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1130:. The lines allow to read off the subconcept-superconcept hierarchy. Each object and attribute name is used as a label exactly once in the diagram, with objects below and attributes above concept circles. This is done in a way that an attribute can be reached from an object via an ascending path if and only if the object has the attribute.
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A bicluster of similar values in a numerical object-attribute data-table is usually defined as a pair consisting of an inclusion-maximal set of objects and an inclusion-maximal set of attributes having similar values for the objects. Such a pair can be represented as an inclusion-maximal rectangle in
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TCA generalizes the above mentioned case by considering temporal data bases with an arbitrary key. That leads to the notion of distributed objects which are at any given time at possibly many places, as for example, a high pressure zone on a weather map. The notions of 'temporal objects', 'time' and
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In the simplest case TCA considers objects that change in time like a particle in physics, which, at each time, is at exactly one place. That happens in those temporal data where the attributes 'temporal object' and 'time' together form a key of the data base. Then the state (of a temporal object at
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Denis
Ponomaryov; Nadezhda Omelianchuk; Victoria Mironova; Eugene Zalevsky; Nikolay Podkolodny; Eric Mjolsness; Nikolay Kolchanov (2011), Karl Erich Wolff; Dmitry E. Palchunov; Nikolay G. Zagoruiko; Urs Andelfinger (eds.), "From Published Expression and Phenotype Data to Structured Knowledge: The
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triclustering include OA-biclustering and OAC-triclustering (here O stands for object, A for attribute, C for condition); to generate patterns these methods use prime operators only once being applied to a single entity (e.g. object) or a pair of entities (e.g. attribute-condition), respectively.
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This aim traces back to the educationalist
Hartmut von Hentig, who in 1972 pleaded for restructuring sciences in view of better teaching and in order to make sciences mutually available and more generally (i.e. also without specialized knowledge) critiqueable. Hence, by its origins formal concept
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Temporal concept analysis (TCA) is an extension of Formal
Concept Analysis (FCA) aiming at a conceptual description of temporal phenomena. It provides animations in concept lattices obtained from data about changing objects. It offers a general way of understanding change of concrete or abstract
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biological processes, the latter should possibly overlap, since a gene may be involved in several processes. The same remark applies for recommender systems where one is interested in local patterns characterizing groups of users that strongly share almost the same tastes for a subset of items.
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There are a number of simple and fast algorithms for generating formal concepts and for constructing and navigating concept lattices. For a survey, see
Kuznetsov and Obiedkov or the book by Ganter and Obiedkov, where also some pseudo-code can be found. Since the number of formal concepts may be
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Since only non-incident object-attribute pairs can be related, these relations can conveniently be recorded in the table representing a formal context. Many lattice properties can be read off from the arrow relations, including distributivity and several of its generalizations. They also reveal
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in a hierarchy called more formally the context's "concept lattice". The concept lattice can be graphically visualized as a "line diagram", which then may be helpful for understanding the data. Often however these lattices get too large for visualization. Then the mathematical theory of formal
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is to group together some objects having similar values of some attributes. For example, in gene expression data, it is known that genes (objects) may share a common behavior for a subset of biological situations (attributes) only: one should accordingly produce local patterns to characterize
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The original formal context can be reconstructed from the labelled diagram, as well as the formal concepts. The extent of a concept consists of those objects from which an ascending path leads to the circle representing the concept. The intent consists of those attributes to which there is an
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as part of general lattice theory. Other previous approaches to the same idea arose from various French research groups, but the
Darmstadt group normalised the field and systematically worked out both its mathematical theory and its philosophical foundations. The latter refer in particular to
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Of course, formal concepts can be considered as "rigid" biclusters where all objects have all attributes and vice versa. Hence, it is not surprising that some bicluster definitions coming from practice are just definitions of a formal concept. Relaxed FCA-based versions of biclustering and
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In his article "Restructuring
Lattice Theory" (1982), initiating formal concept analysis as a mathematical discipline, Wille starts from a discontent with the current lattice theory and pure mathematics in general: The production of theoretical results—often achieved by "elaborate mental
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a time in a view) is formalized as a certain object concept of the formal context describing the chosen view. In this simple case, a typical visualization of a temporal system is a line diagram of the concept lattice of the view into which trajectories of temporal objects are embedded.
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The formal concept analysis can be used as a qualitative method for data analysis. Since the early beginnings of FCA in the early 1980s, the FCA research group at TU Darmstadt has gained experience from more than 200 projects using the FCA (as of 2005). Including the fields of:
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the numerical table, modulo rows and columns permutations. In it was shown that biclusters of similar values correspond to triconcepts of a triadic context where the third dimension is given by a scale that represents numerical attribute values by binary attributes.
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Restructuring lattice theory is an attempt to reinvigorate connections with our general culture by interpreting the theory as concretely as possible, and in this way to promote better communication between lattice theorists and potential users of lattice
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Real-world data is often given in the form of an object-attribute table, where the attributes have "values". Formal concept analysis handles such data by transforming them into the basic type of a ("one-valued") formal context. The method is called
454:
The aim and meaning of Formal
Concept Analysis as mathematical theory of concepts and concept hierarchies is to support the rational communication of humans by mathematically developing appropriate conceptual structures which can be logically
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Beate Kohler-Koch; Frank Vogt; Gerhard Stumme; Rudolf Wille (2000), "Normen- und
Regelgeleitete internationale Kooperationen: Quoted in: Peter Becker et al. The ToscanaJ Suite for Implementing Conceptual Information Systems",
3921:
Dominik Endres; Ruth Adam; Martin A. Giese; Uta
Noppeney (2012), Florent Domenach; Dmitry I. Ignatov; Jonas Poelmans (eds.), "Understanding the Semantic Structure of Human fMRI Brain Recordings with Formal Concept Analysis",
69:, is a useful starting point for translations, but translators must revise errors as necessary and confirm that the translation is accurate, rather than simply copy-pasting machine-translated text into the English Knowledge.
