Fuzzy bases and the fuzzy dimension of fuzzy vector spaces

In this paper, new definitions of a fuzzy basis and a fuzzy dimension for a fuzzy vector space are presented. A fuzzy basis for a fuzzy vector space $(E,mu)$ is a fuzzy set $beta$ on $E$. The cardinality of a fuzzy basis $beta$ is called the fuzzy dimension of $(E,mu)$. The fuzzy dimension of a finite dimensional fuzzy vector space is a fuzzy natural number. For a fuzzy vector space, any two fuzzy bases have the same cardinality. If $widetilde{E}_1$ and $widetilde{E}_2$ are two fuzzy vector spaces, then $dim(widetilde{E}_1+widetilde{E}_2)+dim(widetilde{E}_1cap widetilde{E}_2)=dim(widetilde{E}_1) +dim(widetilde{E}_2)$ and $dim({widetilde{rm{ker }}f})+dim({widetilde{rm{im }}f})=dim(widetilde{E})$ hold without any restricted conditions. end{abstract

Fuzzy Color Clustering for Melanoma Diagnosis in Dermoscopy Images

A fuzzy logic-based color histogram analysis technique is presented for discriminating benign skin lesions from malignant melanomas in dermoscopy images. The approach extends previous research for utilizing a fuzzy set for skin lesion color for a specified class of skin lesions, using alpha-cut and support set cardinality for quantifying a fuzzy ratio skin lesion color feature. Skin lesion discrimination results are reported for the fuzzy clustering ratio over different regions of the lesion over a data set of 517 dermoscopy images consisting of 175 invasive melanomas and 342 benign lesions. Experimental results show that the fuzzy clustering ratio applied over an eight-connected neighborhood on the outer 25% of the skin lesion with an alpha-cut of 0.08 can recognize 92.6% of melanomas with approximately 13.5% false positive lesions. These results show the critical importance of colors in the lesion periphery. Our fuzzy logic-based description of lesion colors offers relevance to clinical descriptions of malignant melanoma

Fuzzy cardinality based evaluation of quanti®ed sentences

Quantified statements are used in the resolution of a great variety of problems. Several methods have been proposed to evaluate statements of types I and II. The objective of this paper is to study these methods, by comparing and generalizing them. In order to do so, we propose a set of properties that must be fulfilled by any method of evaluation of quantified statements, we discuss some existing methods from this point of view and we describe a general approach for the evaluation of quantified statements based on the fuzzy cardinality and fuzzy relative cardinality of fuzzy sets. In addition, we discuss some concrete methods derived from the mentioned approach. These new methods fulfill all the properties proposed and, in some cases, they provide an interpretation or generalization of existing methods

A Mining Algorithm under Fuzzy Taxonomic Structures

Most conventional data-mining algorithms identify the relationships among transactions using binary values and find rules at a single concept level. Transactions with quantitative values and items with taxonomic relations are, however, commonly seen in real-world applications. Besides, the taxonomic structures may also be represented in a fuzzy way. This paper thus proposes a fuzzy multiple-level mining algorithm for extracting fuzzy association rules under given fuzzy taxonomic structures. The proposed algorithm adopts a top-down progressively deepening approach to finding large itemsets. It integrates fuzzy-set concepts, data-mining technologies and multiple-level fuzzy taxonomy to find fuzzy association rules from given transaction data sets. Each item uses only the linguistic term with the maximum cardinality in later mining processes, thus making the number of fuzzy regions to be processed the same as the number of the original items. The algorithm therefore focuses on the most important linguistic terms for reduced time complexit

Counting of finite fuzzy subsets with applications to fuzzy recognition and selection strategies

The counting of fuzzy subsets of a finite set is of great interest in both practical and theoretical contexts in Mathematics. We have used some counting techniques such as the principle of Inclusion-Exclusion and the Mõbius Inversion to enumerate the fuzzy subsets of a finite set satisfying different conditions. These two techniques are interdependent with the M¨obius inversion generalizing the principle of Inclusion-Exclusion. The enumeration is carried out each time we redefine new conditions on the set. In this study one of our aims is the recognition and identification of fuzzy subsets with same features, characteristics or conditions. To facilitate such a study, we use some ideas such as the Hamming distance, mid-point between two fuzzy subsets and cardinality of fuzzy subsets. Finally we introduce the fuzzy scanner of elements of a finite set. This is used to identify elements and fuzzy subsets of a set. The scanning process of identification and recognition facilitates the choice of entities with specified properties. We develop a procedure of selection under the fuzzy environment. This allows us a framework to resolve conflicting issues in the market place

Bell inequalities in cardinality-based similarity measurement

In this thesis a parametric family of cardinality-based similarity measures for ordinary sets (on a finite universe) harbouring numerous well-known similarity measures is introduced. The Lukasiewicz- and product-transitive members of this family are characterized. Their importance derives from the one-to-one correspondence with pseudo-metrics. Also a parametric family of cardinality-based inclusion measures for ordinary sets (on a finite universe) is introduced, and the Lukasiewicz- and product-transitivity properties are also studied. Fuzzification schemes based on a commutative quasi-copula are then used to transform these similarity and inclusion measures for ordinary sets into similarity and inclusion measures for fuzzy sets on a finite universe, rendering them applicable on graded feature set representations of objects. One of the main results of this thesis is that transitivity, and hence the corresponding dual metrical interpretation (for similarity measures only), is preserved along this fuzzification process. It is remarkable that one stumbles across the same inequalities that should be fulfilled when checking these transitivity properties. The inequalities are known as the Bell inequalities. All Bell-type inequalities regarding at most four random events of which not more than two are intersected at the same time are presented in this work and are reformulated in the context of fuzzy scalar cardinalities leading to related inequalities on commutative conjunctors. It is proven that some of these inequalities are fulfilled for commutative (quasi-)copulas and for the most important families of Archimedean t-norms and each of the inequalities, the parameter values such that the corresponding t-norms satisfy the inequality considered, are identified. Meta-theorems, stating general conditions ensuring that certain inequalities for cardinalities of ordinary sets are preserved under fuzzification, when adopting a scalar approach to fuzzy set cardinality, are presented. The conditions pertain to a commutative conjunctor used for modeling fuzzy set intersection. In particular, this conjunctor should fulfill a number of Bell-type inequalities. The advantage of these meta-theorems is that repetitious calculations (for example, when checking the transitivity properties of fuzzy similarity measures) can be avoided