Yukihiro Hamasuna

Fuzzy c-Means Clustering for Uncertain Data Using Quadratic Penalty-Vector Regularization

Clustering – defined as an unsupervised data-analysis classification transforming real-space information into data in pattern space and analyzing it – may require that data be represented by a set, rather than points, due to data uncertainty, e.g., measurement error margin, data regarded as one point, or missing values. These data uncertainties have been represented as interval ranges for which many clustering algorithms are constructed, but the lack of guidelines in selecting available distances in individual cases has made selection difficult and raised the need for ways to calculate dissimilarity between uncertain data without introducing a nearest-neighbor or other distance. The tolerance concept we propose represents uncertain data as a point with a tolerance vector, not as an interval, while this is convenient for handling uncertain data, tolerance-vector constraints make mathematical development difficult. We attempt to remove the tolerance-vector constraints using quadratic penaltyvector regularization similar to the tolerance vector. We also propose clustering algorithms for uncertain data considering optimization and obtaining an optimal solution to handle uncertainty appropriately.

Fuzzy c-Means for Data with Tolerance Defined as Hyper-Rectangle

The paper presents some new clustering algorithms which are based on fuzzy c-means. The algorithms can treat data with tolerance defined as hyper-rectangle. First, the tolerance is introduced into optimization problems of clustering. This is generalization of calculation errors or missing values. Next, the problems are solved and some algorithms are constructed based on the results. Finally, usefulness of the proposed algorithms are verified through numerical examples.

Sequential Extraction by Using Two Types of Crisp Possibilistic Clustering

Possibilistic clustering is well-known as one of the useful clustering methods because it is robust against noise or outlier in data. In the previous study, sparse possibilistic clustering and its variant has been proposed by using L1-regularization. These possibilistic clustering methods with L1-regularization are quite different from the viewpoint of membership function. Two types of new possibilistic approach with L1-regularization named crisp possibilistic clustering are proposed in this paper. Classification function of proposed methods which shows allocation rule in whole space and the way of sequential cluster extraction are also proposed. The effectiveness of proposed methods is, moreover, shown through numerical examples.

On fuzzy c-means clustering for uncertain data using quadratic regularization of penalty vectors

In recent years, data from many natural and social phenomena are accumulated into huge databases in the world wide network of computers. Thus, 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 and both of hard and fuzzy c-means clusterings are the most typical technique of clustering.

Semi-supervised Fuzzy c-Means Clustering Using Clusterwise Tolerance Based Pairwise Constraints

Recently, semi-supervised clustering has been remarked and discussed in many research fields. In semi-supervised clustering, prior knowledge or information are often formulated as pairwise constraints, that is, must-link and cannot-link. Such pairwise constraints are frequently used in order to improve clustering properties. In this paper, we will propose a new semi-supervised fuzzy c-means clustering by using clusterwise tolerance and pairwise constraints. First, the concept of clusterwise tolerance and pairwise constraints are introduced. Second, the optimization problem of fuzzy cmeans clustering using clusterwise tolerance based pairwise constraint is formulated. Especially, must-link constraint is considered and introduced as pairwise constraints. Third, a new clustering algorithm is constructed based on the above discussions. Finally, the effectiveness of proposed algorithm is verified through numerical examples.

Two Clustering Algorithms for Data with Tolerance based on Hard c-Means

Two clustering algorithms that handle data with tolerance are proposed. One is based on hard c-means while the other uses the learning vector quantization. The concept of the tolerance includes. First, the concept of tolerance which implies errors, ranges and the loss of attribute of data is described. Optimization problems that take the tolerance into account are formulated. Since the Kuhn-Tucker condition give a unique and explicit optimal solution, an alternate minimization algorithm and a learning algorithm are constructed. Moreover, the effectiveness of the proposed algorithms is verified through numerical examples.

On sparse possibilistic clustering with crispness — Classification function and sequential extraction

In addition to fuzzy c-means clustering, possibilistic clustering is well-known as one of the useful techniques because it is robust against noise in data. Especially sparse possibilistic clustering is quite different from other possibilistic clustering methods in the point of membership function. We propose a way to induce the crispness in possibilistic clustering by using L1-regularization and show classification function of sparse possibilistic clustering with crispness for understanding allocation rule. We, moreover, show the way of sequential extraction by proposed method. After that, we show the effectiveness of the proposed method through numerical examples.

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 semi-supervised fuzzy c-means clustering with clusterwise tolerance by opposite criteria

This paper presents a new semi-supervised fuzzy c-means clustering for data with clusterwise tolerance by opposite criteria. In semi-supervised clustering, pairwise constraints, that is, must-link and cannot-link, are frequently used in order to improve clustering performances. From the viewpoint of handling pairwise constraints, a new semi-supervised fuzzy c-means clustering is proposed by introducing clusterwise tolerance-based pairwise constraints. First, a concept of clusterwise tolerance-based pairwise constraints is introduced. Second, the optimization problems of the proposed method are formulated. Especially, must-link and cannot-link are handled by opposite criteria in our proposed method. Third, a new clustering algorithm is constructed based on the above discussions. Finally, the effectiveness of the proposed algorithm is verified through numerical examples.

On tolerant fuzzy c -means clustering and tolerant possibilistic clustering

This paper presents two new types of clustering algorithms by using tolerance vector called tolerant fuzzy c-means clustering and tolerant possibilistic clustering. In the proposed algorithms, the new concept of tolerance vector plays very important role. The original concept is developed to handle data flexibly, that is, a tolerance vector attributes not only to each data but also each cluster. Using the new concept, we can consider the influence of clusters to each data by the tolerance. First, the new concept of tolerance is introduced into optimization problems. Second, the optimization problems with tolerance are solved by using Karush–Kuhn–Tucker conditions. Third, new clustering algorithms are constructed based on the optimal solutions for clustering. Finally, the effectiveness of the proposed algorithms is verified through numerical examples and its fuzzy classification function.