4238:
Jens Illig; Andreas Hotho; Robert Jäschke; Gerd Stumme (2011), Karl Erich Wolff; Dmitry E. Palchunov; Nikolay G. Zagoruiko; Urs
Andelfinger (eds.), "A Comparison of Content-Based Tag Recommendations in Folksonomy Systems",
1620:") is not concept forming in the same way as defined above. For this reason, the values 1 and 0 or TRUE and FALSE are usually avoided when representing formal contexts, and a symbol like × is used to express incidence.
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Boumedjout Lahcen and Leonard Kwuida. "Lattice Miner: A Tool for Concept Lattice Construction and Exploration". In: Supplementary Proceeding of International Conference on Formal concept analysis (ICFCA'10),
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exponential in the size of the formal context, the complexity of the algorithms usually is given with respect to the output size. Concept lattices with a few million elements can be handled without problems.
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assumed that negated attributes are available for concept formation. But pairs of attributes which are negations of each other often naturally occur, for example in contexts derived from conceptual scaling.
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Nada Mimouni; Adeline Nazarenko; Sylvie Salotti (2015), Jaume Baixeries; Christian Sacarea; Manuel Ojeda-Aciego (eds.), "A Conceptual Approach for Relational IR: Application to Legal Collections",
4351:
Dieter Eschenfelder; Wolfgang Kollewe; Martin Skorsky; Rudolf Wille (2000), Gerd Stumme; Rudolf Wille (eds.), "Ein Erkundungssystem zum Baurecht: Methoden der Entwicklung Eines TOSCANA-Systems",
2441:'place' are represented as formal concepts in scales. A state is formalized as a set of object concepts. That leads to a conceptual interpretation of the ideas of particles and waves in physics.
407:
had been reduced to its extent. Now again, the philosophy of concepts should become less abstract by considering the intent. Hence, formal concept analysis is oriented towards the categories
2411:. Weak opposition is a dual weak complementation. A (bounded) lattice such as a concept algebra, which is equipped with a weak complementation and a dual weak complementation, is called a
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concept analysis may be helpful, e.g., for decomposing the lattice into smaller pieces without information loss, or for embedding it into another structure which is easier to interpret.
55:
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Aurélie Bertaux; Florence Le Ber; Agnès Braud; Michèle Trémolières (2009), Sébastien Ferré; Sebastian Rudolph (eds.), "Identifying Ecological Traits: A Concrete FCA-Based Approach",
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2579:-dimensional concepts. This reduction allows one to use standard definitions and algorithms from multidimensional concept analysis for computing multidimensional clusters.
79:
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Wolff, Karl Erich (2004), "'Particles' and 'Waves' as Understood by Temporal Concept Analysis.", in Wolff, Karl Erich; Pfeiffer, Heather D.; Delugach, Harry S. (eds.),
4147:
2157:
3248:
Wolff, Karl Erich (2019), "Temporal Concept Analysis with SIENA", in Cristea, Diana; Le Ber, Florence; Missaoui, Rokia; Kwuida, Léonard; Sertkaya, Bariş (eds.),
1654:
by the inclusion of extents, or, equivalently, by the dual inclusion of intents. An order ≤ on the concepts is defined as follows: for any two concepts (
3271:
Conceptual Structures at Work. 12th International Conference on Conceptual Structures, ICCS 2004. Huntsville, AL, USA, July 2004, LNAI 3127. Proceedings
2504:
in that graph. The mathematical and algorithmic results of formal concept analysis may thus be used for the theory of maximal bicliques. The notion of
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replaces the binary incidence relation between objects and attributes by a ternary relation between objects, attributes, and conditions. An incidence
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Do not translate text that appears unreliable or low-quality. If possible, verify the text with references provided in the foreign-language article.
3646:
Kaytoue, M.; Kuznetsov, S.; Macko, J.; Wagner Meira Jr., Napoli A. (2011). "Mining Biclusters of Similar Values with Triadic Concept Analysis".
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172:. Each concept in the hierarchy represents the objects sharing some set of properties; and each sub-concept in the hierarchy represents a
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Wolff, Karl Erich (2010), "Temporal Relational Semantic Systems", in Croitoru, Madalina; Ferré, Sébastien; Lukose, Dickson (eds.),
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Many FCA software applications are available today. The main purpose of these tools varies from formal context creation to formal
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submatrix (not necessarily contiguous) all of whose elements equal 1. It is however misleading to consider a formal context as
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Formal concept analysis has elaborate mathematical foundations, making the field versatile. As a basic example we mention the
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formed by them is much less developed than that of concept lattices, and seems to be difficult. Voutsadakis has studied the
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ascending path from that concept circle (in the diagram). In this diagram the concept immediately to the left of the label
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gymnastics"—were impressive, but the connections between neighboring domains, even parts of a theory were getting weaker.
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Prelic, S.; Bleuler, P.; Zimmermann, A.; Wille, P.; Buhlmann, W.; Gruissem, L.; Hennig, L.; Thiele, E.; Zitzler (2006).
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239:. One such possibility of very general nature is that data tables can be transformed into algebraic structures called
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Ordered Sets. Proceedings of the NATO Advanced Study Institute held at Banff, Canada, August 28 to September 12, 1981
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Content in this edit is translated from the existing German Knowledge article at ]; see its history for attribution.
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Wort und Feld: wortsemantische Fragestellungen mit besonderer Berücksichtigung des Wortfeldbegriffes: Dissertation
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Formal Concept Analysis: 7th International Conference, ICFCA 2009 Darmstadt, Germany, May 21–24, 2009 Proceedings
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objects in continuous, discrete or hybrid space and time. TCA applies conceptual scaling to temporal data bases.
1953:, an irredundant set of implications from which all valid implications can be derived by the natural inference (
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of the objects (as well as a superset of the properties) in the concepts above it. The term was introduced by
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Richard Cole; Gerd Stumme (2000), Bernhard Ganter; Guy W. Mineau (eds.), "CEM – A Conceptual Email Manager",
3554:"Toward the next generation of recommender systems: a survey of the state-of-the-art and possible extensions"
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Kuznetsov, S.; Obiedkov, S. (2002). "Comparing Performance of Algorithms for Generating Concept Lattices".
