Cluster Validity Functions Research Articles

Performance Analysis of Fuzzy C-Means Clustering Methods for MRI Image Segmentation

MRI is one of the most eminent medical imaging techniques and segmentation is a critical stage in investigation of MRI images. FCM is most usually used techniques for image segmentation of medical image applications because of its fuzzy nature, where one pixel can belong to multiple clusters and which lead to better performance than crisp methods. Conventional FCM fail to perform well enough in the presence of noise and intensity inhomogeneity in MRI images. Various FCM variations like BCFCM, PFCM, SFCM, FLICM, MDFCM, FCM_S1, FCM_S2, TEFCM, RFCMK, WIPFCM and KWFLICM, have been proposed to overcome these predicament by using the spatial statistics available in the images. In this paper all these techniques, used for segmentation, are implemented and compared in terms of two classes of cluster validity functions, fuzzy partition and feature structure, on the basis of their performance for the noisy MRI images. All these FCM variants are analyzed in terms of Partition Coefficient, Partition Entropy, Time Complexity and Segmentation Accuracy.

A modified fuzzy C‐means algorithm using scale control spatial information for MRI image segmentation in the presence of noise

AbstractThe fuzzy C‐means (FCM) algorithm does not fully utilize the spatial information for image segmentation and is sensitive to the presence of noise and intensity inhomogeneity in magnetic resonance imaging (MRI) images. The underlying reason is that using a single fuzzy membership function the FCM algorithm cannot properly represent pattern associations to all clusters. In this paper, we present a modified FCM (mFCM) algorithm by incorporating scale control spatial information for segmentation of MRI images in the presence of high levels of noise and intensity inhomogeneity. The algorithm utilizes scale controlled spatial information from the neighbourhood of each pixel under consideration in the form of a probability function. Using this probability function, a local membership function is introduced for each pixel. Finally, new clustering centre and weighted joint membership functions are introduced based on the local membership and global membership functions. The resulting mFCM algorithm is robust to the noise and intensity inhomogeneity in MRI image data and thereby improves the segmentation results. The experimental results on a synthetic image, four volumes of simulated and one volume of real‐patient MRI brain images show that the mFCM algorithm outperforms k‐means, FCM and some other recently proposed FCM‐based algorithms for image segmentation in terms of qualitative and quantitative studies such as cluster validity functions, segmentation accuracy and tissue segmentation accuracy. Copyright © 2015 John Wiley & Sons, Ltd.

Conditional spatial fuzzy C-means clustering algorithm for segmentation of MRI images

The fuzzy C-means (FCM) algorithm has got significant importance due to its unsupervised form of learning and more tolerant to variations and noise as compared to other methods in medical image segmentation. In this paper, we propose a conditional spatial fuzzy C-means (csFCM) clustering algorithm to improve the robustness of the conventional FCM algorithm. This is achieved through the incorporation of conditioning effects imposed by an auxiliary (conditional) variable corresponding to each pixel, which describes a level of involvement of the pixel in the constructed clusters, and spatial information into the membership functions. The problem of sensitivity to noise and intensity inhomogeneity in magnetic resonance imaging (MRI) data is effectively reduced by incorporating local and global spatial information into a weighted membership function. The experimental results on four volumes of simulated and one volume of real-patient MRI brain images, each one having 51 images, show that the csFCM algorithm has superior performance in terms of qualitative and quantitative studies such as, cluster validity functions, segmentation accuracy, tissue segmentation accuracy and receiver operating characteristic (ROC) curve on the image segmentation results than the k-means, FCM and some other recently proposed FCM-based algorithms.

Fuzzy Clustering Algorithms for Effective Medical Image Segmentation

Medical image segmentation demands a segmentation algorithm which works against noise. The most popular algorithm used in image segmentation is Fuzzy C-Means clustering. It uses only intensity values for clustering which makes it highly sensitive to noise. The comparison of the three fundamental image segmentation methods based on fuzzy logic namely Fuzzy C-Means (FCM), Intuitionistic Fuzzy C-Means (IFCM), and Type-II Fuzzy C-Means (T2FCM) is presented in this paper. These algorithms are executed in two scenarios- both in the absence and in the presence of noise and on two kinds of images- Bacteria and CT scan brain image. In the bacteria image, clustering differentiates the bacteria from the background and in the brain CT scan image, clustering is used to identify the abnormality region. Performance is analyzed on the basis cluster validity functions, execution time and convergence rate. Misclassification error is also calculated for brain image analysis.

