Yuchi Kanzawa

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 Algorithms for Data with Tolerance Based on Opposite Criterions

In this paper, two new clustering algorithms are proposed for the data with some errors. In any of these algorithms, the error is interpreted as one of decision variables — called “tolerance” — of a certain optimization problem like the previously proposed algorithm, but the tolerance is determined based on the opposite criterion to its corresponding previously proposed one. Applying our each algorithm together with its corresponding previously proposed one, a reliability of the clustering result is discussed. Through some numerical experiments, the validity of this paper is discussed.

Fuzzy c-Means Algorithms for Data with Tolerance Using Kernel Functions

In this paper, two new clustering algorithms based on fuzzy c-means for data with tolerance using kernel functions are proposed. Kernel functions which map the data from the original space into higher dimensional feature space are introduced into the proposed algorithms. Nonlinear boundary of clusters can be easily found by using the kernel functions. First, two clustering algorithms for data with tolerance are introduced. One is based on standard method and the other is on entropy-based one. Second, the tolerance in feature space is discussed taking account into soft margin algorithm in Support Vector Machine. Third, two objective functions in feature space are shown corresponding to two methods, respectively. Fourth, Karush-Kuhn-Tucker conditions of two objective functions are considered, respectively, and these conditions are re-expressed with kernel functions as the representation of an inner product for mapping from the original pattern space into a higher dimensional feature space. Fifth, two iterative algorithms are proposed for the objective functions, respectively. Through some numerical experiments, the proposed algorithms are discussed.

Indefinite kernel fuzzy c-means clustering algorithms

This paper proposes two types of kernel fuzzy c-means algorithms with an indefinite kernel. Both algorithms are based on the fact that the relational fuzzy c-means algorithm is a special case of the kernel fuzzy c-means algorithm. The first proposed algorithm adaptively updated the indefinite kernel matrix such that the dissimilarity between each datum and each cluster center in the feature space is non-negative, instead of subtracting the minimal eigenvalue of the given kernel matrix as its preprocess. This derivation follows the manner in which the non-Euclidean relational fuzzy c-means algorithm is derived from the original relational fuzzy c-means one. The second proposed method produces the memberships by solving the optimization problem in which the constraint of non-negative memberships is added to the one of K-sFCM. This derivation follows the manner in which the non-Euclidean fuzzy relational clustering algorithm is derived from the original relational fuzzy c-means one. Through a numerical example, the proposed algorithms are discussed.

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.

On Possibilistic Clustering Methods Based on Shannon/Tsallis-Entropy for Spherical Data and Categorical Multivariate Data

In this paper, four possibilistic clustering methods are proposed. First, we propose two possibilistic clustering methods for spherical data — one based on Shannon entropy, and the other on Tsallis entropy. These methods are derived by subtracting the cosine correlation between an object and a cluster center from 1, to obtain the object-cluster dissimilarity. These methods are derived from the proposed spherical data methods by considering analogies between the spherical and categorical multivariate fuzzy clustering methods, in which the fuzzy methods’ object-cluster similarity calculation is modified to accommodate the proposed possibilistic methods. The validity of the proposed methods is verified through numerical examples.

On Bezdek-type fuzzy clustering for categorical multivariate data

In this study, five co-clustering algorithms based on Bezdek-type fuzzification of fuzzy clustering are propsoed for categorical multivariate data. The algorithms are motivated the fact that, there are only two fuzzy co-clustering methods - entropy-regularization and quadratic regularization - whereas there are three fuzzy clustering methods for vectorial data: entropy-regularization, quadratic regularization, and Bezdek-type fuzzification. The first algorithm proposed forms the basis of other two algorithms. By interpreting the first algorithm as a variant of a maximizing model of fuzzy multi-medoids, the second algorithm, a spectral clustering approach is obtained. Further, by slightly revising the objective function of the first algorithm, the third algorithm, another spectral clustering approach, is also obtained. The fourth algorithm is obtained by Bezdek-type fuzzification for row-membership whereas entropy-regularization for column-mebership. The fifth algorithm is a spectral clustering approach to the fourth algorithm. Numerical examples demonstrate that the proposed algorithms can produce satisfactory results when suitable parameter values are selected.

On Kernelization for a Maximizing Model of Bezdek-Like Spherical Fuzzy c-Means Clustering

In this study, we propose three modifications for a maximizing model of spherical Bezdek-type fuzzy c-means clustering (msbFCM). First, we kernelize msbFCM (K-msbFCM). The original msbFCM can only be applied to objects on the first quadrant of the unit hypersphere, whereas its kernelized form can be applied to a wider class of objects. The second modification is a spectral clustering approach to K-msbFCM using a certain assumption. This approach solves the local convergence problem in the original algorithm. The third modification is to construct a model providing the exact solution of the spectral clustering approach. Numerical examples demonstrate that the proposed methods can produce good results for clusters with nonlinear borders when an adequate parameter value is selected.

A maximizing model of Bezdek-like spherical fuzzy c-means clustering

In this study, a maximizing model of Bezdek-type spherical fuzzy c-means clustering is proposed, which is based on the regularization of the maximizing model of spherical hard c-means. Using theoretical analysis and numerical experiments, it is shown that the proposed method is not equivalent to the minimizing model of Bezdek-type spherical fuzzy c-means, because the effect of its fuzzifier parameter is different from that found in conventional methods.

On FNM-based and RFCM-based fuzzy co-clustering algorithms

In this paper, some types of fuzzy co-clustering algorithms are proposed. First, it is shown that the common base of the objective function for quadratic-regularized fuzzy co-clustering and entropy-regularized fuzzy co-clustering is very similar to the base for quadratic-regularized fuzzy nonmetric model and entropy-regularized fuzzy nonmetric model, respectively. Next, it is shown that the above mentioned non-sense clustering problem in previously proposed fuzzy co-clustering algorithms is identical to that in fuzzy nonmetric model algorithms, in the case that all dissimilarities among rows and columns are zero. Based on the above discussion, a method is proposed applying fuzzy nonmetric model after all dissimilarities among rows and columns are non-zero. Furthermore, since relational fuzzy c-means is similar to fuzzy nonmetric model, in the sense that both methods are designed for homogenous relational data, a method is proposed applying relational fuzzy c-means after setting all dissimilarities among rows and columns to some non-zero value. An illustrative numerical example is presented for the proposed methods.