243:, and that these can be utilized for data visualization and interpretation. A data table that represents a
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2230:) is in general not a concept. However, since the concept lattice is complete one can consider the join (
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between sets of objects and of attributes. This is why in French a concept lattice is sometimes called a
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The original motivation of formal concept analysis was the search for real-world meaning of mathematical
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structural information and can be used for determining, e.g., the congruence relations of the lattice.
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in which the rows correspond to the objects, the columns correspond to the attributes, and each entry
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Wille, Rudolf. "Formal Concept Analysis as Mathematical Theory of Concepts and Concept Hierarchies".
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and generating the concepts lattice of a given formal context and the corresponding implications and
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The above line diagram consists of circles, connecting line segments, and labels. Circles represent
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Susanne Motameny; Beatrix Versmold; Rita Schmutzler (2008), Raoul Medina; Sergei Obiedkov (eds.),
3759:"Assessment of discretization techniques for relevant pattern discovery from gene expression data"
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were systematically categorized by their attributes. For the purpose here it has been simplified.
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1785:, the intent of which comprises all attributes which all objects of that set of concepts have.
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With these derivation operators, Wille gave an elegant definition of a formal concept: a pair (
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3889:"Formal Concept Analysis for the Identification of Combinatorial Biomarkers in Breast Cancer"
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2992:"Familles minimales d'implications informatives résultant d'un tableau de données binaires"
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2350:. The concept lattice equipped with the two additional operations Δ and 𝛁 is known as the
1777:, or meet. Its extent consists of those objects that are common to all extents of the set.
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1973:, which are simple and easy to compute, but very useful. They are defined as follows: For
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Proceedings of the 4th ACM SIGKDD Workshop on Data Mining in Bioinformatics (BIOKDD 2004)
3596:"A Systematic Comparison and Evaluation of Biclustering Methods for Gene Expression Data"
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objects. In his late philosophy, Peirce assumed that logical thinking aims at perceiving
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Formal concept analysis aims at the clarity of concepts according to Charles S. Peirce's
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3508:"Discovery of optimal factors in binary data via a novel method of matrix decomposition"
3273:, Lecture Notes in Artificial Intelligence, vol. 3127, Springer, pp. 126–141,
3219:, Lecture Notes in Artificial Intelligence, vol. 6208, Springer, pp. 165–180,
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The theory in its present form goes back to the early 1980s and a research group led by
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2202:: Modelling negation of formal concepts is somewhat problematic because the complement
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The data in the example is taken from a semantic field study, where different kinds of
352:. Its basic mathematical definitions, however, were already introduced in the 1930s by
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2516:(of the concept lattice) and has applications e.g. for Boolean matrix factorization.
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4192:, vol. 40, no. 1, Medford, NJ 09855: Information Today, pp. 521–543,
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Wille, Rudolf (2000), "Boolean Concept Logic", in Ganter, B.; Mineau, G. W. (eds.),
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3718:"Triadic Formal Concept Analysis and triclustering: searching for optimal patterns"
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4511:"International Conferences On Conceptual Structures – Conferences and Workshops"
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4013:"Mining gene expression data with pattern structures in formal concept analysis"
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Magier oder Magister? Über die Einheit der Wissenschaft im Verständigungsprozeß
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4315:, LNAI, vol. 1867, Berlin Heidelberg: Springer-Verlag, pp. 438–452,
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4068:, LNAI, vol. 5548, Berlin Heidelberg: Springer-Verlag, pp. 224–236,
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3482:"GALACTIC GAlois LAttices, Concept Theory, Implicational system and Closures"
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One can find a non exhaustive list of FCA tools in the FCA software website:
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Conceptual Structures: From Information to Intelligence. ICCS 2010. LNAI 6208
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ICCS 2000 Conceptual Structures: Logical, Linguistic and Computational Issues
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2194:: Extensive work has been done on a fuzzy version of formal concept analysis.
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of attributes and expresses that every object possessing each attribute from
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Davey, B.A.; Priestley, H. A. (2002), "Chapter 3. Formal Concept Analysis",
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Mehdi Kaytoue; Sergei Kuznetsov; Amedeo Napoli; Sébastien Duplessis (2011),
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2796:"Restructuring lattice theory: An approach based on hierarchies of concepts"
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For computing purposes, a formal context may be naturally represented as a
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It corrects the starting point of lattice theory during the development of
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can be defined in analogy to the formal concepts above, the theory of the
1949:. For each finite formal context, the set of all valid implications has a
4387:, LNAI, vol. 9113, Heidelberg New York: Springer, pp. 303–318,
4243:, LNCS, vol. 6581, Heidelberg New York: Springer, pp. 136–149,
3975:, LNCS, vol. 6581, Heidelberg New York: Springer, pp. 101–120,
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analysis aims at interdisciplinarity and democratic control of research.
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Dicomplemented Lattices. A contextual generalization of Boolean algebras
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Formal concept analysis finds practical application in fields including
3895:, LNAI, vol. 4933, Berlin Heidelberg: Springer, pp. 229–240,
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3057:
1600:." In this matrix representation, each formal concept corresponds to a
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Applying either derivation operator and then the other constitutes two
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108:
to the source of your translation. A model attribution edit summary is
4542:, Lecture Notes in Artificial Intelligence, vol. 3626, Springer,
3926:, LNCS, vol. 7278, Berlin Heidelberg: Springer, pp. 96–111,
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is union-closed. The complements of knowledge states therefore form a
2461:. Most of these tools are academic open-source applications, such as:
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between objects and attributes, tabulating pairs of the form "object
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Formal Concept Analysis. ICFCA International Conference Proceedings
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Conceptual Structures: Logical, Linguistic, and Computational Issues
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International Workshop on Knowledge Discovery in Inductive Databases
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Wille, R. (1995). "The basic theorem of triadic concept analysis"".
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The formal concepts of any formal context can—as explained below—be
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Arabidopsis Gene Net Supplementary Database and Its Applications",
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2802:. Nato Science Series C. Vol. 83. Springer. pp. 445–470.
