基于模糊聚类的模糊神经网络控制

对于复杂的非线性离散系统,提出将模糊聚类算法同神经网络相结合,使用衡量聚类有效性的S函数确定模糊规则数目,进而确定模糊神经网络的结构;控制器的设计应用LMI方法。以典型的非线性系统二级倒立摆为例,在Matlab中进行仿真实验,结果表明,基于聚类算法的神经网络控制能够在较大范围的初始状态下使系统获得稳定。     

Cluster validity for fuzzy criterion clustering

In this paper, we define a validity measure for fuzzy criterion clustering which is a novel approach to fuzzy clustering that in addition to being non-distance-based, addresses the cluster validity problem. The model is then recast as a bilevel fuzzy criterion clustering problem. We propose an algorithm for this model that solves both the validity and clustering problems. Our approach is validated via some sample problems.     

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.     

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.     

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.     

基于层次分析法的模糊分类优选模型

不同的模糊分类算法在同一个数据集合上常会产生不同的模糊分类．究竟哪种方法最能揭示数据的真实结构,对此,以模糊分类有效性指标为评价指标,应用层次分析法对各模糊分类进行综合评价,建立了一个模糊分类优选模型．大量实验表明,该优选模型所选出的最优模糊分类,其模式识别率高,能揭示数据的真实结构．     

A multiobjective spatial fuzzy clustering algorithm for image segmentation

This article describes a multiobjective spatial fuzzy clustering algorithm for image segmentation. To obtain satisfactory segmentation performance for noisy images, the proposed method introduces the non-local spatial information derived from the image into fitness functions which respectively consider the global fuzzy compactness and fuzzy separation among the clusters. After producing the set of non-dominated solutions, the final clustering solution is chosen by a cluster validity index utilizing the non-local spatial information. Moreover, to automatically evolve the number of clusters in the proposed method, a real-coded variable string length technique is used to encode the cluster centers in the chromosomes. The proposed method is applied to synthetic and real images contaminated by noise and compared with k-means, fuzzy c-means, two fuzzy c-means clustering algorithms with spatial information and a multiobjective variable string length genetic fuzzy clustering algorithm. The experimental results show that the proposed method behaves well in evolving the number of clusters and obtaining satisfactory performance on noisy image segmentation.     

Algorithms of fuzzy clustering with partial supervision

The paper is concerned with methods of fuzzy clustering based minimization of a scalar performance index with the aid of some labelled patterns. Some modifications of the performance index that take into account the results of partial supervision are proposed. The results of the algorithms introduced are exemplified by the use of numerical examples (Gustafson's cross and EKG data set).     

Note on the relationship between probabilistic and fuzzy clustering

In this short communication, based on Renyi entropy measure, a new Renyi information based clustering algorithm A is presented. Algorithm A and the well-known fuzzy clustering algorithm FCM have the same clustering track. This fact builds the very bridge between probabilistic clustering and fuzzy clustering, and fruitful research results on Renyi entropy measure may help us to further understand the essence of fuzzy clustering.This work was supported in part by the RGC CERG grant under project Hong Kong PolyU 5065/98E.     

Note on the relationship between probabilistic and fuzzy clustering

In this short communication, based on Renyi entropy measure, a new Renyi information based clustering algorithm A is presented. Algorithm A and the well-known fuzzy clustering algorithm FCM have the same clustering track. This fact builds the very bridge between probabilistic clustering and fuzzy clustering, and fruitful research results on Renyi entropy measure may help us to further understand the essence of fuzzy clustering.This work was supported in part by the RGC CERG grant under project HongKong PolyU 5065/98E     

Evolutionary fuzzy clustering of relational data

This paper is concerned with the computational efficiency of fuzzy clustering algorithms when the data set to be clustered is described by a proximity matrix only (relational data) and the number of clusters must be automatically estimated from such data. A fuzzy variant of an evolutionary algorithm for relational clustering is derived and compared against two systematic (pseudo-exhaustive) approaches that can also be used to automatically estimate the number of fuzzy clusters in relational data. An extensive collection of experiments involving 18 artificial and two real data sets is reported and analyzed.     

Partitioning elements of the Periodic Table via fuzzy clustering technique

In this paper we present fuzzy clustering results of the elements of the Periodic Table according to several physical properties related directly to their electronic structure (ionization potential, cohesion energy, conductivity, atomic number). We find that the resulting classes reproduce the Periods of the Periodic Table. Using additional physical properties not directly related to the electronic structure of the elements (compressibility, density) in the fuzzy clustering procedure, we find that the elements of the Periodic Table are partitioned in another way.The authors are grateful to the referees for their valuable suggestions that contributed to improve the presentation of their work.     

Spatial dependence-based fuzzy regression clustering

Fuzzy clustering based regression analysis is a novel hybrid approach to capture the linear structure while considering the classification structure of the measurement. Using the concept that weights provided via the fuzzy degree of clustering, some regression models have been proposed in literature. In these models, membership values derived from clustering or some weights obtained from geometrical functions are employed as the weights of regression system. This paper addresses a weighted fuzzy regression analysis based on spatial dependence measure of the memberships. By the methodology presented in this paper, the relative weights are used in fuzzy regression models instead of direct membership values or their geometrical transforms. The experimental studies indicate that the spatial dependence based analyses yield more reliable results to show the correlation of the independent variables into the dependent variable. In addition, it has been observed that spatial dependence based models have high estimation and generalization capacities.     

