also can derive other new algorithms such as FGFCM_S1 and FGFCM_S2 proposed in the rest of this paper. The major characteristics of FGFCM are: (1) to use a new factor S_ij as a local (both spatial and gray) similarity measure aiming to guarantee both noise-immunity and detail-preserving for image, and meanwhile remove the empirically-adjusted parameter α; (2) fast clustering or segmenting image, the segmenting time is only dependent on the number of the gray-levels q rather than the size N(?q) of the image, and consequently its computational complexity is reduced from O(NcI₁) to O(qcI₂), where c is the number of the clusters, I₁ and are the numbers of iterations, respectively, in the standard FCM and our proposed fast segmentation method. The experiments on the synthetic and real-world images show that FGFCM algorithm is effective and efficient.     

Suppressed possibilistic c-means clustering algorithm

The possibilistic c-means (PCM) clustering algorithm always suffers from a coincident clustering problem since it relaxes the probabilistic constraint in the fuzzy c-means (FCM) clustering algorithm. In this paper, to overcome the shortcoming of the PCM, a novel suppressed possibilistic c-means (S-PCM) clustering algorithm by introducing a suppressed competitive learning strategy into the PCM so as to improve the between-cluster relationships is proposed. Specifically, in the updating process the new algorithm searches for the biggest typicality which is regarded as winner by a competitive mechanism. Then it suppresses the non-winner typicalities with a suppressed rate which is used to control the learning strength. Moreover, the parameter setting problems of the suppressed rate and the penalty parameter in the S-PCM are also discussed in detail. In addition, the suppressed competitive learning strategy is still introduced into the possibilistic Gustafson–Kessel (PGK) clustering algorithm and a novel suppressed possibilistic Gustafson–Kessel (S-PGK) clustering model is proposed, which is more applicable to the ellipsoidal data clustering. Finally, experiments on several synthetic and real datasets with noise injection demonstrate the effectiveness of the proposed algorithms.     

A possibilistic clustering approach to novel fault detection and isolation

In this paper, a new approach for fault detection and isolation that is based on the possibilistic clustering algorithm is proposed. Fault detection and isolation (FDI) is shown here to be a pattern classification problem, which can be solved using clustering and classification techniques. A possibilistic clustering based approach is proposed here to address some of the shortcomings of the fuzzy c-means (FCM) algorithm. The probabilistic constraint imposed on the membership value in the FCM algorithm is relaxed in the possibilistic clustering algorithm. Because of this relaxation, the possibilistic approach is shown in this paper to give more consistent results in the context of the FDI tasks. The possibilistic clustering approach has also been used to detect novel fault scenarios, for which the data was not available while training. Fault signatures that change as a function of the fault intensities are represented as fault lines, which have been shown to be useful to classify faults that can manifest with different intensities. The proposed approach has been validated here through simulations involving a benchmark quadruple tank process and also through experimental case studies on the same setup. For large scale systems, it is proposed to use the possibilistic clustering based approach in the lower dimensional approximations generated by algorithms such as PCA. Towards this end, finally, we also demonstrate the key merits of the algorithm for plant wide monitoring study using a simulation of the benchmark Tennessee Eastman problem.     

On cluster validity index for estimation of the optimal number of fuzzy clusters

A new cluster validity index is proposed that determines the optimal partition and optimal number of clusters for fuzzy partitions obtained from the fuzzy c-means algorithm. The proposed validity index exploits an overlap measure and a separation measure between clusters. The overlap measure, which indicates the degree of overlap between fuzzy clusters, is obtained by computing an inter-cluster overlap. The separation measure, which indicates the isolation distance between fuzzy clusters, is obtained by computing a distance between fuzzy clusters. A good fuzzy partition is expected to have a low degree of overlap and a larger separation distance. Testing of the proposed index and nine previously formulated indexes on well-known data sets showed the superior effectiveness and reliability of the proposed index in comparison to other indexes.     

Global optimization in clustering using hyperbolic cross points

Erich Novak and Klaus Ritter developed in 1996 a global optimization algorithm that uses hyperbolic cross points (HCPs). In this paper we develop a hybrid algorithm for clustering called CMHCP that uses a modified version of this HCP algorithm for global search and the alternating optimization for local search. The program has been tested extensively with very promising results and high efficiency. This provides a nice addition to the arsenal of global optimization in clustering. In the process, we also analyze the smoothness of some reformulated objective functions.     

