scholarly journals Four-Fold Formal Concept Analysis Based on Complete Idempotent Semifields

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 173
Author(s):  
Francisco José Valverde-Albacete ◽  
Carmen Peláez-Moreno
Keyword(s):  
Data Mining ◽  
Formal Concept Analysis ◽  
Order Theory ◽  
Concept Analysis ◽  
Galois Connection ◽  
Formal Concept ◽  
Data Mining Technique ◽  
Full Spectrum ◽  
Mining Technique ◽  
Ordered Pairs

Formal Concept Analysis (FCA) is a well-known supervised boolean data-mining technique rooted in Lattice and Order Theory, that has several extensions to, e.g., fuzzy and idempotent semirings. At the heart of FCA lies a Galois connection between two powersets. In this paper we extend the FCA formalism to include all four Galois connections between four different semivectors spaces over idempotent semifields, at the same time. The result is K¯-four-fold Formal Concept Analysis (K¯-4FCA) where K¯ is the idempotent semifield biasing the analysis. Since complete idempotent semifields come in dually-ordered pairs—e.g., the complete max-plus and min-plus semirings—the basic construction shows dual-order-, row–column- and Galois-connection-induced dualities that appear simultaneously a number of times to provide the full spectrum of variability. Our results lead to a fundamental theorem of K¯-four-fold Formal Concept Analysis that properly defines quadrilattices as 4-tuples of (order-dually) isomorphic lattices of vectors and discuss its relevance vis-à-vis previous formal conceptual analyses and some affordances of their results.

scholarly journals Efficient Closure Operators for FCA-Based Classification

2020 ◽  
Vol 10 (2) ◽  
pp. 79-98
Author(s):  
Nida Meddouri ◽  
Mondher Maddouri
Keyword(s):  
Data Mining ◽  
Formal Concept Analysis ◽  
Concept Lattice ◽  
Concept Analysis ◽  
Nearest Neighbor Search ◽  
Knowledge Discovery In Databases ◽  
Support Vector ◽  
Formal Concept ◽  
Closure Operators ◽  
Nominal Data

Knowledge discovery in databases (KDD) aims to exploit the large amounts of data collected every day in various fields of computing application. The idea is to extract hidden knowledge from a set of data. It gathers several tasks that constitute a process, such as: data selection, pre-processing, transformation, data mining, visualization, etc. Data mining techniques include supervised classification and unsupervised classification. Classification consists of predicting the class of new instances with a classifier built on learning data of labeled instances. Several approaches were proposed such as: the induction of decision trees, Bayes, nearest neighbor search, neural networks, support vector machines, and formal concept analysis. Learning formal concepts always refers to the mathematical structure of concept lattice. This article presents a state of the art on formal concept analysis classifier. The authors present different ways to calculate the closure operators from nominal data and also present new approach to build only a part of the lattice including the best concepts. This approach is based on Dagging (ensemble method) that generates an ensemble of classifiers, each one represents a formal concept, and combines them by a voting rule. Experimental results are given to prove the efficiency of the proposed method.

Lower and Upper Concept Lattices

2014 ◽  
Vol 60 (2) ◽  
pp. 337-352
Author(s):  
Cristian Vaideanu
Keyword(s):  
Formal Concept Analysis ◽  
Mathematical Theory ◽  
Concept Analysis ◽  
Galois Connection ◽  
Knowledge Systems ◽  
Domain Theory ◽  
Formal Concept ◽  
Concept Lattices ◽  
New Models ◽  
Formal Concepts

Abstract Formal Concept Analysis is a mathematical theory of data analysis using formal contexts and concept lattices. In this paper, two new types of concept lattices are introduced by using notions from domain theory (in particular, Hoare and Smyth powerdomains). Based on a Galois connection, we prove the fundamental theorem of the Formal Concept Analysis, as well as other properties of lower and upper formal concepts. In this way, we provide new models to represent and retrieve the information in data and knowledge systems.

Formal Concept Analysis Based Clustering

2011 ◽  
pp. 514-518
Author(s):  
Jamil M. Saquer
Keyword(s):  
Data Mining ◽  
Formal Concept Analysis ◽  
Lattice Theory ◽  
Applied Mathematics ◽  
Concept Analysis ◽  
Grocery Store ◽  
Association Mining ◽  
Formal Concept ◽  
Area Of Application

Formal concept analysis (FCA) is a branch of applied mathematics with roots in lattice theory (Wille, 1982; Ganter & Wille, 1999). It deals with the notion of a concept in a given universe, which it calls context. For example, consider the context of transactions at a grocery store where each transaction consists of the items bought together. A concept here is a pair of two sets (A, B). A is the set of transactions that contain all the items in B and B is the set of items common to all the transactions in A. A successful area of application for FCA has been data mining. In particular, techniques from FCA have been successfully used in the association mining problem and in clustering (Kryszkiewicz, 1998; Saquer, 2003; Zaki & Hsiao, 2002). In this article, we review the basic notions of FCA and show how they can be used in clustering.

Interactive Data Mining for Large-Scale Image Databases Based on Formal Concept Analysis

2010 ◽  
Vol 14 (3) ◽  
pp. 303-308 ◽  
Author(s):  
Takanari Tanabata ◽  
◽  
Kazuhito Sawase ◽  
Hajime Nobuhara ◽  
Barnabas Bede ◽  
...  
Keyword(s):  
Data Mining ◽  
Formal Concept Analysis ◽  
Large Scale ◽  
Lattice Structure ◽  
Concept Analysis ◽  
Image Features ◽  
Image Databases ◽  
Formal Concept ◽  
Interactive Data Mining ◽  
Interactive Data

In order to perform an interactive data-mining for huge image databases efficiently, a visualization interface based on Formal Concept Analysis (FCA) is proposed. The proposed interface system provides an intuitive lattice structure enabling users freely and easily to select FCA attributes and to view different aspects of the Hasse diagram of the lattice of a given image database. The investigation environment is implemented using C++ and the OpenCV library on a personal computer (CPU = 2.13 GHz, MM = 2 GB). In visualization experiments using 1,000 Corel Image Gallery images, we test image features such as color, edge, and face detectors as FCA attributes. Experimental analysis confirms the effectiveness of the proposed interface and its potential as an efficient datamining tool.

scholarly journals ESTABLISHING LOGICAL RULES FROM EMPIRICAL DATA

2008 ◽  
Vol 17 (05) ◽  
pp. 985-1001 ◽  
Author(s):  
JOHN L. PFALTZ
Keyword(s):  
Data Mining ◽  
Empirical Data ◽  
Formal Concept Analysis ◽  
Statistical Significance ◽  
Concept Analysis ◽  
Formal Concept ◽  
Closed Set ◽  
Large Sets ◽  
Logical Rules

We review a method of generating logical rules, or axioms, from empirical data. This method, using closed set properties of formal concept analysis, has been previously described and tested on rather large sets of deterministic data. In spite of the fact that formal concept techniques have been used to prune frequent set data mining results, frequency and/or statistical significance are totally irrelevant to this method. It is strictly logical and deterministic. The contribution of this paper is a completely new extension of this method to create implications involving numeric inequalities. That is, numerical inequalities such as "age > 39" can be treated as logical predicates that have been extracted from the data itself and not postulated apriori.

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