A fuzzy clustering algorithm enhancing local model interpretability

In this work, simple modifications on the cost index of particular local-model fuzzy clustering algorithms are proposed in order to improve the readability of the resulting models. The final goal is simultaneously providing local linear models (reasonably close to the plant’s Jacobian) and clustering in the input space so that desirable characteristics (regarding final model accuracy, and convexity and smoothness of the cluster membership functions) are improved with respect to other proposals in literature. Some examples illustrate the proposed approach.     

A method of relational fuzzy clustering based on producing feature vectors using FastMap

The first stage of organizing objects is to partition them into groups or clusters. The clustering is generally done on individual object data representing the entities such as feature vectors or on object relational data incorporated in a proximity matrix.This paper describes another method for finding a fuzzy membership matrix that provides cluster membership values for all the objects based strictly on the proximity matrix. This is generally referred to as relational data clustering. The fuzzy membership matrix is found by first finding a set of vectors that approximately have the same inter-vector Euclidian distances as the proximities that are provided. These vectors can be of very low dimension such as 5 or less. Fuzzy c-means (FCM) is then applied to these vectors to obtain a fuzzy membership matrix. In addition two-dimensional vectors are also created to provide a visual representation of the proximity matrix. This allows comparison of the result of automatic clustering to visual clustering. The method proposed here is compared to other relational clustering methods including NERFCM, Rouben’s method and Windhams A-P method. Various clustering quality indices are also calculated for doing the comparison using various proximity matrices as input. Simulations show the method to be very effective and no more computationally expensive than other relational data clustering methods. The membership matrices that are produced by the proposed method are less crisp than those produced by NERFCM and more representative of the proximity matrix that is used as input to the clustering process.     

Multi-variable fuzzy forecasting based on fuzzy clustering and fuzzy rule interpolation techniques

In this paper, we present a new method for multi-variable fuzzy forecasting based on fuzzy clustering and fuzzy rule interpolation techniques. First, the proposed method constructs training samples based on the variation rates of the training data set and then uses the training samples to construct fuzzy rules by making use of the fuzzy C-means clustering algorithm, where each fuzzy rule corresponds to a given cluster. Then, we determine the weight of each fuzzy rule with respect to the input observations and use such weights to determine the predicted output, based on the multiple fuzzy rules interpolation scheme. We apply the proposed method to the temperature prediction problem and the Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) data. The experimental results show that the proposed method produces better forecasting results than several existing methods.     

Interactive clustering and merging with a new fuzzy expected value

Major problems exist in both crisp and fuzzy clustering algorithms. The fuzzy c-means type of algorithms use weights determined by a power m of inverse distances that remains fixed over all iterations and over all clusters, even though smaller clusters should have a larger m. Our method uses a different “distance” for each cluster that changes over the early iterations to fit the clusters. Comparisons show improved results. We also address other perplexing problems in clustering: (i) find the optimal number K of clusters; (ii) assess the validity of a given clustering; (iii) prevent the selection of seed vectors as initial prototypes from affecting the clustering; (iv) prevent the order of merging from affecting the clustering; and (v) permit the clusters to form more natural shapes rather than forcing them into normed balls of the distance function. We employ a relatively large number K of uniformly randomly distributed seeds and then thin them to leave fewer uniformly distributed seeds. Next, the main loop iterates by assigning the feature vectors and computing new fuzzy prototypes. Our fuzzy merging then merges any clusters that are too close to each other. We use a modified Xie-Bene validity measure as the goodness of clustering measure for multiple values of K in a user-interaction approach where the user selects two parameters (for eliminating clusters and merging clusters after viewing the results thus far). The algorithm is compared with the fuzzy c-means on the iris data and on the Wisconsin breast cancer data.     

Type-2 fuzzy neural networks with fuzzy clustering and differential evolution optimization

In many real-world problems involving pattern recognition, system identification and modeling, control, decision making, and forecasting of time-series, available data are quite often of uncertain nature. An interesting alternative is to employ type-2 fuzzy sets, which augment fuzzy models with expressive power to develop models, which efficiently capture the factor of uncertainty. The three-dimensional membership functions of type-2 fuzzy sets offer additional degrees of freedom that make it possible to directly and more effectively account for model’s uncertainties. Type-2 fuzzy logic systems developed with the aid of evolutionary optimization forms a useful modeling tool subsequently resulting in a collection of efficient “If-Then” rules.The type-2 fuzzy neural networks take advantage of capabilities of fuzzy clustering by generating type-2 fuzzy rule base, resulting in a small number of rules and then optimizing membership functions of type-2 fuzzy sets present in the antecedent and consequent parts of the rules. The clustering itself is realized with the aid of differential evolution.Several examples, including a benchmark problem of identification of nonlinear system, are considered. The reported comparative analysis of experimental results is used to quantify the performance of the developed networks.