Definition of Management Zones for Enhancing Cultivated Land Conservation Using Combined Spatial Data

The loss of cultivated land has increasingly become an issue of regional and national concern in China. Definition of management zones is an important measure to protect limited cultivated land resource. In this study, combined spatial data were applied to define management zones in Fuyang city, China. The yield of cultivated land was first calculated and evaluated and the spatial distribution pattern mapped; the limiting factors affecting the yield were then explored; and their maps of the spatial variability were presented using geostatistics analysis. Data were jointly analyzed for management zone definition using a combination of principal component analysis with a fuzzy clustering method, two cluster validity functions were used to determine the optimal number of cluster. Finally one-way variance analysis was performed on 3,620 soil sampling points to assess how well the defined management zones reflected the soil properties and productivity level. It was shown that there existed great potential for increasing grain production, and the amount of cultivated land played a key role in maintaining security in grain production. Organic matter, total nitrogen, available phosphorus, elevation, thickness of the plow layer, and probability of irrigation guarantee were the main limiting factors affecting the yield. The optimal number of management zones was three, and there existed significantly statistical differences between the crop yield and field parameters in each defined management zone. Management zone I presented the highest potential crop yield, fertility level, and best agricultural production condition, whereas management zone III lowest. The study showed that the procedures used may be effective in automatically defining management zones; by the development of different management zones, different strategies of cultivated land management and practice in each zone could be determined, which is of great importance to enhance cultivated land conservation, stabilize agricultural production, promote sustainable use of cultivated land and guarantee food security.

Algorithm Design for Cluster Validity Based on Geometric Probability

Determining optimum cluster number is a key research topic included in cluster validity. Based on geometric probability, this article proposes a new cluster validity algorithm to determine optimum cluster number for two-dimension datasets. The algorithm measure the cluster structure of the data set according to the distributive feature of the point set in the characteristic space. The structure information of the point set has been stored in a line segment set generated by connecting each pair points in the point set and the cluster validity function is formed by comparing the values of line segment direction in the line segment set with those resulted from completely random condition. Experiments prove that the pattern of the function curve generated with a given example data set effectively enables determining the optimum cluster number of the data set.

지역 가중치 적용 퍼지 클러스터링을 이용한 효과적인 이미지 분할

본 논문에서는 기존의 퍼지 클러스터링 기반 이미지 분할의 성능과 계산 효율을 개선하기 위해 퍼지 클러스터링의 목적 함수를 수정하는 이미지 분할 프레임워크를 제안한다. 제안하는 이미지 분할 프레임워크는 주변 픽셀들에 가중치를 부여함으로써 현재 센터 픽셀 연산을 위해 주변 픽셀들의 중요성을 고려하는 지역 가중치 적용 퍼지 클러스터링 기법을 포함한다. 이러한 가중치들은 각 멤버쉽들의 중요성을 표시하기 위해 현재 픽셀과 대응되는 각 주변 픽셀들 사이의 거리차에 의해 결정되어 지며, 이러한 프로세서는 향상된 클러스터링 성능을 보장한다. 제안하는 방법의 성능을 평가하기 위해 분할 계수, 분할 엔트로피, Xie-Bdni 함수, Fukuyzma-Sugeno 함수와 같은 네 가지 클러스터 유효성 함수를 이용하여 분석하였다. 모의실험 결과, 제안한 방법은 기존의 다른 퍼지 클러스터링 기법들보다 클러스터 유효성 함수들뿐만 아니라 분할과 조밀도 측면에서 우수한 성능을 보였다. This paper proposes an image segmentation framework that modifies the objective function of Fuzzy C-Means (FCM) to improve the performance and computational efficiency of the conventional FCM-based image segmentation. The proposed image segmentation framework includes a locally weighted fuzzy c-means (LWFCM) algorithm that takes into account the influence of neighboring pixels on the center pixel by assigning weights to the neighbors. Distance between a center pixel and a neighboring pixels are calculated within a window and these are basis for determining weights to indicate the importance of the memberships as well as to improve the clustering performance. We analyzed the segmentation performance of the proposed method by utilizing four eminent cluster validity functions such as partition coefficient (<TEX>$V_{pc}$</TEX>), partition entropy (<TEX>$V_{pe}$</TEX>), Xie-Bdni function (<TEX>$V_{xb}$</TEX>) and Fukuyama-Sugeno function (<TEX>$V_{fs}$</TEX>). Experimental results show that the proposed LWFCM outperforms other FCM algorithms (FCM, modified FCM, and spatial FCM, FCM with locally weighted information, fast generation FCM) in the cluster validity functions as well as both compactness and separation.