Fuzzy clustering and enumeration of target type based on sonar returns

The fuzzy c-means (FCM) clustering algorithm is used in conjunction with a cluster validity criterion, to determine the number of different types of targets in a given environment, based on their sonar signatures. The class of each target and its location are also determined. The method is experimentally verified using real sonar returns from targets in indoor environments. A correct differentiation rate of 98% is achieved with average absolute valued localization errors of and 0.8° in range and azimuth, respectively.     

Evaluation of the performance of clustering algorithms in kernel-induced feature space

By using a kernel function, data that are not easily separable in the original space can be clustered into homogeneous groups in the implicitly transformed high-dimensional feature space. Kernel k-means algorithms have recently been shown to perform better than conventional k-means algorithms in unsupervised classification. However, few reports have examined the benefits of using a kernel function and the relative merits of the various kernel clustering algorithms with regard to the data distribution. In this study, we reformulated four representative clustering algorithms based on a kernel function and evaluated their performances for various data sets. The results indicate that each kernel clustering algorithm gives markedly better performance than its conventional counterpart for almost all data sets. Of the kernel clustering algorithms studied in the present work, the kernel average linkage algorithm gives the most accurate clustering results.     

改进的核可能性C-均值聚类算法研究

提出了一种改进的核可能性C-均值聚类算法,它是对PCM聚类模型的推广。通过限制PCM聚类模型中解的可行域,利用全局优化技术（以模拟退火（SA）为例）来求解,使其保持了PCM对噪声鲁棒的优点,又避免了重合聚类的产生,且能较好地找到问题的全局最优解,减少了全局优化方法的搜索范围,加快了收敛速度。     

基于空间信息的可能性模糊C均值聚类遥感图像分割

可能性模糊C均值(PFCM)聚类算法作为模糊C均值(FCM)聚类算法的一种改进算法,能在一定程度上克服FCM算法对噪声的敏感性;但由于PFCM没有考虑像元间的空间信息,对含有较大噪声的图像分割效果依然不理想。为此,提出一种新的基于空间信息的PFCM算法(SPFCM),克服了PFCM算法对含有较大噪声的图像分割效果不佳的缺点。通过对人工图像和IKONOS遥感图像进行分析,结果表明,SPFCM算法无论是在视觉上还是在分割正确率上都优于传统的FCM算法、PFCM算法及两种加入空间信息的FCM算法;对于含有高斯噪声和盐椒噪声的图像,平均分割正确率高达99.71%,是一种去噪效果较好的图像分割算法。     

A stability based validity method for fuzzy clustering

An important goal in cluster analysis is the internal validation of results using an objective criterion. Of particular relevance in this respect is the estimation of the optimum number of clusters capturing the intrinsic structure of your data. This paper proposes a method to determine this optimum number based on the evaluation of fuzzy partition stability under bootstrap resampling. The method is first characterized on synthetic data with respect to hyper-parameters, like the fuzzifier, and spatial clustering parameters, such as feature space dimensionality, clusters degree of overlap, and number of clusters. The method is then validated on experimental datasets. Furthermore, the performance of the proposed method is compared to that obtained using a number of traditional fuzzy validity rules based on the cluster compactness-to-separation criteria. The proposed method provides accurate and reliable results, and offers better generalization capabilities than the classical approaches.     

Cluster validity index for estimation of fuzzy clusters of different sizes and densities

Cluster validity indices are used for estimating the quality of partitions produced by clustering algorithms and for determining the number of clusters in data. Cluster validation is difficult task, because for the same data set more partitions exists regarding the level of details that fit natural groupings of a given data set. Even though several cluster validity indices exist, they are inefficient when clusters widely differ in density or size. We propose a clustering validity index that addresses these issues. It is based on compactness and overlap measures. The overlap measure, which indicates the degree of overlap between fuzzy clusters, is obtained by calculating the overlap rate of all data objects that belong strongly enough to two or more clusters. The compactness measure, which indicates the degree of similarity of data objects in a cluster, is calculated from membership values of data objects that are strongly enough associated to one cluster. We propose ratio and summation type of index using the same compactness and overlap measures. The maximal value of index denotes the optimal fuzzy partition that is expected to have a high compactness and a low degree of overlap among clusters. Testing many well-known previously formulated and proposed indices on well-known data sets showed the superior reliability and effectiveness of the proposed index in comparison to other indices especially when evaluating partitions with clusters that widely differ in size or density.     

