GenAll Algorithm: Decorating Galois lattice with minimal generators (original) (raw)
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A partition-based approach towards constructing Galois (concept) lattices
Discrete Mathematics, 2002
Galois lattices and formal concept analysis of binary relations have proved useful in the resolution of many problems of theoretical or practical interest. Recent studies of practical applications in data mining and software engineering have put the emphasis on the need for both e cient and exible algorithms to construct the lattice. Our paper presents a novel approach for lattice construction based on the apposition of binary relation fragments. We extend the existing theory to a complete characterization of the global Galois (concept) lattice as a substructure of the direct product of the lattices related to fragments. The structural properties underlie a procedure for extracting the global lattice from the direct product, which is the basis for a full-scale lattice construction algorithm implementing a divide-and-conquer strategy. The paper provides a complexity analysis of the algorithm together with some results about its practical performance and describes a class of binary relations for which the algorithm outperforms the most e cient lattice-constructing methods.
Lecture Notes in Computer Science, 2005
The fixpoints of Galois Connections form patterns in binary relational data, such as objectattribute relations, that are important in a number of data analysis fields, including Formal Concept Analysis (FCA), Boolean factor analysis and frequent itemset mining. However, the large number of such fixpoints present in a typical dataset requires efficient computation to make analysis tractable, particularly since any particular fixpoint may be computed many times. Because they can be computed in a canonical order, testing the canonicity of fixpoints to avoid duplicates has proven to be a key factor in the design of efficient algorithms. The most efficient of these algorithms have been variants of the Close-By-One (CbO) algorithm. In this article, the algorithms CbO, FCbO, In-Close, In-Close2 and a new variant, In-Close3, are presented together for the first time, with in-Close2 and In-Close3 being the results of breeding In-Close with FCbO. To allow them to be easily compared, the algorithms are presented in the same style and notation. The important advances in CbO are described and compared graphically using a simple example. For the first time, the algorithms are implemented using the same structures and techniques to provide a level playing field for evaluation. Their performance is tested and compared using a range of data sets and the most important features identified for a CbO 'Best-of-Breed'. This article also presents, for the first time, the 'partial-closure' canonicity test.
VIE_MGB: A Visual Interactive Exploration of Minimal Generic Basis of Association Rules
2005
Mining association rules is an important task, even though the number of rules discovered is often huge. A possible solution to this problem, is to use the Formal Concept Analysis (FCA) mathematical settings to restrict rules extraction to a generic basis of association rules. This one is considered as a reduced set to which we can apply appropriate inference mechanisms to derive redundant rules. In this paper, we introduce a new minimal generic basis MGB of non-redundant association rules based on the augmented Iceberg Galois lattice. The proposed approach involves the inference mechanisms used and a set of experiments applied to several real and synthetic databases. Carried out experiments showed important benefits in terms of reduction in the number of generic rules extracted. We present also a new framework for generating and visually exploring the minimal generic basis MGB.
Building Concept (Galois) Lattices from Parts: Generalizing the Incremental Methods
Lecture Notes in Computer Science, 2001
Formal concept analysis and Galois lattices in general are increasingly used as a framework for the resolution of practical problems from software engineering, knowledge engineering and data mining. Recent applications have put the emphasis on the need for both e cient, scalable and exible algorithms to build the lattice. Such features are sought in the development of incremental algorithms. However, the major known incremental algorithm lacks clear theoretical foundations and shows some design aws which strongly a ect its practical performances. Our paper presents a general theoretical framework for the assembly of lattices sharing a same set of attributes based on existing theory on subposition of contexts. The framework underlies the design of a generic procedure for lattice assembly from parts, a new lattice building approach which is more general than the existing incremental and batch ones. As an argument for its theoretical strength, we describe the way our procedure reduces to an improved version of the major known incremental algorithm, which both corrects existing bugs and increases its overall e ciency.
