A Weight Possibilistic Fuzzy C-Means Clustering Algorithm

Possibilistic fuzzy c-means (PFCM) clustering is a kind of hybrid clustering method based on fuzzy c-means (FCM) and possibilistic c-means (PCM), which not only has the stability of FCM but also partly inherits the robustness of PCM. However, as an extension of FCM on the objective function, PFCM tends to find a suboptimal local minimum, which affects its performance. In this paper, we rederive PFCM using the majorization-minimization (MM) method, which is a new derivation approach not seen in other studies. In addition, we propose an effective optimization method to solve the above problem, called MMPFCM. Firstly, by eliminating the variable V∈Rp×c, the original optimization problem is transformed into a simplified model with fewer variables but a proportional term. Therefore, we introduce a new intermediate variable s∈Rc to convert the model with the proportional term into an easily solvable equivalent form. Subsequently, we design an iterative sub-problem using the MM method. The complexity analysis indicates that MMPFCM and PFCM share the same computational complexity. However, MMPFCM requires less memory per iteration. Extensive experiments, including objective function value comparison and clustering performance comparison, demonstrate that MMPFCM converges to a better local minimum compared to PFCM.

The identification of fuzzy models for classification is a very complex task. Often, real world databases have a large number of features and the most relevant ones must be chosen. Recently, a new automatic feature selection for classification problems was proposed to construct compact fuzzy classification models. This technique used the classical fuzzy c-means algorithm. However, other fuzzy clustering algorithms, such as possibilistic c-means, fuzzy possibilistic c-means or possibilistic fuzzy c-means can be used to cluster the data. An open topic of research is what clustering algorithms can be used to derive fuzzy models for classification. This paper addresses this topic, by comparing fuzzy clustering algorithms in terms of computational efficiency and accuracy in classification problems. The algorithms were tested in well-known data sets: iris plant, wine, hepatitis, breast cancer and in a difficult real-world problem: the prediction of bankruptcy.

In this paper, a new Hopfield-model net based on fuzzy possibilistic reasoning is proposed for the classification of multispectral images. The main purpose is to modify the Hopfield network embedded with fuzzy possibilistic C-means (FPCM) method to construct a classification system named fuzzy-possibilistic Hopfield net (FPHN). The classification system is a paradigm for the implementation of fuzzy logic systems in neural network architecture. Instead of one state in a neuron for the conventional Hopfield nets, each neuron occupies 2 states called membership state and typicality state in the proposed FPHN. The proposed network not only solves the noise sensitivity fault of Fuzzy C-means (FCM) but also overcomes the simultaneous clustering problem of possibilistic C-means (PCM) strategy. In addition to the same characteristics as the FPCM algorithm, the simple features of this network are clear potential in optimal problem. The experimental results show that the proposed FPHN can obtain better solutions in the classification of multispectral images.

Fuzzy clustering partitions data points of a dataset into clusters in which one data point can belong to more than one cluster. In the literature, a number of fuzzy clustering algorithms have been proposed. This paper reviews various fuzzy clustering algorithms such as Fuzzy C-Means (FCM), Possibilistic C-Means (PCM), Possibilistic Fuzzy C-Means (PFCM), Intuitionistic Fuzzy C-Means (IFCM), Kernel Fuzzy C-Means (KFCM), and Density-Oriented Fuzzy C-Means (DOFCM). We have demonstrated the experimental performance of these algorithms on some standard and synthetic datasets which include—Bensaid, Square (DUNN), D15, and D45 dataset. Then, the results are analyzed and compared to see the effectiveness of these algorithms in presence of noise and outliers.

We justify the need for computing both membership and typicality values when clustering unlabeled data. Then we propose a new model called fuzzy-possibilistic c-means (FPCM). Unlike the fuzzy and possibilistic c-means (FCM/PCM) models, FPCM simultaneously produces both memberships and possibilities, along with the usual point prototypes or cluster centers for each cluster We show that FPCM solves the noise sensitivity defect of FCM, and also overcomes the coincident clusters problem of PCM. Then we derive first order necessary conditions for extrema of the PFCM objective function, and use them as the basis for a standard alternating optimization approach to finding local minima. Three numerical examples are given that compare FCM to FPCM. Our calculations show that FPCM compares favorably to FCM.

