Soft ranking in clustering

Due to the diffusion of large-dimensional data sets (e.g., in DNA microarray or document organization and retrieval applications), there is a growing interest in clustering methods based on a proximity matrix. These have the advantage of being based on a data structure whose size only depends on cardinality, not dimensionality. In this paper, we propose a clustering technique based on fuzzy ranks. The use of ranks helps to overcome several issues of large-dimensional data sets, whereas the fuzzy formulation is useful in encoding the information contained in the smallest entries of the proximity matrix. Comparative experiments are presented, using several standard hierarchical clustering techniques as a reference

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 Mathematical Theory of Big Data

This article presents a cardinality approach to big data, a fuzzy logicbased approach to big data, a similarity-based approach to big data, and a logical approach to the marketing strategy of social networking services. All these together constitute a mathematical theory of big data. This article also examines databases with infinite attributes. The research results reveal that relativity and infinity are two characteristics of big data. The relativity of big data is based on the theory of fuzzy sets. The relativity of big data leads to the continuum from small data to big data, big data-driven small data analytics to become statistical significance. The infinity of big data is based on the calculus and cardinality theory. The infinity of big data leads to the infinite similarity of big data. The proposed theory in this article might facilitate the mathematical research and development of big data, big data analytics, big data computing, and data science with applications in intelligent business analytics and business intelligence

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