tailieunhanh - Theoretical analysis of picture fuzzy clustering
Recently, picture fuzzy clustering (FC-PFS) has been introduced as a new computational intelligence tool for various problems in knowledge discovery and pattern recognition. However, an important question that was lacked in the related researches is examination of mathematical properties behind the picture fuzzy clustering algorithm such as the convergence, the boundary or the convergence rate, etc. In this paper, we will prove that FC-PFS converges to at least one local minimum. Analysis on the loss function is also considered. | Journal of Computer Science and Cybernetics, , (2018), 17–31 DOI THEORETICAL ANALYSIS OF PICTURE FUZZY CLUSTERING PHAM THI MINH PHUONG1 , PHAM HUY THONG, LE HOANG SON Vietnam National University, Ha Noi, Viet Nam 1 phamthiminhphuong t60@ Abstract. Recently, picture fuzzy clustering (FC-PFS) has been introduced as a new computational intelligence tool for various problems in knowledge discovery and pattern recognition. However, an important question that was lacked in the related researches is examination of mathematical properties behind the picture fuzzy clustering algorithm such as the convergence, the boundary or the convergence rate, etc. In this paper, we will prove that FC-PFS converges to at least one local minimum. Analysis on the loss function is also considered. Keywords. Convergence analysis, picture fuzzy sets, picture fuzzy clustering. 1. INTRODUCTION One of the most efficient tools in pattern recogntion and knowledge discovery is fuzzy clustering in which the uncertainty and vagueness of data can be handled sucessfully. Fuzzy clustering, as its reminiscent names recalled, uses a membership function to assign for each data elements in the original dataset. The decision of an appropriate cluster depends on the membership values, that is to say, a greater one implies the inclusion. Fuzzy clustering sucessfully handle the problem of crisp clustering in which a data element can belong to many clusters at the same time [1, 2]. However, it was deployed on the traditional fuzzy set, which shows some limitations in dealing with practical scenarios like voting [3]. A new extension of the fuzzy set called the Picture Fuzzy Set (PFS) was presented by Cuong in [3, 4] to handle such the problem. A PFS is characterized by three membership degrees: positive, neutral, and negative degrees. In the real case of voting applications, ‘positive’ refers to the support for a candidate, ‘negative’ in reverse shows the .
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