A fast and general algorithm for Galois lattices building
2005
Standard Galois Lattices are effective tools for data analysis and knowledge discovery. They allow structuring data sets, by extracting concepts and rules to deduce concepts from other concepts. They focus on binary data arrays, called contexts. Several algorithms were proposed to generate concepts or concept lattices on a data context. However, the mining of large databases needs more efficient algorithms.
Mining Succinct Systems of Minimal Generators of Formal Concepts
Lecture Notes in Computer Science, 2005
Formal concept analysis has become an active field of study for data analysis and knowledge discovery. A formal concept C is determined by its extent (the set of objects that fall under C) and its intent (the set of properties or attributes covered by C). The intent for C, also called a closed itemset, is the maximum set of attributes that characterize C. The minimal generators for C are the minimal subsets of C's intent which can similarly characterize C. This paper introduces the succinct system of minimal generators (SSMG) as a minimal representation of the minimal generators of all concepts, and gives an efficient algorithm for mining SSMGs. The SSMGs are useful for revealing the equivalence relationship among the minimal generators, which may be important for medical and other scientific discovery; and for revealing the extent-based semantic equivalence among associations. The SSMGs are also useful for losslessly reducing the size of the representation of all minimal generators, similar to the way that closed itemsets are useful for losslessly reducing the size of the representation of all frequent itemsets. The removal of redudancies will help human users to grasp the structure and information in the concepts.
Yet a Faster Algorithm for Building the Hasse Diagram of a Concept Lattice
Lecture Notes in Computer Science, 2009
Formal concept analysis (FCA) is increasingly applied to data mining problems, essentially as a formal framework for mining reduced representations (bases) of target pattern families. Yet most of the FCA-based miners, closed pattern miners, would only extract the patterns themselves out of a dataset, whereas the generality order among patterns would be required for many bases. As a contribution to the topic of the (precedence) order computation on top of the set of closed patterns, we present a novel method that borrows its overall incremental approach from two algorithms in the literature. The claimed innovation consists of splitting the update of the precedence links into a large number of lower-cover list computations (as opposed to a single uppercover list computation) that unfold simultaneously. The resulting method shows a good improvement with respect to its counterpart both on its theoretical complexity and on its practical performance. It is therefore a good starting point for the design of efficient and scalable precedence miners.
A fast algorithm for building the Hasse diagram of a Galois lattice
2000
Formal concept analysis and Galois lattices in general are increasingly used for large contexts that are automatically generated. As the size of the resulting datasets may grow considerably, it becomes essential to keep the algorithmic complexity of the analysis procedures as low as possible. This paper presents an e cient algorithm that computes the Hasse diagram of a Galois lattice from the lattice ground set, i.e., the set of all concepts. The algorithm performs an element-wise completion of the lattice according to a linear extension of the lattice order. This requires only a limited number of comparisons between concepts and therefore makes the global algorithm very e cient. In fact, its asymptotic time complexity i s almost linear in the number of concepts. Consequently, the joint use of our algorithm with an e cient procedure for concept generation yields a complete procedure for building the Galois lattice.
Concept Lattice Simplification in Formal Concept Analysis Using Attribute Clustering
Journal of Ambient Intelligence and Humanized Computing, 2019
In Formal Concept Analysis (FCA), a concept lattice graphically portrays the underlying relationships between the objects and attributes of an information system. One of the key complexity problems of concept lattices lies in extracting the useful information. The unorganized nature of attributes in huge contexts often does not yield an informative lattice in FCA. Moreover, understanding the collective relationships between attributes and objects in a larger many valued context is more complicated. In this paper, we introduce a novel approach for deducing a smaller and meaningful concept lattice from which excerpts of concepts can be inferred. In existing attribute-based concept lattice reduction methods for FCA, mostly either the attribute size or the context size is reduced. Our approach involves in organizing the attributes into clusters using their structural similarities and dissimilarities, which is commonly known as attribute clustering, to produce a derived context. We have observed that the deduced concept lattice inherits the structural relationship of the original one. Furthermore, we have mathematically proved that a unique surjective inclusion mapping from the original lattice to the deduced one exists.