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. By the way, 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. These uncertainties of data have been represented as interval range and many clustering algorithms for these interval ranges of data have been constructed.However, the guideline to select an available distance in each case has not been shown so that this selection problem is difficult. Therefore, methods to calculate the dissimilarity between such uncertain data without introducing a particular distance, e.g., nearest neighbor one and so on, have been strongly desired. From this viewpoint, we have proposed a concept of tolerance.The concept represents a uncertain data not as an interval but as a point with a tolerance vector. In this paper, we try to remove the constraint for tolerance vectors by using quadratic regularization of penalty vector which is similar to tolerance vector and propose new clustering algorithms for uncertain data through considering the optimization problems and obtaining the optimal solution, to handle such uncertainty more appropriately.