Naohiko Kinoshita

Objective‐Based Rough c‐Means Clustering

Conventional clustering algorithms classify a set of objects into some clusters with clear boundaries, that is, one object must belong to one cluster. However, many objects belong to more than one cluster in real world since the boundaries of clusters generally overlap with each other. Fuzzy‐set representation of clusters makes it possible for each object to belong to more than one cluster. On the other hand, the fuzzy degree is sometimes regarded as too descriptive for interpreting clustering results. Rough‐set representation could deal with such cases. Clustering based on rough set could provide a solution that is less restrictive than conventional clustering and less descriptive than fuzzy clustering. This paper proposes a rough clustering algorithm which is based on optimization of an objective function and the calculation formula of cluster centers is the same as one by Lingras et al. Moreover, it shows effectiveness of our proposed clustering algorithm in comparison with other algorithms.

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.

Controlled-sized clustering based on optimization

Clustering is one of unsupervised classification method, that is, it classifies a data set into some clusters without any external criterion. Typical clustering methods, e.g. k-means (KM) and fuzzy c-means (FCM) are constructed based on optimization of the given objective function. Many clustering methods as well as KM and FCM are formulated as optimization problems with typical objective functions and constraints. The objective function itself is also an evaluation guideline of results of clustering methods. Considering together with its theoretical extensibility, there is the great advantage to construct clustering methods in the framework of optimization. From the viewpoint of optimization, some of the authors proposed an even-sized clustering method based on optimization (ECBO), which is with strengthened constraints of cluster size, and constructed some variations of ECBO. The constraint considered in ECBO is that each cluster size is K or K + 1. ECBO is based on KM and its algorithm is constructed as iterative optimization. The belongingness of each object to clusters are calculated by the simplex method in each iteration. The numerical experiments show that ECBO has higher classification accuracy than other similar clustering methods. It is considered that ECBO has the advantage in the viewpoint of clustering accuracy, cluster size, and optimization framework than other similar methods. However, the constraint of cluster sizes of ECBO is strict so that it may be inconvenient in case that the partition results, of which each cluster size need not be strictly even, but uneven, is desirable. Moreover, it is expected that new clustering algorithms of which each cluster size can be controlled can deal with more various datasets. In this paper, we first propose two new clustering algorithms based on ECBO. Each cluster size can be controlled in the proposed algorithms. Next, we estimate the new clustering algorithms through some numerical experiments.

EM-Based Clustering Algorithm for Uncertain Data

In recent years, advanced data analysis techniques to get valuable knowledge from data using computing power of today are required. Clustering is one of the unsupervised classification technique of the data analysis. Information on a real space is transformed to data in a pattern space and analyzed in clustering. However, the data should be often represented not by a point but by a set because of uncertainty of the data, e.g., measurement error margin, data that cannot be regarded as one point, and missing values in data.

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.

Rough c-Regression Based on Optimization of Objective Function

Clustering which is one of the pattern recognition methods is a technique automatically classifying data into some clusters. Various types of clustering are divided broadly into hierarchical and non-hierarchical clustering and crisp and fuzzy set theories have been applied to non-hierarchical clustering. Recently, clustering based on rough set theory has been attracted. Rough clustering represents a cluster by using two layers, i.e., upper and lower approximations. This paper proposes a c-regression method based on rough set representation which does regression analysis and clustering at the same time. Moreover, its effectiveness is shown through numerical examples.

On objective-based rough c-means clustering

Conventional clustering algorithms classify a set of objects into some clusters with clear boundaries, that is, one object must belong to one cluster. However, many objects belong to more than one cluster in real world, since the boundaries of clusters overlap with each other. Fuzzy set representation of clusters makes it possible for each object to belong to more than one cluster. On the other hand, the fuzzy degree sometimes may be too descriptive for interpreting clustering results. Rough set representation could deal with such cases. Clustering based on rough set could provide a solution that is less restrictive than conventional clustering and less descriptive than fuzzy clustering. This paper proposes a rough clustering algorithm which is based on optimization of an objective function and the calculation formula of cluster centers is the same as one by Lingras. Moreover, it shows effectiveness of our proposed clustering algorithm in comparison with other algorithms.

On various types of controlled-sized clustering based on optimization

Clustering is one of unsupervised classification method, that is, it classifies a data set into some clusters without any external criterion. Typical clustering methods, e.g. k-means (KM) or fuzzy c-means (FCM) are constructed based on optimization of the given objective function. Many clustering methods as well as KM and FCM are formulated as optimization problems with typical objective functions and constraints. The objective function itself is also an evaluation guideline of results of clustering methods. Considered together with its theoretical extensibility, there is the great advantage to construct clustering methods in the framework of optimization. From the viewpoint of optimization, some of the authors proposed an Even-sized Clustering method Based on Optimization (ECBO), which is with tight constraints of cluster size, and constructed some variations of ECBO. The constraint considered in ECBO is that each cluster size is K or K + 1, and the belongingness of each object to clusters is calculated by the simplex method in each iteration. It is considered that ECBO has the advantage in the viewpoint of clustering accuracy, cluster size, and optimization framework than other similar methods. However, the constraint of cluster sizes of ECBO is tight in the meaning of cluster size so that it may be inconvenient in case that some extra margin of cluster size is allowed. Moreover, it is expected that new clustering algorithms in which each cluster size can be controlled deal with more various datasets. From the above view point, we proposed two new clustering algorithms based on ECBO. One is COntrolled-sized Clustering Based on Optimization (COCBO), and the other is an extended COCBO, which is referred to as COntrolled-sized Clustering Based on Optimization++ (COCBO++). Each cluster size can be controlled in the algorithms. However, these algorithms have some problems. In this paper, we will describe various types of COCBO to solve the above problems and estimate the methods in some numerical examples.

Various Types of Objective-Based Rough Clustering

Conventional clustering algorithms classify a set of objects into some clusters with clear boundaries, that is, one object must belong to one cluster. However, many objects belong to more than one cluster in real world since the boundaries of clusters generally overlap with each other. Fuzzy set representation of clusters makes it possible for each object to belong to more than one cluster. On the other hand, it is pointed out that the fuzzy degree is sometimes regarded as too descriptive for interpreting clustering results. Instead of fuzzy representation, rough set one could deal with such cases. Clustering based on rough set could provide a solution that is less restrictive than conventional clustering and less descriptive than fuzzy clustering. Therefore, Lingras et al. (Lingras and Peters, Wiley Interdiscip Rev: Data Min Knowl Discov 1(1):64–72, 1207–1216, 2011, [1] and Lingras and West, J Intell Inf Syst 23(1):5–16, 2004, [2]) proposed a clustering method based on rough set, rough K-means (RKM). RKM is almost only one algorithm inspired by KM and some assumptions of RKM are very natural, however it is not useful from the viewpoint that the algorithm is not based on any objective functions. Outputs of non-hierarchical clustering algorithms strongly depend on initial values and the “better” output among many outputs from different initial values should be chosen by comparing the value of the objective function of the output with each other. Therefore the objective function plays very important role in clustering algorithms. From the standpoint, we have proposed some rough clustering algorithms based on objective functions. This paper shows such rough clustering algorithms which is based on optimization of an objective function.