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Priss, Uta, "Linguistic Applications of Formal Concept Analysis",
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Gerd Stumme; Alexander Maedche (2001), Universität Leipzig (ed.),
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Supplementary Proceedings of ICFCA 2019, Conference and Workshops
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Besson, J.; Robardet, C.; Raedt, L.D.; Boulicaut, J.-F. (2007).
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2012 IEEE 12th International Conference on Data Mining Workshops
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Journal of Experimental and Theoretical Artificial Intelligence
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442:. Mathematics is an abstraction of logic, develops patterns of
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Pensa, R.G.; Leschi, C.; Besson, J.; Boulicaut, J.-F. (2004).
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Ignatov, D.; Gnatyshak, D.; Kuznetsov, S.; Mirkin, B. (2015).
2964:, Linguistische Arbeiten 103 (in German), Tübingen: Niemeyer,
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and may be represented as the extents of some formal context.
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A bicluster in a binary object-attribute data-table is a pair
1820:, the extent of which is just the complement of the extent of
255:", is considered as a basic data type. It is referred to as a
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Cerf, L.; Besson, J.; Robardet, C.; Boulicaut, J.-F. (2009).
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These meet and join operations satisfy the axioms defining a
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Ganter, Bernhard; Stumme, Gerd; Wille, Rudolf, eds. (2005),
4355:(in German), Berlin Heidelberg: Springer, pp. 254–272,
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4112:, vol. 23, no. 6, New York: ACM, pp. 99–110,
4022:, vol. 181, no. 10, Elsevier, pp. 1989–2001,
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Given an object-attribute numerical data-table, the goal of
2508:(of the complemented bipartite graph) translates to that of
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Begriffliche Wissensverarbeitung – Methoden und Anwendungen
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Begriffliche Wissenverarbeitung – Methoden und Anwendungen
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Ferré, Sébastien; Rudolph, Sebastian, eds. (12 May 2009).
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For possible negations of formal concepts see the section
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to this template: there are already 1,886 articles in the
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2415:. Weakly dicomplemented lattices generalize distributive
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4106:"Reengineering class hierarchies using concept analysis"
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it is assumed that in any knowledge space the family of
1961:, a knowledge acquisition method based on implications.
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3673:"Concept-Based Biclustering for Internet Advertisement"
3764:. In Zaki, M.J.; Morishita, S.; Rigoutsos, I. (eds.).
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2290:, respectively. This can be expressed in terms of the
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by unfolding observable, elementary properties of the
4568:, translated by C. Franzke, Springer-Verlag, Berlin,
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Formal Concept Analysis: Foundations and Applications
4466:"International Conference on Formal Concept Analysis"
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Formal Concept Analysis. Foundations and Applications
2500:. The formal concepts then correspond to the maximal
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consists of all objects that share the attributes in
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In this way, formal concept analysis formalizes the
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consists of all attributes shared by the objects in
62:
4190:
Annual Review of Information Science and Technology
3561:
IEEE Transactions on Knowledge and Data Engineering
3547:
3474:
3341:"Formal Concept Analysis Software and Applications"
2872:
2870:
2665:
International Conference on Formal Concept Analysis
2496:A formal context can naturally be interpreted as a
1315:
attributes shared by all objects from A, and dually
58:
a machine-translated version of the German article.
3799:. LNCS. Vol. 4747. Springer. pp. 11–23.
3505:
2880:Attribute Exploration of Gene Regulatory Processes
2536:consisting of an inclusion-maximal set of objects
2151:
1773:In this order, every set of formal concepts has a
180:in 1981, and builds on the mathematical theory of
4566:Formal Concept Analysis: Mathematical Foundations
4535:
4425:
4279:Claudio Carpineto; Giovanni Romano, eds. (2004),
3875:
3840:ACM Transactions on Knowledge Discovery from Data
3671:Ignatov, D.; Poelmans, J.; Kuznetsov, S. (2012).
3301:
2945:
2920:Formal Concept Analysis: Mathematical Foundations
2673:International Conference on Conceptual Structures
474:Line diagram corresponding to the formal context
4784:
4609:SoftStat'93: Advances in Statistical Software 4.
4183:"Formal Concept Analysis in Information Science"
3014:
3012:
2867:
2603:Hands-on experience with formal concept analysis
2486:
1799:
505:Example for a formal context: "bodies of water"
4581:
3448:
2989:
2789:
2787:
1188:
4615:
4584:Concept Data Analysis: Theory and Applications
4490:"CLA: Concept Lattices and Their Applications"
4282:Concept Data Analysis: Theory and Applications
3418:
3332:
3077:
3075:
3018:
104:accompanying your translation by providing an
49:Click for important translation instructions.
36:expand this article with text translated from
4582:Carpineto, Claudio; Romano, Giovanni (2004),
3826:
3824:
3641:
3639:
3009:
1179:puddle, maar, lake, pond, tarn, pool, lagoon,
446:realities and therefore may support rational
4148:"FCA-Merge: Bottom-up merging of ontologies"
2823:
2784:
2520:Biclustering and multidimensional clustering
2427:
4602:"A first course in Formal Concept Analysis"
4563:
4157:, Leipzig, pp. 225–230, archived from
3506:Belohlavek, Radim; Vychodil, Vilem (2010).
3081:
3072:
3019:Ganter, Bernhard; Obiedkov, Sergei (2016).
2917:
2848:
2571:-dimensional clusters of similar values in
2540:and an inclusion-maximal set of attributes
2110:
3821:
3636:
3408:
3173:
2659:Many more examples are e.g. described in:
2354:of a context. Concept algebras generalize
4611:, Gustav Fischer Verlag, pp. 429–438
4445:(in German), Springer, pp. 325–340,
4027:
3871:
3869:
3733:
3655:
3619:
3295:
3159:, LNAI 1867, Springer, pp. 317–331,
3125:"Formal Concept Analysis and Fuzzy Logic"
2891:
1608:, because the negated incidence ("object
116:{{Translated|de|Formale Begriffsanalyse}}
4564:Ganter, Bernhard; Wille, Rudolf (1998),
3541:
2959:
2918:Ganter, Bernhard; Wille, Rudolf (1999).