Clustering is the process of dividing data clusters into fragmented clusters so that the same set of data is similar, but the data of different clusters is different. The basis of the fuzzy clustering algorithms is the c-means families and the strongest algorithm is the fuzzy c-means algorithm. However, the fuzzy c-means algorithm is sensitive to outliers. In this study, on the real data set we examined three different algorithms -possibilistic c-means algorithm (PCM), fuzzy possibilistic c-means (FPCM) and possibilistic fuzzy c- means algorithm (PFCM)- which are developed to overcome the unfavorable side of the FCM algorithm. To compare these algorithms, iteration numbers and completion times were calculated.

본 논문에서는 interval type-2 possibilistic fuzzy C-means(IT2PFCM) 클러스터링 방법에 multiple Gaussian kernels을 기반으로 한 possibilistic fuzzy C-means multiple kernels(PFCM-MK) 알고리즘을 결합하여 적응적인 하이브리드 클러스터링 방법인 multiple kernels interval type-2 possibilistic fuzzy C-means(IT2PFCM-MK) 방법을 제안 하였다. 일반적으로 possibilistic fuzzy C-means(PFCM) 알고리즘은 fuzzy C-means(FCM) 알고리즘의 단점인 노이즈 민감성 및 특이점 문제와 알고리즘 초기 클러스터의 Prototype에 따라 위치가 겹치는 문제를 해결하기 위해 제안 되었다. 하지만 이 방법 역시 퍼지화 파라미터 값에 따라 위와 같은 문제를 여전히 가지고 있기 때문에 이와 같은 문제를 보완하기 위해 interval type-2 퍼지 접근 방법을 이용 하는 interval type-2 possibilistic fuzzy C-means(IT2PFCM) 알고리즘을 제안 하였다. 또한 multiple kernels 함수를 interval type-2 possibilistic fuzzy C-means(IT2PFCM) 알고리즘에 적용하여 분류하기 복잡한 형태의 데이터와 노이즈가 있는 데이터에 대하여 보다 정확하고, 향상된 클러스터링을 수행할 수 있다. In this paper, we propose a hybrid approach towards multiple kernels interval type-2 possibilistic fuzzy C-means(PFCM) based on interval type-2 possibilistic fuzzy c-means(IT2PFCM) and possibilistic fuzzy c-means using multiple kernels( PFCM-MK). In case of noisy data or overlapping cluster prototypes, fuzzy C-means gives poor performance in comparison to possibilistic fuzzy C-means(PFCM). Moreover, to address the uncertainty associated with fuzzifier parameter m, interval type-2 possibilistic fuzzy C-means(PFCM) is used. Most of the practical data available are complex and non-linearly separable. In such cases using Gaussian kernels proves helpful. Therefore, in order to overcome all these issues, we have integrated multiple kernels possibilistic fuzzy C-means(PFCM) into interval type-2 possibilistic fuzzy C-means(IT2PFCM) and propose the idea of multiple kernels based interval type-2 possibilistic fuzzy C-means(IT2PFCM-MK).