Self-updating clustering algorithm for estimating the parameters in mixtures of von Mises distributions

The EM algorithm is the standard method for estimating the parameters in finite mixture models. Yang and Pan [25] proposed a generalized classification maximum likelihood procedure, called the fuzzy c-directions (FCD) clustering algorithm, for estimating the parameters in mixtures of von Mises distributions. Two main drawbacks of the EM algorithm are its slow convergence and the dependence of the solution on the initial value used. The choice of initial values is of great importance in the algorithm-based literature as it can heavily influence the speed of convergence of the algorithm and its ability to locate the global maximum. On the other hand, the algorithmic frameworks of EM and FCD are closely related. Therefore, the drawbacks of FCD are the same as those of the EM algorithm. To resolve these problems, this paper proposes another clustering algorithm, which can self-organize local optimal cluster numbers without using cluster validity functions. These numerical results clearly indicate that the proposed algorithm is superior in performance of EM and FCD algorithms. Finally, we apply the proposed algorithm to two real data sets.

Flood Season Division with an Improved Fuzzy C-mean Clustering Method in the Taihu Lake Basin in China

The division of the flood season in a basin is a multi-dimension time series clustering problem with unknown clustering numbers. So the clustering method is required to have the capability of dealing with multi-dimensions, time series and clustering validity. A dynamic fuzzy C-means clustering method (simplified as DFCCM) with clustering validity function based time series was developed in the paper. The real coding based accelerating genetic algorithm (simplified as RAGA) is used for solving DFCCM, which is very effective and can overcome the initial value sensitivity from the regular iterative optimal method generally used for the fuzzy C-means clustering. The newly developed method was applied in the Taihu lake basin in China for the division of the basin flood season, and 11 schemes were designed for removing the uncertainty from the initial value of flood season, which could exert influence on the results of the flood seasons division. Application results show that the method is very effective and the division conclusion of flood season in the Taihu Lake Basin is also reasonable.

Parameter identification of TSK neuro-fuzzy models

In the framework of the TSK neuro-fuzzy model a combination of the two well-known identification methods are employed for parameter estimation of the neuro-fuzzy inference system, namely the series–parallel and the parallel configurations. The presented paper proposes two new possible configurations for identifying the parameters of the TSK neuro-fuzzy model using the combinations of these two existing configurations. One of the proposed configurations constitutes the series–parallel configuration to the premise part and the parallel configuration to the consequent part of the neuro-fuzzy model, termed as PS-P configuration. The second one is composed of the series–parallel configuration to the consequent part and the parallel configuration to the premise part of the neuro-fuzzy model, termed as CS-P configuration. The presented work mainly deals with a comparative study of the proposed configurations and the existing configurations in the context of parameter identification of the TSK neuro-fuzzy model on three different benchmark examples. Moreover, it investigates upper bound of the learning rates, using the Lyapunov stability theorem, to assure the stability and the convergence of the model learning process. Implementation of the modified mountain clustering (MMC) and the cluster validity function yields initial models. To restrict the upper bound during the learning process it also presents a two-phase adaptive learning rate.