2876:
2322:, and weak opposition can be written as
469:
3709:
3515:Journal of Computer and System Sciences
2669:Concept Lattices and their Applications
2444:
399:in the 19th century. Then—and later in
371:Motivation and philosophical background
230:
4785:
3866:
3833:"Closed patterns meet n-ary relations"
3795:. In Dzeroski, S.; Struyf, J. (eds.).
3664:
3179:
2886:(PhD). University of Jena. p. 9.
1526:) is a formal concept precisely when:
1357:objects sharing all attributes from B.
1133:In the diagram shown, e.g. the object
4599:
4180:
3268:
3247:
3212:
3154:
3043:
2793:
2575:-dimensional data are represented by
2358:. Weak negation on a concept lattice
1781:, every set of formal concepts has a
1147:temporary, running, natural, maritime
509:
450:. On this background, Wille defines:
2990:Guigues, J.L.; Duquenne, V. (1986).
2913:
2911:
2909:
2907:
2282:. These two operations are known as
2197:
1848:
1518:Equivalently and more intuitively, (
18:
4808:Formal semantics (natural language)
4104:Gregor Snelting; Frank Tip (1998),
3456:"FcaBedrock Formal Context Creator"
2898:urn:nbn:de:gbv:27-20120103-132627-0
2736:Formal semantics (natural language)
2582:
1624:Concept lattice of a formal context
1110:
1101:
1094:
1078:
1069:
1062:
1046:
1039:
1032:
1012:
1003:
996:
980:
973:
966:
959:
939:
930:
923:
905:
896:
889:
871:
862:
855:
837:
830:
808:
801:
794:
776:
769:
762:
740:
733:
724:
708:
701:
694:
674:
667:
660:
642:
635:
628:
621:
601:
590:
572:
561:
501:. Formal definitions follow below.
13:
4648:A Formal Concept Analysis Homepage
4619:Introduction to Lattices and Order
3790:"Mining bi-sets in numerical data"
2999:Mathématiques et Sciences Humaines
2544:such that almost all objects from
2294:. Weak negation can be written as
1964:
1568:that does not have that attribute.
1434:The derivation operators define a
14:
4829:
4641:
4422:
2942:
2904:
2853:. Klett (1972), Suhrkamp (1974).
2563:This fact can be generalized to
2548:have almost all attributes from
1153:has exactly the characteristics
1111:
1102:
1095:
1079:
1070:
1063:
1047:
1040:
1033:
1013:
1004:
997:
981:
974:
967:
960:
940:
931:
924:
906:
897:
890:
872:
863:
856:
838:
831:
809:
802:
795:
777:
770:
763:
741:
734:
725:
709:
702:
695:
675:
668:
661:
643:
636:
629:
622:
602:
591:
573:
562:
419:and classical conceptual logic.
350:Technische Universität Darmstadt
275:is a set of objects (called the
23:
4503:
4482:
4458:
4433:
4426:Ganter, Stumme & Wille 2005
4416:
4376:
4344:
4304:
4272:
4231:
4174:
4139:
4097:
4057:
4004:
3963:
3914:
3880:
3876:Ganter, Stumme & Wille 2005
3781:
3750:
3587:
3499:
3262:
3241:
3206:
3148:
3117:
2946:Ganter, Stumme & Wille 2005
2766:Statistical relational learning
2512:(of the formal context) and of
1854:
1270:of attributes, one defines two
4813:Ontology (information science)
3037:
2983:
2953:
2936:
2842:
2146:
2128:
1553:that the object does not have,
1250:which attributes. For subsets
114:You may also add the template
16:Method of deriving an ontology
1:
4528:
4513:. New Mexico State University
4110:Proceeding. SIGSOFT '98/FSE-6
3612:10.1093/bioinformatics/btl060
2960:Lutzeier, Peter Rolf (1981),
2849:Hentig, von, Hartmut (1972).
2487:Related analytical techniques
2413:weakly dicomplemented lattice
2268:) of all concepts satisfying
1884:also has each attribute from
1812:The negation of an attribute
1800:Attribute values and negation
1549:, there is some attribute in
1246:that expresses which objects
1193:A formal context is a triple
4393:10.1007/978-3-319-19545-2_19
4361:10.1007/978-3-642-57217-3_12
2491:
1242:is a binary relation called
1189:Formal contexts and concepts
489:The data table represents a
348:and Peter Burmeister at the
283:is a set of attributes (the
7:
3805:10.1007/978-3-540-75549-4_2
3279:10.1007/978-3-540-27769-9_8
3084:"Polyadic Concept Analysis"
2877:Wollbold, Johannes (2011).
2771:Schema (genetic algorithms)
2756:Inductive logic programming
2678:
2381:which satisfies the axioms
1112:
1103:
1096:
1080:
1071:
1064:
1048:
1041:
1034:
1014:
1005:
998:
982:
975:
968:
961:
941:
932:
925:
907:
898:
891:
873:
864:
857:
839:
832:
810:
803:
796:
778:
771:
764:
742:
735:
726:
710:
703:
696:
676:
669:
662:
644:
637:
630:
623:
603:
592:
574:
563:
86:will aid in categorization.
10:
4834:
4624:Cambridge University Press
4600:Wolff, Karl Erich (1994),
3527:10.1016/j.jcss.2009.05.002
3257:, Springer, pp. 94–99
2417:orthocomplemented lattices
1775:greatest common subconcept
1564:, there is some object in
1397:⊆ G (extent closure), and
549:
478:shown in the example table
465:
61:Machine translation, like
4775:10.1007/978-3-030-77867-5
4765:10.1007/978-3-030-21462-3
4755:10.1007/978-3-319-59271-8
4745:10.1007/978-3-319-19545-2
4735:10.1007/978-3-319-07248-7
4725:10.1007/978-3-642-38317-5
4715:10.1007/978-3-642-29892-9
4705:10.1007/978-3-642-20514-9
4695:10.1007/978-3-642-11928-6
4685:10.1007/978-3-642-01815-2
4675:10.1007/978-3-540-78137-0
4665:10.1007/978-3-540-70901-5
4548:10.1007/978-3-540-31881-1
4285:, John Wiley & Sons,
4249:10.1007/978-3-642-22140-8
4074:10.1007/978-3-642-01815-2
4038:10.1016/j.ins.2010.07.007
3981:10.1007/978-3-642-22140-8
3932:10.1007/978-3-642-29892-9
3735:10.1007/s10994-015-5487-y
3318:10.1080/09528130210164170
3225:10.1007/978-3-642-14197-3
2828:. Springer. p. 314.
