This paper considers competitive learning networks using three types of hard, soft, and fuzzy learning schemes. The hard competitive learning algorithm is with the winner-take-all. The soft competition learning algorithm is with a stochastic relaxation technique using the Gibbs distribution as a dynamic neighborhood function. The fuzzy competition learning algorithm is with a fuzzy relaxation technique using fuzzy membership functions as kernel type neighborhood interaction functions. Some numerical examples are made for these three types of competitive learning schemes. The numerical results show that the fuzzy learning has better performance than hard and soft learning under the normal mixture data. We then present an application to magnetic resonance image segmentation. A real case of ophthalmology recommended by a neurologist with MR image data is examined in this paper. These competitive learning algorithms are used in segmenting the ophthalmological MRI data for reducing medical image noise effects with a learning mechanism. Based on the segmentation results, the fuzzy learning gives better performance than hard and soft learning so that the fuzzy competitive learning algorithm is recommended for use in MRI segmentation as an aid for support diagnoses. © 2010 Wiley Periodicals, Inc.

In the traditional fuzzy c-means clustering algorithm, nearly no data points have a membership value one. Oumlzdemir and Akarum proposed a partition index maximization (PIM) algorithm which allows the data points can whole belonging to one cluster. This modification can form a core for each cluster and data points inside the core will have membership value {0,1}. In this paper, we will discuss the parameter selection problems and robust properties of the PIM algorithm.

We know that the partition coefficient index has the monotonic tendency with cluster number c. Moreover, they always select the smallest cluster number c=2 as a optimal cluster number estimate when data contains some noise points. In this paper, we will discuss this problem by defining the validity measure of each single data point. We then define the singular point that has equal memberships to each cluster. By analyzing the influence of the singular point on the validity index, we can then give some guidelines for designing the fuzzy c-partitions based validity indexes that can avoid the influence of the noise.

In the fuzzy c-means (FCM) clustering algorithm, almost none of the data points have a membership value of 1. Moreover, noise and outliers may cause difficulties in obtaining appropriate clustering results from the FCM algorithm. The embedding of FCM into switching regressions, called the fuzzy c-regressions (FCRs), still has the same drawbacks as FCM. In this paper, we propose the alpha-cut implemented fuzzy clustering algorithms, referred to as FCMalpha, which allow the data points being able to completely belong to one cluster. The proposed FCMalpha algorithms can form a cluster core for each cluster, where data points inside a cluster core will have a membership value of 1 so that it can resolve the drawbacks of FCM. On the other hand, the fuzziness index m plays different roles for FCM and FCMalpha. We find that the clustering results obtained by FCMalpha are more robust to noise and outliers than FCM when a larger m is used. Moreover, the cluster cores generated by FCMalpha are workable for various data shape clusters, so that FCMalpha is very suitable for embedding into switching regressions. The embedding of FCMalpha into switching regressions is called FCRalpha. The proposed FCRalpha provides better results than FCR for environments with noise or outliers. Numerical examples show the robustness and the superiority of our proposed methods.

In fuzzy clustering, the fuzzy c-mean (FCM) is a most used algorithm. The embedding of FCM into switching regressions, called the fuzzy c-regression (FCR), was proposed by Hathaway and Bezdek in 1993. However, these FCRs always heavily depend on the initial values. In this paper, we propose mountain c-regressions (MCR) to solve the initial-value problem where the MCR is based on the modified mountain clustering method. We first transform the data set into a parameter space and then take a random sample from the transformed data set. We implement the modified mountain clustering on the random samples for switching regressions. The proposed MCR can form c estimated regression models for switching regression data sets without giving initials. Several examples show the accuracy and effectiveness of the proposed MCR method.

This paper presents an alternating optimization clustering procedure called a similarity-based clustering method (SCM). It is an effective and robust approach to clustering on the basis of a total similarity objective function related to the approximate density shape estimation. We show that the data points in SCM can self-organize local optimal cluster number and volumes without using cluster validity functions or a variance-covariance matrix. The proposed clustering method is also robust to noise and outliers based on the influence function and gross error sensitivity analysis. Therefore, SCM exhibits three robust clustering characteristics: 1) robust to the initialization (cluster number and initial guesses), 2) robust to cluster volumes (ability to detect different volumes of clusters), and 3) robust to noise and outliers. Several numerical data sets and actual data are used in the SCM to show these good aspects. The computational complexity of SCM is also analyzed. Some experimental results of comparing the proposed SCM with the existing methods show the superiority of the SCM method.

Uncertainty always exists in nature and real systems. It is known that probability has been used traditionally in modeling uncertainty. Since a belief function was proposed as an another type of measuring uncertainty, Dempster-Shafer theory (DST) has been widely studied and applied in diverse areas. Because of the advent of computer technology, the representation of human knowledge can be processed by a computer in complex systems. The analysis of fuzzy data becomes increasingly important. Up to date, there are several generalizations of DST to fuzzy sets proposed in the literature. In this article, we propose another generalization of belief function, plausibility function, and Dempster's combinational rule to fuzzy sets. We then make the comparisons of the proposed extension with some existing generalizations and show its effectiveness. © 2003 Wiley Periodicals, Inc.

The alternative fuzzy c-means (AFCM) clustering algorithm proposed by Wu and Yang (2001) has shown more robustness than the fuzzy c-means (FCM) on the basis of the robust statistic and the influence function. We propose a possibilistic type of AFCM by relaxing the restriction /spl Sigma//sub i=1//sup c/ /spl mu//sub i/(x)=1 for all data points x. The resulting cluster memberships constitute a possibilistic partition which is different to a fuzzy partition from AFCM. The comparisons of the proposed method to FCM, AFCM and possibilistic c-means are made.

This paper presents a new validity index for fuzzy clustering called a partition separation (PS) index. It uses the factors of a normalized partition coefficient and an exponential separation measure. According to the numerical comparisons with the other five cluster-validity indexes, the proposed PS index shows its high ability to produce a valid clustering of the data set.