Yasunori Endo

On even-sized clustering algorithm based on optimization

Clustering methods to divide a data set into some clusters of which the size is more than a given constant K, are very useful in many applications. The methods are called K-member clustering (KMC). As a natural result, clustering methods to divide a data set into even-sized clusters can be considered. However, there are no algorithms of such methods based on optimization. That is why the conventional algorithms often output inadequate results. Therefore we should consider an algorithm based on optimization. In this paper, we propose evensized clustering algorithm using simplex method which is one of optimization method, and verify the proposed method through some numerical examples.

On Various Types of Even-Sized Clustering Based on Optimization

Clustering is a very useful tool of data mining. A clustering method which is referred to as K-member clustering is to classify a dataset into some clusters of which the size is more than a given constant K. The K-member clustering is useful and it is applied to many applications. Naturally, clustering methods to classify a dataset into some even-sized clusters can be considered and some even-sized clustering methods have been proposed. However, conventional even-sized clustering methods often output inadequate results. One of the reasons is that they are not based on optimization. Therefore, we proposed Even-sized Clustering Based on Optimization (ECBO) in our previous study. The simplex method is used to calculate the belongingness of each object to clusters in ECBO. In this study, ECBO is extended by introducing some ideas which were introduced in k-means or fuzzy c-means to improve problems of initial-value dependence, robustness against outliers, calculation cost, and nonlinear boundaries of clusters. Moreover, we reconsider the relation between the dataset size, the cluster number, and K in ECBO.

On Some Models of Objective-Based Rough Clustering

Fuzzy clustering which can implement flexible classification is very useful but sometimes calculates the degrees of belongingness of an objects to a cluster too exactly. To solve this problem, a new clustering method called rough k-means (RKM) is proposed by Lingras et al. RKM which is an extended method by using rough set representation can classify more roughly than fuzzy clustering without lack of flexibility. Generally, non-hierarchical clustering methods including RKM are strongly dependent on initial values. Therefore we need some indicator to evaluate outputs of a method. Many methods define an objective function as the indicator. However conventional rough clustering methods including RKM are not based on an objective function. Thus we cannot evaluate the outputs of them. We solve this problem by defining objective functions clearly and proposing some new objective-based rough clustering methods. In this paper, we introduce these methods classified into three categories, that is metric model, non-metric model and regression model. Moreover we discuss the features of these methods.

A Note on Even-Sized Clustering Based on Optimization

Some clustering methods to classify a dataset into some clusters of which the size is more than a constant K have been proposed until now. The methods are called K-member clustering and very useful for many applications. It is natural to consider clustering methods to classify a dataset into even-sized clusters, and actually, such methods have been also proposed. However, they often output inadequate results. It is considered that the reason is that they are not based on optimization. Therefore, in the previous study, we proposed Even-sized Clustering Based on Optimization (ECBO). We improved the clustering results by the simplex method to calculate the membership grade. In this study, we propose some types of extended ECBO by introducing some concept of medoid and kernel to improve ECBO.

Fuzzy non-metric model for data with tolerance and its application to incomplete data clustering

Clustering is a technique of unsupervised classification. The methods are classified into two types, one is hierarchical and the other is non-hierarchical. Fuzzy non-metric model (FNM) is a representative method of non-hierarchical clustering. FNM is very useful because belongingness or the membership degree of each datum to each cluster is calculated directly from dissimilarities between data, and cluster centers are not used. However FNM cannot handle data with uncertainty, called uncertain data, e.g. incomplete data, or data which have errors. In order to handle such data, concept of tolerance vector has been proposed. The clustering methods using the concept can handle the uncertain data in the framework of optimization, e.g. fuzzy c-means for data with tolerance (FCM-T). In this paper, we will first propose new clustering algorithm to apply the concept of tolerance to FNM, called fuzzy non-metric model for data with tolerance (FNM-T). Second, we will show that the proposed algorithm handle incomplete data sets. Third, we will verify the effectiveness of the proposed algorithm in comparison with conventional ones for incomplete data sets through some numerical examples.

On new sequential hard c-medoids

This paper presents a new sequential cluster extraction algorithm based on hard c-medoids clustering. The word sequential cluster extraction means that the algorithm extract one cluster at a time. The hard c-medoids is one of the variants of hard c-means clustering. The cluster medoid which is referred to as representative of each cluster is an object in hard c-medoids. The sequential clustering algorithms are based on Daves noise clustering approach. A characteristic parameter which is called noise parameter is used in noise clustering. We construct a new sequential hard c-medoids algorithm by considering the noise parameter as a variables in optimization problem. First, the optimization problem of new sequential hard c-medoids clustering is introduced. Next, the sequential clustering algorithm is constructed based on the optimization problem. Moreover, the effectiveness of proposed method is shown through numerical experiments.

Semi-Supervised Hard and Fuzzy c-Means with Assignment Prototype Term

Semi-supervised learning is an important task in the field of data mining. Pairwise constraints such as must-link and cannot-link are used in order to improve clustering properties. This paper proposes a new type of semi-supervised hard and fuzzy c-means clustering with assignment prototype term. The assignment prototype term is based on the Windham’s assignment prototype algorithm which handles pairwise constraints between objects in the proposed method. First, an optimization problem of the proposed method is formulated. Next, a new clustering algorithm is constructed based on the above discussions. Moreover, the effectiveness of the proposed method is shown through numerical experiments.

Hard and Fuzzy c -means Algorithms with Pairwise Constraints by Non-metric Terms

Recently, semi-supervised clustering has been focused, e.g., Refs. [2–5]. The semi-supervised clustering algorithms improve clustering results by incorporating prior information with the unlabeled data. This paper proposes three new clustering algorithms with pairwise constraints by introducing non-metric term to objective functions of the well-known clustering algorithms. Moreover, its effectiveness is verified through some numerical examples.

A Note on Objective-Based Rough Clustering with Fuzzy-Set Representation

Clustering is a method of data analysis. Rough k-means (RKM) by Lingras et al. is one of rough clustering algorithms [3]. The method does not have a clear indicator to determine the most appropriate result because it is not based on objective function. Therefore we proposed a rough clustering algorithm based on optimization of an objective function [7]. This paper will propose a new rough clustering algorithm based on optimization of an objective function with fuzzy-set representation to obtain better lower approximation, and estimate the effectiveness through some numerical examples.

On new sequential hard c-means and its kernelization

This paper presents a new sequential clustering algorithm based on sequential hard c-means clustering. The word sequential cluster extraction means that the algorithm extract one cluster at a time. The sequential hard c-means is one of the typical and conventional sequential clustering methods. The proposed new sequential clustering algorithm is based on Daves noise clustering approach. A characteristic parameter which is called noise parameter is applied in Daves approach. We construct a new sequential hard c-means algorithm by introducing another new parameter which controls a number of extracting objects and considering the noise parameter as a variables in optimization problem. First, the optimization problem of new sequential hard c-means clustering is introduced. Next, the sequential clustering algorithm and its kernelization are constructed based on above optimization problem. Moreover, the effectiveness of proposed method is shown through numerical experiments.