2808:10.1007/978-94-009-7798-3
2686:Association rule learning
2567:-dimensional case, where
2428:Temporal concept analysis
1783:least common superconcept
1145:, but not the attributes
512:
434:, by the triade concept,
263:is defined to be a pair (
192:and others in the 1930s.
38:the corresponding article
4607:, in F. Faulbaum (ed.),
4223:: CS1 maint: location (
3395:Toscanaj.sourceforge.net
3180:Kwuida, Léonard (2004),
3082:Voutsadakis, G. (2002).
2970:10.1515/9783111678726.fm
2798:. In Rival, Ivan (ed.).
2777:
2117:Triadic concept analysis
2111:Extensions of the theory
1924:), then the implication
1904:is a formal context and
4198:10.1002/aris.1440400120
3852:10.1145/1497577.1497580
3103:10.1023/A:1021252203599
2741:General Concept Lattice
2721:Correspondence analysis
2152:{\displaystyle (g,m,c)}
1912:are subsets of the set
1592:equals to 1 if "object
1556:for every attribute in
1534:has every attribute in
1260:of objects and subsets
146:formal concept analysis
125:For more guidance, see
3685:10.1109/ICDMW.2012.100
3370:Conexp.sourceforge.net
3366:"The Concept Explorer"
3021:Conceptual Exploration
2794:Wille, Rudolf (1982).
2260:; or dually the meet (
2192:Fuzzy concept analysis
2153:
1430:⊆ M (intent closure).
479:
463:
389:
245:heterogeneous relation
188:that was developed by
4118:10.1145/291252.288273
2706:Conceptual clustering
2696:Commonsense reasoning
2642:office administration
2222:of a formal concept (
2154:
1959:attribute exploration
1916:of attributes (i.e.,
497:next to it shows its
473:
452:
378:
164:from a collection of
127:Knowledge:Translation
98:copyright attribution
4020:Information Sciences
3679:. pp. 123–130.
3573:10.1109/TKDE.2005.99
2626:software engineering
2445:Algorithms and tools
2364:weak complementation
2292:derivation operators
2171:under the condition
2161:then expresses that
2125:
1541:for every object in
1272:derivation operators
403:—a concept as unary
259:. In this theory, a
231:Overview and history
217:software development
209:knowledge management
4241:Kont 2007, KPP 2007
4181:Priss, Uta (2006),
3973:Kont 2007, KPP 2007
3486:galactic.univ-lr.fr
3391:"ToscanaJ: Welcome"
2701:Conceptual analysis
2506:bipartite dimension
2238:) of all concepts (
1957:). This is used in
1839:. It is in general
1652:(partially) ordered
1155:temporary, stagnant
1137:has the attributes
506:
142:information science
4818:Semantic relations
4429:, pp. 149–160
3621:20.500.11850/23740
3426:"The Coron System"
3058:10.1007/BF01108624
2167:has the attribute
2149:
1807:conceptual scaling
1442:(Galois lattice).
1440:treillis de Galois
504:
480:
361:, but also to the
106:interlanguage link
4633:978-0-521-78451-1
4593:978-0-470-85055-8
4452:978-3-540-66391-1
4402:978-3-319-19544-5
4258:978-3-642-22139-2
4083:978-3-642-01814-5
3990:978-3-642-22139-2
3941:978-3-642-29891-2
3902:978-3-540-78136-3
3814:978-3-540-75549-4
3694:978-1-4673-5164-5
3548:Adomavicius, C.;
3436:on 16 August 2022
3288:978-3-540-22392-4
3234:978-3-642-14196-6
3200:978-3-8322-3350-1
3166:978-3-540-67859-5
3030:978-3-662-49290-1
2817:978-94-009-7800-3
2726:Description logic
2654:political science
2587:In the theory of
2510:Ferrers dimension
2459:association rules
1872:relates two sets
1816:is an attribute ¬
1728:. Equivalently, (
1714:) precisely when
1436:Galois connection
1363:closure operators
1353:, i.e., a set of
1311:, i.e., a set of
1121:
1120:
359:Charles S. Peirce
241:complete lattices
158:concept hierarchy
138:
137:
50:
46:
4825:
4793:Machine learning
4636:
4612:
4606:
4596:
4578:
4560:
4522:
4521:
4519:
4518:
4507:
4501:
4500:
4498:
4497:
4486:
4480:
4479:
4477:
4476:
4462:
4456:
4455:
4437:
4431:
4430:
4420:
4414:
4413:
4380:
4374:
4373:
4348:
4342:
4341:
4321:10.1007/10722280
4308:
4302:
4301:
4300:
4299:
4276:
4270:
4269:
4235:
4229:
4228:
4222:
4214:
4213:
4212:
4187:
4178:
4172:
4171:
4170:
4169:
4163:
4152:
4143:
4137:
4136:
4135:
4134:
4101:
4095:
4094:
4061:
4055:
4054:
4053:
4052:
4031:
4017:
4008:
4002:
4001:
3967:
3961:
3960:
3918:
3912:
3911:
3910:
3909:
3884:
3878:
3873:
3864:
3863:
3837:
3828:
3819:
3818:
3794:
3785:
3779:
3778:
3776:
3775:
3770:. pp. 24–30
3763:
3754:
3748:
3747:
3737:
3728:(1–3): 271–302.
3713:
3707:
3706:
3668:
3662:
3661:
3659:
3643:
3634:
3633:
3623:
3591:
3585:
3584:
3558:
3545:
3539:
3538:
3512:
3503:
3497:
3496:
3494:
3492:
3478:
3472:
3471:
3469:
3467:
3452:
3446:
3445:
3443:
3441:
3432:. Archived from
3422:
3416:
3412:
3406:
3405:
3403:
3401:
3387:
3381:
3380:
3378:
3376:
3362:
3356:
3355:
3353:
3352:
3343:. Archived from
3336:
3330:
3329:
3312:(2–3): 189–216.