Fuzzy clustering algorithms for unsupervised change detection in remote sensing images

In this paper, we propose a context-sensitive technique for unsupervised change detection in multitemporal remote sensing images. The technique is based on fuzzy clustering approach and takes care of spatial correlation between neighboring pixels of the difference image produced by comparing two images acquired on the same geographical area at different times. Since the ranges of pixel values of the difference image belonging to the two clusters (changed and unchanged) generally have overlap, fuzzy clustering techniques seem to be an appropriate and realistic choice to identify them (as we already know from pattern recognition literatures that fuzzy set can handle this type of situation very well). Two fuzzy clustering algorithms, namely fuzzy c-means (FCM) and Gustafson-Kessel clustering (GKC) algorithms have been used for this task in the proposed work. For clustering purpose various image features are extracted using the neighborhood information of pixels. Hybridization of FCM and GKC with two other optimization techniques, genetic algorithm (GA) and simulated annealing (SA), is made to further enhance the performance. To show the effectiveness of the proposed technique, experiments are conducted on two multispectral and multitemporal remote sensing images. A fuzzy cluster validity index (Xie-Beni) is used to quantitatively evaluate the performance. Results are compared with those of existing Markov random field (MRF) and neural network based algorithms and found to be superior. The proposed technique is less time consuming and unlike MRF does not require any a priori knowledge of distributions of changed and unchanged pixels.     

Further reduced form of wavelet feature for text independent speaker recognition

A text independent speaker recognition system based on wavelet transform derived from fuzzy c-means clustering is proposed. The fuzzy c-means clustering is applied to the speaker data compression in spectrum domain. A set of experiments are conducted, which gives a 95% recognition rate for 100 Mandarin speakers.     

Application of fuzzy ARTMAP and fuzzy <Emphasis Type="Italic">c</Emphasis>-means clustering to pattern classification with incomplete data

In this paper, a hybrid neural network that is capable of incremental learning and classification of patterns with incomplete data is proposed. Fuzzy ARTMAP (FAM) is employed as the constituting network for pattern classification while fuzzy c-means (FCM) clustering is used as the underlying algorithm for processing training as well as test samples with missing features. To handle an incomplete training set, FAM is first trained using complete samples only. Missing features of the training samples are estimated and replaced using two FCM-based strategies. Then, network training is conducted using all the complete and estimated samples. To handle an incomplete test set, a non-substitution FCM-based strategy is employed so that a predicted output can be produced rapidly. The performance of the proposed hybrid network is evaluated using a benchmark problem, and its practical applicability is demonstrated using a medical diagnosis task. The results are compared, analysed and quantified statistically with the bootstrap method. Implications of the proposed network for pattern classification tasks with incomplete data are discussed.     

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.     

On relational possibilistic clustering

This paper initially describes the relational counterpart of possibilistic c-means (PCM) algorithm, called relational PCM (or RPCM). RPCM is then improved to better handle arbitrary dissimilarity data. First, a re-scaling of the PCM membership function is proposed in order to obtain zero membership values when the distance to prototype equals the maximum value allowed in bounded dissimilarity measures. Second, a heuristic method of reference distance initialisation is provided which diminishes the known PCM tendency of producing coincident clusters. Finally, RPCM improved with our initialisation strategy is tested on both synthetic and real data sets with satisfactory results.     

A time-efficient pattern reduction algorithm for k-means clustering

This paper presents an efficient algorithm, called pattern reduction (PR), for reducing the computation time of k-means and k-means-based clustering algorithms. The proposed algorithm works by compressing and removing at each iteration patterns that are unlikely to change their membership thereafter. Not only is the proposed algorithm simple and easy to implement, but it can also be applied to many other iterative clustering algorithms such as kernel-based and population-based clustering algorithms. Our experiments—from 2 to 1000 dimensions and 150 to 10,000,000 patterns—indicate that with a small loss of quality, the proposed algorithm can significantly reduce the computation time of all state-of-the-art clustering algorithms evaluated in this paper, especially for large and high-dimensional data sets.     

一种基于核的快速可能性聚类算法

传统的快速聚类算法大多基于模糊C均值算法（Fuzzy C-means,FCM）,而FCM对初始聚类中心敏感,对噪音数据敏感并且容易收敛到局部极小值,因而聚类准确率不高。可能性C-均值聚类较好地解决了FCM对噪声敏感的问题,但容易产生一致性聚类。将FCM和可能性C-均值聚类结合的聚类算法较好地解决了一致性聚类问题。为进一步提高算法收敛速度和鲁棒性,提出一种基于核的快速可能性聚类算法。该方法引入核聚类的思想,同时使用样本方差对目标函数中参数η进行优化。标准数据集和人造数据集的实验结果表明这种基于核的快速可能性聚类算法提高了算法的聚类准确率,加快了收敛速度。