Genetic fuzzy clustering for the definition of fuzzy sets

Author(s):  
J.R. Velasco ◽  
S. Lopez ◽  
L. Magdalena
2021 ◽  
Vol 12 (4) ◽  
pp. 1-20
Author(s):  
Nicolás Enrique Salgado Guitiérrez ◽  
Sergio Andrés Valencia Ramírez ◽  
José Soriano Méndez

This paper proposes a definition of a fuzzy partition element based on the homomorphism between type-1 fuzzy sets and the three-valued Kleene algebra. A new clustering method based on the C-means algorithm, using the defined partition, is presented in this paper, which will be validated with the traditional iris clustering problem by measuring its petals.


Author(s):  
Pedro Huidobro ◽  
Pedro Alonso ◽  
Vladimír Janis ◽  
Susana Montes

Convexity is one of the most important geometric properties of sets and a useful concept in many fields of mathematics, like optimization. As there are also important applications making use of fuzzy optimization, it is obvious that the studies of convexity are also frequent. In this paper we have extended the notion of convexity for hesitant fuzzy sets in order to fulfill some necessary properties. Namely, we have found an appropriate definition of convexity for hesitant fuzzy sets on any ordered universe based on aggregation functions such that it is compatible with the intersection, that is, the intersection of two convex hesitant fuzzy sets is a convex hesitant fuzzy set and it fulfills the cut worthy property.


Author(s):  
Witold Pedrycz ◽  
Athanasios Vasilakos

In contrast to numeric models, granular models produce results coming in a form of some information granules. Owing to the granularity of information these constructs dwell upon, such models become highly transparent and interpretable as well as operationally effective. Given also the fact that information granules come with a clearly defined semantics, granular models are often referred to as linguistic models. The crux of the design of the linguistic models studied in this paper exhibits two important features. First, the model is constructed on a basis of information granules which are assembled in the form of a web of associations between the granules formed in the output and input spaces. Given the semantics of information granules, we envision that a blueprint of the granular model can be formed effortlessly and with a very limited computing overhead. Second, the interpretability of the model is retained as the entire construct dwells on the conceptual entities of a well-defined semantics. The granulation of available data is accomplished by a carefully designed mechanism of fuzzy clustering which takes into consideration specific problem-driven requirements expressed by the designer at the time of the conceptualization of the model. We elaborate on a so-called context – based (conditional) Fuzzy C-Means (cond-FCM, for brief) to demonstrate how the fuzzy clustering is engaged in the design process. The clusters formed in the input space become induced (implied) by the context fuzzy sets predefined in the output space. The context fuzzy sets are defined in advance by the designer of the model so this design facet provides an active way of forming the model and in this manner becomes instrumental in the determination of a perspective at which a certain phenomenon is to be captured and modeled. This stands in a sharp contrast with most modeling approaches where the development is somewhat passive by being predominantly based on the existing data. The linkages between the fuzzy clusters induced by the given context fuzzy set in the output space are combined by forming a blueprint of the overall granular model. The membership functions of the context fuzzy sets are used as granular weights (connections) of the output processing unit (linear neuron) which subsequently lead to the granular output of the model thus identifying a feasible region of possible output values for the given input. While the above design is quite generic addressing a way in which information granules are assembled in the form of the model, we discuss further refinements which include (a) optimization of the context fuzzy sets, (b) inclusion of bias in the linear neuron at the output layer.


Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 515 ◽  
Author(s):  
Aykut Emniyet ◽  
Memet Şahin

In this paper, the concept of fuzzy normed ring is introduced and some basic properties related to it are established. Our definition of normed rings on fuzzy sets leads to a new structure, which we call a fuzzy normed ring. We define fuzzy normed ring homomorphism, fuzzy normed subring, fuzzy normed ideal, fuzzy normed prime ideal, and fuzzy normed maximal ideal of a normed ring, respectively. We show some algebraic properties of normed ring theory on fuzzy sets, prove theorems, and give relevant examples.


2019 ◽  
Vol 0 (9/2019) ◽  
pp. 5-11
Author(s):  
Andrzej Ameljańczyk

The paper concerns the mathematical modeling of patient’s disease states and disease unit patterns for the needs of algorithms supporting medical decisions. Due to the specificity of medical data and assessments in the modeling of patient’s disease states as well as diseases, the fuzzy set methodology was used. The paper presents a number of new characteristics of fuzzy sets allowing to assess the quality of medical diagnosis. In addition, a definition of a multi-aspect fuzzy set is presented, which may be useful in supporting medical diagnostics based on multi-criteria similarity models. The presented results can be used in the construction of algorithms for assessing the patient's state of health and mainly in the construction of algorithms for supporting diagnostic processes.


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