3299:
3293:
3291:
3266:
3260:
3258:
3256:
3245:
3239:
3237:
3210:
3204:
3203:
3188:
3177:
3171:
3169:
3152:
3146:
3145:
3143:
3142:
3136:
3130:. Archived from
3129:
3121:
3115:
3114:
3088:
3079:
3070:
3069:
3041:
3035:
3034:
3016:
3007:
3006:
2996:
2987:
2981:
2980:
2957:
2951:
2950:
2940:
2934:
2933:
2915:
2902:
2901:
2895:
2885:
2874:
2865:
2864:
2846:
2840:
2839:
2821:
2791:
2716:Concept learning
2711:Conceptual space
2691:Cluster analysis
2638:library sciences
2593:knowledge states
2589:knowledge spaces
2583:Knowledge spaces
2552:and vice versa.
2421:Boolean algebras
2410:
2380:
2349:
2321:
2281:
2259:
2221:
2201:
2199:Concept algebras
2178:triadic concepts
2174:
2170:
2166:
2160:
2158:
2156:
2155:
2150:
2102:
2091:
2087:
2080:
2073:
2044:
2033:
2029:
2022:
1992:
1982:
1947:
1940:
1903:
1849:concept algebras
1837:
1832:= G \
1831:
1794:complete lattice
1530:every object in
1508:
1497:
1472:
1425:
1420:
1414:
1410:
1392:
1387:
1381:
1377:
1352:
1325:
1310:
1283:
1269:
1259:
1241:
1211:
1115:
1114:
1106:
1105:
1099:
1098:
1083:
1082:
1074:
1073:
1067:
1066:
1051:
1050:
1044:
1043:
1037:
1036:
1017:
1016:
1008:
1007:
1001:
1000:
985:
984:
978:
977:
971:
970:
964:
963:
944:
943:
935:
934:
928:
927:
910:
909:
901:
900:
894:
893:
876:
875:
867:
866:
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859:
842:
841:
835:
834:
813:
812:
806:
805:
799:
798:
781:
780:
774:
773:
767:
766:
745:
744:
738:
737:
729:
728:
713:
712:
706:
705:
699:
698:
679:
678:
672:
671:
665:
664:
647:
646:
640:
639:
633:
632:
626:
625:
606:
605:
595:
594:
577:
576:
566:
565:
507:
503:
461:
387:
364:Port-Royal Logic
354:Garrett Birkhoff
205:machine learning
190:Garrett Birkhoff
117:
111:
85:
84:|topic=
82:, and specifying
67:Google Translate
48:
44:
27:
26:
19:
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4102:
4098:
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4058:
4050:
4048:
4029:10.1.1.457.8879
4015:
4009:
4005:
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3504:
3500:
3490:
3488:
3480:
3479:
3475:
3465:
3463:
3460:SourceForge.net
3454:
3453:
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2883:
2875:
2868:
2861:
2847:
2843:
2836:
2835:978-364201814-5
2822:, reprinted in
2818:
2792:
2785:
2780:
2775:
2751:Grounded theory
2746:Graphical model
2731:Factor analysis
2681:
2605:
2585:
2522:
2514:order dimension
2498:bipartite graph
2494:
2489:
2447:
2430:
2382:
2371:
2368:order-reversing
2352:concept algebra
2323:
2295:
2288:weak opposition
2269:
2247:
2246:) that satisfy
2203:
2172:
2168:
2164:
2126:
2123:
2122:
2120:
2113:
2089:
2085:
2078:
2071:
2052:
2031:
2027:
2020:
1997:
1984:
1974:
1971:arrow relations
1967:
1965:Arrow relations
1955:Armstrong rules
1951:canonical basis
1945:
1938:
1889:
1857:
1835:
1829:
1824:, i.e., with (¬
1802:
1769:
1762:
1755:
1748:
1741:
1734:
1727:
1720:
1713:
1706:
1699:
1692:
1686:, we say that (
1681:
1674:
1667:
1660:
1646:) of a context
1645:
1636:
1626:
1616:have attribute
1591:
1560:that is not in
1545:that is not in
1506:
1495:
1473:provided that:
1458:
1423:
1418:
1412:
1408:
1390:
1385:
1379:
1375:
1323:
1320:
1281:
1278:
1261:
1251:
1229:
1194:
1191:
1177:and the extent
1169:has the intent
1149:. Accordingly,
1128:formal concepts
1123:
552:
510:bodies of water
499:concept lattice
484:bodies of water
476:bodies of water
468:
462:
459:
424:pragmatic maxim
388:
385:
373:
346:Bernhard Ganter
233:
134:
133:
132:
115:
109:
83:
51:
45:(February 2012)
28:
24:
17:
12:
11:
5:
4831:
4821:
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4815:
4810:
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4800:
4798:Lattice theory
4795:
4781:
4780:
4779:
4778:
4768:
4758:
4748:
4738:
4728:
4718:
4708:
4698:
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4678:
4668:
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4643:
4642:External links
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3663:
3635:
3600:Bioinformatics
3586:
3567:(6): 734–749.
3540:
3498:
3473:
3462:. 12 June 2014
3447:
3430:Coron.loria.fr
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3097:(3): 295–304.
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3052:(2): 149–158.
