Faster Algorithms for Constructing a Concept (Galois) Lattice

Perhaps more than ever, new economic and enterprise needs have increased the interest and utility of the methods of the grouping process based on the theory of uncertainty. Numerous studies made in this field until recently have contributed with a great number of techniques and algorithms that proved their efficiency in resolving challenging problems that arise in an economic management framework of modern companies. A fuzzy grouping (clustering) process is a key phase of knowledge acquisition and reduction complexity regarding different groups of objects. In this study, we consider some elements of the theory of affinities and uncertain pretopology which form a significant support tool for a fuzzy clustering process. A Galois lattice is also introduced in order to provide a clearer vision of the results we obtained in a previous process. We implement this useful set of mathematical elements to make an homogeneous grouping process of the economic regions of Russian Federation and Ukraine. The obtained results give us a large situational panorama of a regional economic situation of two countries as well as the key guidelines to follow in a decision-making process. We offer the calculus for two different countries to demonstrate the significant differences that could be obtained taking into account the minimum changes of an input data basis. The mathematical method is very sensible to any changes that the economic situation of the regions can have. In this study, we pretend to give an alternative method of the grouping process for any modern company that performs its activity under uncertainty.

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Conceptual Coverage Driven by Essential Concepts: A Formal Concept Analysis Approach

Mathematics ◽

10.3390/math9212694 ◽

2021 ◽

Vol 9 (21) ◽

pp. 2694

Author(s):

Amira Mouakher ◽

Axel Ragobert ◽

Sébastien Gerin ◽

Andrea Ko

Keyword(s):

Formal Concept Analysis ◽

Concept Analysis ◽

Order Structure ◽

Formal Concept ◽

Massive Datasets ◽

Interestingness Measures ◽

Galois Lattice ◽

Formal Concepts ◽

Representation Method

Formal concept analysis (FCA) is a mathematical theory that is typically used as a knowledge representation method. The approach starts with an input binary relation specifying a set of objects and attributes, finds the natural groupings (formal concepts) described in the data, and then organizes the concepts in a partial order structure or concept (Galois) lattice. Unfortunately, the total number of concepts in this structure tends to grow exponentially as the size of the data increases. Therefore, there are numerous approaches for selecting a subset of concepts to provide full or partial coverage. In this paper, we rely on the battery of mathematical models offered by FCA to introduce a new greedy algorithm, called Concise, to compute minimal and meaningful subsets of concepts. Thanks to its theoretical properties, the Concise algorithm is shown to avoid the sluggishness of its competitors while offering the ability to mine both partial and full conceptual coverage of formal contexts. Furthermore, experiments on massive datasets also underscore the preservation of the quality of the mined formal concepts through interestingness measures agreed upon by the community.

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