Theoretical Analysis of Picture Fuzzy Clustering: Convergence and Property

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. The similarities and differences between this algorithm and other clustering methods are compared. Analysis on the loss function is also considered.

Degree Approximation-Based Fuzzy Partitioning Algorithm and Applications in Wheat Production Prediction

“…While the former uses another artificial neural network to forecast production based on energy inputs, another decision making analysis has been done in [66]. Several prediction procedures on case basis has been done by authors in [67][68][69][70]. The above stated method provided prediction using support vector machines based on soil properties.…”

Some picture fuzzy Bonferroni mean operators with their application to multicriteria decision making

“…Wei 11 proposed picture fuzzy aggregation operators: picture fuzzy weighted averaging (PFWA) operator, picture fuzzy weighted geometric (PFWG) operator, picture fuzzy hybrid average (PFHA) operator, picture fuzzy hybrid geometric (PFHG) operator, picture fuzzy ordered weighted averaging (PFOWA) operator, and picture fuzzy ordered weighted geometric (PFOWG) operator. For other researches on PFS, the readers are referred to recent works 12–21 …”

“…For other researches on PFS, the readers are referred to recent works. [12][13][14][15][16][17][18][19][20][21] The large diversity of possible correlations between the criterions in multicriteria decision making (MCDM) problems motivates excellent interest in purposing aggregation functions that can be used to model these possible relationships. Bonferroni mean (BM), which reflects the correlations of input arguments, was originally presented by Bonferroni 22 and then interpreted by Yager 23 as joined averaging and "anding" operator, which allows replacing the average operator by other means, such as Choquet integral 24 and ordered weighted average operator.…”

New Dissimilarity Measures on Picture Fuzzy Sets and Applications

“…Picture fuzzy set is a generality of fuzzy set [42] and intuitionistic fuzzy set [1]. Today, picture fuzzy set has been studied and applied widely in many fields [2,6,8,11,24,25,26,37], especially in clustering problems [13,15,27,28,29,32,33,31,36]. Hoa et al [13] used picture fuzzy sets to apply for Geographic Data Clustering.…”