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2841:
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2761:Pattern theory
2758:
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2708:
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2597:closure system
2584:
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2493:
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2455:concept mining
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1628:The concepts (
1625:
1622:
1596:has attribute
1583:
1570:
1569:
1554:
1539:
1516:
1515:
1455:formal concept
1432:
1431:
1399:
1398:
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491:formal context
467:
464:
460:Rudolf Wille,
457:
386:Rudolf Wille,
383:
372:
369:
315:
314:
303:
261:formal concept
257:formal context
251:has attribute
232:
229:
156:of deriving a
154:principled way
136:
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90:
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75:adding a topic
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4620:
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4603:
4598:
4595:
4589:
4585:
4580:
4577:
4575:3-540-62771-5
4571:
4567:
4562:
4559:
4557:3-540-27891-5
4553:
4549:
4545:
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4540:
4534:
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4506:
4491:
4485:
4471:
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4461:
4454:
4448:
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4428:
4427:
4419:
4412:
4408:
4404:
4398:
4394:
4390:
4386:
4379:
4372:
4370:3-540-66391-6
4366:
4362:
4358:
4354:
4347:
4340:
4336:
4332:
4330:3-540-67859-X
4326:
4322:
4318:
4314:
4307:
4294:
4292:0-470-85055-8
4288:
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4242:
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4191:
4184:
4177:
4164:on 2016-02-13
4160:
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4129:
4127:1-58113-108-9
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3653:
3649:
3642:
3640:
3631:
3627:
3622:
3617:
3613:
3609:
3606:(9): 1122–9.
3605:
3601:
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3590:
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3562:
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3361:
3347:on 2010-04-16
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3202:
3196:
3192:
3191:Shaker Verlag
3185:
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3137:on 2017-12-09
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2472:Lattice Miner
2470:
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2331:
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2299:
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2289:
2285:
2284:weak negation
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2263:
2258:
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2250:
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1457:of a context
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80:main category
77:
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71:
68:
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47:
41:
39:
34:You can help
30:
21:
20:
4618:
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4565:
4538:
4515:. Retrieved
4505:
4494:. Retrieved
4484:
4473:. Retrieved
4460:
4442:
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4154:
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4099:
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3906:, retrieved
3892:
3882:
3843:
3839:
3796:
3783:
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1855:Implications
1846:
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1787:
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1271:
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1234:
1230:
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1224:is a set of
1221:
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1216:is a set of
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495:line diagram
494:
490:
488:
481:
475:
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421:
401:model theory
397:formal logic
394:
390:
379:
374:
362:
342:Rudolf Wille
339:
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316:
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296:
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287:) such that
284:
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276:
272:
268:
264:
260:
256:
252:
248:
240:
237:order theory
234:
213:semantic web
194:
186:ordered sets
178:Rudolf Wille
157:
149:
145:
139:
102:edit summary
93:
73:
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4803:Data mining
3846:(1): 1–36.
3722:Mach. Learn
3650:: 175–190.
3521:(1): 3–20.
3466:27 December
3440:27 December
3400:27 December
3375:27 December
2650:linguistics
2634:information
2182:trilattices
2176:. Although
2163:the object
2048:and dually
1862:implication
1756:) whenever
513:attributes
417:linguistics
321:notions of
305:the intent
291:the extent
201:text mining
197:data mining
4787:Categories
4529:References
4517:2016-02-14
4496:2015-11-14
4475:2016-02-14
4385:Icfca 2015
4298:2016-02-04
4211:2016-02-04
4168:2016-02-13
4133:2016-02-04
4066:Icfca 2009
4051:2016-02-13
3924:Icfca 2012
3908:2016-01-29
3893:Icfca 2008
3774:2022-07-20
3491:2 February
3351:2010-06-10
3141:2017-12-08
2671:(CLA), or
2479:FcaBedrock
2366:, i.e. an
2356:power sets
2188:-ary case.
1226:attributes
455:activated.
440:conclusion
170:properties
168:and their
160:or formal
4777:2021 16th
4767:2019 15th
4757:2017 14th
4747:2015 13th
4737:2014 12th
4727:2013 11th
4717:2012 10th
4586:, Wiley,
4411:0302-9743
4267:0302-9743
4206:0066-4200
4046:215797283
4024:CiteSeerX
3999:0302-9743
3950:0302-9743
3744:254738363
3657:1111.3270
3581:206742345
3066:122657534
2893:1204.1995
2667:(ICFCA),
2502:bicliques
2492:Bicliques
1244:incidence
1167:reservoir
1135:reservoir
821:reservoir
519:temporary
436:judgement
413:intension
409:extension
405:predicate
327:intension
323:extension
271:), where
221:chemistry
120:talk page
72:Consider
40:in German
4707:2011 9th
4697:2010 8th
4687:2009 7th
4677:2008 6th
4667:2007 5th
4219:citation
4092:26304023
3860:11148363
3703:32701053
3630:16500941
3552:(2005).
3535:15659185
3326:10784843
3111:17738011
2679:See also
2675:(ICCS).
2630:ontology
2618:genetics
2610:medicine
2482:GALACTIC
2468:ToscanaJ
2093:, then (
2035:, then (
1933:is valid
1343:for all
1301:for all
1212:, where
1171:stagnant
1143:constant
1139:stagnant
1090:trickle
1058:torrent
544:maritime
539:constant
534:stagnant
458:—
444:possible
428:subsumed
384:—
319:semantic
182:lattices
162:ontology
96:provide
4339:5942241
3958:6256292
3005:: 5–18.
2978:8205166
2622:ecology
2419:, i.e.
2159:
2121:
2068:and if
2013:and if
1888:. When
1851:below.
1790:lattice
1668:) and (
1650:can be
1606:boolean
1602:maximal
1503:, and
1453:) is a
1218:objects
1175:natural
1159:natural
884:rivulet
585:channel
551:objects
529:natural
524:running
466:Example
432:reality
334:ordered
225:biology
166:objects
152:) is a
118:to the
100:in the
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1151:puddle
991:stream
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721:puddle
614:lagoon
493:, the
381:theory
301:dually
299:, and
285:intent
279:) and
277:extent
174:subset
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4492:. CLA
4335:S2CID
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4162:(PDF)
4155:IJCAI
4151:(PDF)
4088:S2CID
4042:S2CID
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3954:S2CID
3856:S2CID
3836:(PDF)
3793:(PDF)
3762:(PDF)
3740:S2CID
3699:S2CID
3652:arXiv
3577:S2CID
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3531:S2CID
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3322:S2CID
3255:(PDF)
3187:(PDF)
3135:(PDF)
3128:(PDF)
3107:S2CID
3091:Order
3087:(PDF)
3062:S2CID
3046:Order
2995:(PDF)
2888:arXiv
2884:(PDF)
2778:Notes
2534:(A,B)
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