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] The possibilistic c-means (PCM) was developed as an extension of the fuzzy c-means (FCM) clustering algorithm by abandoning the membership sum-to-one constraint. In the PCM, each cluster is independent of the other clusters and can be processed separately. Because of this separability, the sequential possibilistic one-mean (SP1M) was proposed to find clusters sequentially by running the possibilistic one-mean (P1M) c times. One critical problem in both PCM and SP1M is how to determine the parameter [subscript]. The sequential possibilistic one-mean with adaptive eta (SP1M-AE) was developed to allow [subscript] to change during iterations. In this thesis, a new dynamic adaptation mechanism for the parameter [subscript] in each cluster is inserted into SP1M. The resultant algorithm, called the sequential possibilistic one-mean with dynamic eta (SP1M-DE) is shown to provide superior performance over PCM, SP1M, and SP1M-AE in determining correct clustering results. In this thesis, the family of the SP1M clustering algorithm is also extended to combine local spatial information in an image. The resultant algorithm, called sequential possibilistic local information one-mean (SPLI1M) is shown to have a better performance on image segmentation over the fuzzy local information c-means (FLICM), the possibilistic local information c-means (PLICM), and the family of SP1M without combing local spatial information in image.

Questions: Does fuzzy clustering provide an appropriate numerical framework to manage vegetation classifications? What is the best fuzzy clustering method to achieve this? Material: We used 531 releves from Catalonia (Spain), belonging to two syntaxonomic alliances of mesophytic and xerophytic montane pastures, and originally classified by experts into nine and 13 associations, respectively. Methods: We compared the performance of fuzzy C-means (FCM), noise clustering (NC) and possibilistic C-means (PCM) on four different management tasks: (1) assigning new releve data to existing types; (2) updating types incorporating new data; (3) defining new types with unclassified releves; and (4) reviewing traditional vegetation classifications. Results: As fuzzy classifiers, FCM fails to indicate when a given releve does not belong to any of the existing types; NC might leave too many releves unclassified; and PCM membership values cannot be compared. As unsupervised clustering methods, FCM is more sensitive than NC to transitional releves and therefore produces fuzzier classifications. PCM looks for dense regions in the space of species composition, but these are scarce when vegetation data contain many transitional releves. Conclusions: All three models have advantages and disadvantages, although the NC model may be a good compromise between the restricted FCM model and the robust but impractical PCM model. In our opinion, fuzzy clustering might provide a suitable framework to manage vegetation classifications using a consistent operational definition of vegetation type. Regardless of the framework chosen, national/regional vegetation classification panels should promote methodological standards for classification practices with numerical tools.

Possibilistic fuzzy c-means with partial supervision

Osteoarthritis is a chronic joint disease that occurs when the protective cartilage that cushions the ends of bones wears down over time and fails to be repaired. The common form of the disease is knee osteoarthritis while it can affect all body parts with joints, such as hands, ankles, hips, and spine. The major cause of knee osteoarthritis is the continuous depletion of its cartilage. During the diagnosis, machine learning is used because early prevention is necessary for proper treatment. This study, therefore, considers classification methods of Support Vector Machine (SVM) and clustering methods using fuzzy clusterings such as Fuzzy C-Means (FCM), Fuzzy Possibilistic C-Means (FPCM), and Fuzzy Possibilistic C-Means based on kernel (FPCMK) to analyze of knee osteoarthritis. SVM is a machine learning technique that works based on the principle of structural risk minimization (SRM) to obtain the best hyperplane to separate two or more classes in input space. Otherwise, the fuzzy clustering is to determine the value of a distance and to know and measure the similarity of each object to be observed. FPCMK uses the kernel Radial Base Function (RBF) in the fuzzy clustering method. The kernel function is applicable for handling non-separable data problems. This method will be compared to the level of the measured parameter; their accuracy, recall, precision, and f1 score. The greatest level of accuracy is generated from SVM with an accuracy value of 86.7%, then followed by FPCMK with an accuracy value of 85.5%.