Relational Duals of Cluster-Validity Functions for the $c$ -Means Family

Clustering aims to identify groups of similar objects. To evaluate the results of cluster algorithms, an investigator uses cluster-validity indices. While the theory of cluster validity is well established for vector object data, little effort has been made to extend it to relationship-based data. As such, this paper proposes a theory of reformulation for object-data validity indices so that they can be used to rank the results produced by the relational -means clustering algorithms. More specifically, we create a class of relational validity indices, which is called dual-relational indices, that are guaranteed under certain, but easily met, constraints to produce the same results and, hence, the same cluster counts, as their object-data counterparts.

Segmentation of Shoppers using their Behavioral Patterns Based on Store Images

The main theme of this paper is to portray the segmentation of the shoppers based on their behavioural pattern towards the retail store images. Primarily, this article triggers off the various behavioural outcomes of the shoppers which have been influenced by the different store attributes. Based on the various store attributes, the shoppers have been segmented by the use of k-means Clustering Algorithm. The segment was well separated by using cluster validity function by Xie and Beni. The resulted segments are Pre-Determined, Variety-Seekers, Economy and Familiar. It has been found that every segment has its own impacts over the various store images and the managerial implications of these attributes have been highlighted.

Dynamic Niching Genetic Algorithm with Data Attraction for Automatic Clustering

A genetic clustering algorithm was developed based on dynamic niching with data attraction. The algorithm uses the concept of Coulomb attraction to model the attraction between data points. Then, the niches with data attraction are dynamically identified in each generation to automatically evolve the optimal number of clusters as well as the cluster centers of the data set without using cluster validity functions or a variance-covariance matrix. Therefore, this clustering scheme does not need to pre-specify the number of clusters as in existing methods. Several data sets with widely varying characteristics are used to demonstrate the superiority of this algorithm. Experimental results show that the performance of this clustering algorithm is high, effective, and flexible.

A robust dynamic niching genetic algorithm with niche migration for automatic clustering problem

In this paper, a genetic clustering algorithm based on dynamic niching with niche migration (DNNM-clustering) is proposed. It is an effective and robust approach to clustering on the basis of a similarity function relating to the approximate density shape estimation. In the new algorithm, a dynamic identification of the niches with niche migration is performed at each generation to automatically evolve the optimal number of clusters as well as the cluster centers of the data set without invoking cluster validity functions. The niches can move slowly under the migration operator which makes the dynamic niching method independent of the radius of the niches. Compared to other existing methods, the proposed clustering method exhibits the following robust characteristics: (1) robust to the initialization, (2) robust to clusters volumes (ability to detect different volumes of clusters), and (3) robust to noise. Moreover, it is free of the radius of the niches and does not need to pre-specify the number of clusters. Several data sets with widely varying characteristics are used to demonstrate its superiority. An application of the DNNM-clustering algorithm in unsupervised classification of the multispectral remote sensing image is also provided.

A new point symmetry based fuzzy genetic clustering technique for automatic evolution of clusters

In this paper a fuzzy point symmetry based genetic clustering technique (Fuzzy-VGAPS) is proposed which can automatically determine the number of clusters present in a data set as well as a good fuzzy partitioning of the data. The clusters can be of any size, shape or convexity as long as they possess the property of symmetry. Here the membership values of points to different clusters are computed using the newly proposed point symmetry based distance. A variable number of cluster centers are encoded in the chromosomes. A new fuzzy symmetry based cluster validity index, FSym-index is first proposed here and thereafter it is utilized to measure the fitness of the chromosomes. The proposed index can detect non-convex, as well as convex-non-hyperspherical partitioning with variable number of clusters. It is mathematically justified via its relationship to a well-defined hard cluster validity function: the Dunn’s index, for which the condition of uniqueness has already been established. The results of the Fuzzy-VGAPS are compared with those obtained by seven other algorithms including both fuzzy and crisp methods on four artificial and four real-life data sets. Some real-life applications of Fuzzy-VGAPS to automatically cluster the gene expression data as well as segmenting the magnetic resonance brain image with multiple sclerosis lesions are also demonstrated.