Fuzzy c‐means (FCMs) with spatial constraints have been considered as an effective algorithm for image segmentation. The well‐known Gaussian mixture model (GMM) has also been regarded as a useful tool in several image segmentation applications. In this study, the authors propose a new algorithm to incorporate the merits of these two approaches and reveal some intrinsic relationships between them. In the authors model, the new objective function pays more attention on spatial constraints and adopts Gaussian distribution as the distance function. Thus, their model can degrade to the standard GMM as a special case. Our algorithm is fully free of the empirically pre‐defined parameters that are used in traditional FCM methods to balance between robustness to noise and effectiveness of preserving the image sharpness and details. Furthermore, in their algorithm, the prior probability of an image pixel is influenced by the fuzzy memberships of pixels in its immediate neighbourhood to incorporate the local spatial information and intensity information. Finally, they utilise the mean template instead of the traditional hidden Markov random field (HMRF) model for estimation of prior probability. The mean template is considered as a spatial constraint for collecting more image spatial information. Compared with HMRF, their method is simple, easy and fast to implement. The performance of their proposed algorithm, compared with state‐of‐the‐art technologies including extensions of possibilistic fuzzy c‐means (PFCM), GMM, FCM, HMRF and their hybrid models, demonstrates its improved robustness and effectiveness.

In the last decade, breast cancer has become the focus of world attention as this disease is one of the primary leading cause of death for women. Therefore, it is necessary to have the correct precautions and treatment. In previous studies, Fuzzy Kennel K-Medoid algorithm has been used for multi-class data. This paper proposes an algorithm to classify the high dimensional data of breast cancer using Fuzzy Possibilistic C-means (FPCM) and a new method based on clustering analysis using Normed Kernel Function-Based Fuzzy Possibilistic C-Means (NKFPCM). The objective of this paper is to obtain the best accuracy in classification of breast cancer data. In order to improve the accuracy of the two methods, the features candidates are evaluated using feature selection, where Laplacian Score is used. The results show the comparison accuracy and running time of FPCM and NKFPCM with and without feature selection.

Gray matter segmentation and delineation from positron emission tomography (PET) are a very essential requirement in medical applications due to low spatial resolution, presence of noise, and larger variability in the shape as well as in texture. PET images provide the functional details of the brain which are very much essential to detect brain disorders. The diagnosis of dementia, particularly in the early stages, are very much helpful with PET image processing. In this present work, we implement three essential steps, which significantly help during dementia analysis. The first step is skull stripping from PET image based on K-means and connected component analysis, the second step is to segment gray matter from skull-stripped PET image using fuzzy possibilistic C-means (FPCM), and the last step is to delineate the gray matter based on cellular automata. The Indian PET dataset collected from Apollo Gleneagles Hospitals, Kolkata, India, has been used in our experiment as a standard dataset. Experimental results show that the proposed method has achieved promising results for automatic segmentation and delineation of gray matter on PET brain images.

Clustered microcalcifications (MCs) in mammograms are an important early sign of breast cancer in women. Their accurate detection is important in computer-aided detection (CADe). In this paper, we integrated the possibilistic fuzzy c-means (PFCM) clustering algorithm and weighted support vector machine (WSVM) for the detection of MC clusters in full-field digital mammograms (FFDM). For each image, suspicious MC regions are extracted with region growing and active contour segmentation. Then geometry and texture features are extracted for each suspicious MC, a mutual information-based supervised criterion is used to select important features, and PFCM is applied to cluster the samples into two clusters. Weights of the samples are calculated based on possibilities and typicality values from the PFCM, and the ground truth labels. A weighted nonlinear SVM is trained. During the test process, when an unknown image is presented, suspicious regions are located with the segmentation step, selected features are extracted, and the suspicious MC regions are classified as containing MC or not by the trained weighted nonlinear SVM. Finally, the MC regions are analyzed with spatial information to locate MC clusters. The proposed method is evaluated using a database of 410 clinical mammograms and compared with a standard unweighted support vector machine (SVM) classifier. The detection performance is evaluated using response receiver operating (ROC) curves and free-response receiver operating characteristic (FROC) curves. The proposed method obtained an area under the ROC curve of 0.8676, while the standard SVM obtained an area of 0.8268 for MC detection. For MC cluster detection, the proposed method obtained a high sensitivity of 92 % with a false-positive rate of 2.3 clusters/image, and it is also better than standard SVM with 4.7 false-positive clusters/image at the same sensitivity.