Validation criteria for enhanced fuzzy clustering

We introduce two new criterions for validation of results obtained from recent novel-clustering algorithm, improved fuzzy clustering (IFC) to be used to find patterns in regression and classification type datasets, separately. IFC algorithm calculates membership values that are used as additional predictors to form fuzzy decision functions for each cluster. Proposed validity criterions are based on the ratio of compactness to separability of clusters. The optimum compactness of a cluster is represented with average distances between every object and cluster centers, and total estimation error from their fuzzy decision functions. The separability is based on a conditional ratio between the similarities between cluster representatives and similarities between fuzzy decision surfaces of each cluster. The performance of the proposed validity criterions are compared to other structurally similar cluster validity indexes using datasets from different domains. The results indicate that the new cluster validity functions are useful criterions when selecting parameters of IFC models.

An adaptive fuzzy c-means clustering-based mixtures of experts model for unlabeled data classification

Compared with labeled data, unlabeled data are more readily available. Currently, classification of unlabeled data is an open issue, especially for the case of unknown class number. In this paper, we propose an adaptive fuzzy c-means (FCM)-based mixtures of experts model to deal with the problem. In this model, each mixture of experts (ME) consists of two expert networks and a gating network. Two experts, namely, Gaussian neural network (GNN) and sigmoid neural network (SNN), are selected as two candidates. Two phases are employed to construct the proposed model. First, the whole input space is partitioned into several clusters using the FCM clustering algorithm. The number of clusters can be determined adaptively by a cluster validity function. Second, the proposed model is trained by a small fraction of samples which are closer to their corresponding cluster centers. A numerical study is made on several synthetic and real-world data sets. Compared with the other four models, the proposed model exhibits better generalization ability in dealing with problems of unsupervised classification. The experimental results also show that the extension version of the proposed model for semi-supervised classification is comparable to the CVS 3 VM approach.

A fuzzy CLOPE algorithm and its optimal parameter choice

Among the available clustering algorithms in data mining, the CLOPE algorithm attracts much more attention with its high speed and good performance. However, the proper choice of some parameters in the CLOPE algorithm directly affects the validity of the clustering results, which is still an open issue. For this purpose, this paper proposes a fuzzy CLOPE algorithm, and presents a method for the optimal parameter choice by defining a modified partition fuzzy degree as a clustering validity function. The experimental results with real data set illustrate the effectiveness of the proposed fuzzy CLOPE algorithm and optimal parameter choice method based on the modified partition fuzzy degree.

A Weighted Sum Validity Function for Clustering With a Hybrid Niching Genetic Algorithm

Clustering is inherently a difficult problem, both with respect to the construction of adequate objective functions as well as to the optimization of the objective functions. In this paper, we suggest an objective function called the Weighted Sum Validity Function (WSVF), which is a weighted sum of the several normalized cluster validity functions. Further, we propose a Hybrid Niching Genetic Algorithm (HNGA), which can be used for the optimization of the WSVF to automatically evolve the proper number of clusters as well as appropriate partitioning of the data set. Within the HNGA, a niching method is developed to preserve both the diversity of the population with respect to the number of clusters encoded in the individuals and the diversity of the subpopulation with the same number of clusters during the search. In addition, we hybridize the niching method with the k-means algorithm. In the experiments, we show the effectiveness of both the HNGA and the WSVF. In comparison with other related genetic clustering algorithms, the HNGA can consistently and efficiently converge to the best known optimum corresponding to the given data in concurrence with the convergence result. The WSVF is found generally able to improve the confidence of clustering solutions and achieve more accurate and robust results.

A new fuzzy clustering algorithm for optimally finding granular prototypes

Prototype Reasoning using granular objects is an important technology for knowledge discovery. Fuzzy clustering can be used to generate prototypes with different granularities. In order to find optimal granular prototypes through fuzzy clustering, for given data, two conditions are necessary: a good cluster validity function, which can be applied to evaluate the goodness of cluster schemes for varying number of clusters (different granularity); a good cluster algorithm that can produce an optimal solution for a fixed number of clusters. To satisfy the first condition, a new validity measure called granularity–dissimilarity (GD) measure is proposed, which is stable in evaluating granularities and works well even when the number of clusters is very large. For the second condition, we propose a new algorithm called multi-step maxmin and merging algorithm (3M algorithm). Experiments show that, when used in conjunction with the new cluster validity measure, 3M algorithm produces better results on the experimental data sets than several alternatives.