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Method of deriving an ontology

In information scientific discipline, formal concept analysis (FCA) is a principled manner of deriving a concept hierarchy or formal ontology from a collection of objects and their properties. Each concept in the hierarchy represents the objects sharing some set of properties; and each sub-concept in the bureaucracy represents a subset of the objects (likewise equally a superset of the properties) in the concepts above it. The term was introduced by Rudolf Wille in 1981, and builds on the mathematical theory of lattices and ordered sets that was developed past Garrett Birkhoff and others in the 1930s.

Formal concept analysis finds applied awarding in fields including information mining, text mining, car learning, knowledge management, semantic spider web, software development, chemistry and biological science.

Overview and history [edit]

The original motivation of formal concept analysis was the search for real-globe meaning of mathematical gild theory. 1 such possibility of very general nature is that data tables can exist transformed into algebraic structures called complete lattices, and that these tin be utilized for data visualization and interpretation. A information table that represents a heterogeneous relation between objects and attributes, tabulating pairs of the form "object g has attribute m", is considered every bit a bones data type. Information technology is referred to as a formal context. In this theory, a formal concept is divers to exist a pair (A, B), where A is a ready of objects (called the extent) and B is a gear up of attributes (the intent) such that

  • the extent A consists of all objects that share the attributes in B, and dually
  • the intent B consists of all attributes shared by the objects in A.

In this way, formal concept analysis formalizes the semantic notions of extension and intension.

The formal concepts of any formal context can—as explained below—be ordered in a hierarchy chosen more than formally the context'due south "concept lattice." The concept lattice tin be graphically visualized as a "line diagram", which so may be helpful for understanding the data. Oft nevertheless these lattices get too big for visualization. Then the mathematical theory of formal concept analysis may be helpful, e.g., for decomposing the lattice into smaller pieces without data loss, or for embedding information technology into another structure which is easier to translate.

The theory in its present form goes back to the early 1980s and a inquiry grouping led by Rudolf Wille, Bernhard Ganter and Peter Burmeister at the Technische Universität Darmstadt. Its basic mathematical definitions, however, were already introduced in the 1930s by Garrett Birkhoff as part of full general lattice theory. Other previous approaches to the same idea arose from various French enquiry groups, but the Darmstadt group normalised the field and systematically worked out both its mathematical theory and its philosophical foundations. The latter refer in item to Charles S. Peirce, simply also to the Port-Regal Logic.

Motivation and philosophical groundwork [edit]

In his article "Restructuring Lattice Theory" (1982),[ane] initiating formal concept analysis as a mathematical subject, Wille starts from a discontent with the current lattice theory and pure mathematics in general: The product of theoretical results—oft achieved by "elaborate mental gymnastics"—were impressive, only the connections between neighboring domains, even parts of a theory were getting weaker.

Restructuring lattice theory is an attempt to reinvigorate connections with our full general civilisation by interpreting the theory every bit concretely equally possible, and in this manner to promote better communication between lattice theorists and potential users of lattice theory

—Rudolf Wille, [1]

This aim traces back to the educationalist Hartmut von Hentig, who in 1972 pleaded for restructuring sciences in view of amend teaching and in order to make sciences mutually available and more than more often than not (i.due east. also without specialized knowledge) critiqueable.[2] Hence, by its origins formal concept assay aims at interdisciplinarity and democratic command of inquiry.[three]

It corrects the starting point of lattice theory during the development of formal logic in the 19th century. Then—and subsequently in model theory—a concept as unary predicate had been reduced to its extent. At present again, the philosophy of concepts should go less abstruse by considering the intent. Hence, formal concept analysis is oriented towards the categories extension and intension of linguistics and classical conceptual logic.[4]

Formal concept analysis aims at the clarity of concepts according to Charles South. Peirce's pragmatic saying by unfolding observable, elementary backdrop of the subsumed objects.[3] In his tardily philosophy, Peirce assumed that logical thinking aims at perceiving reality, by the triade concept, judgement and decision. Mathematics is an abstraction of logic, develops patterns of possible realities and therefore may support rational communication. On this background, Wille defines:

The aim and meaning of Formal Concept Analysis every bit mathematical theory of concepts and concept hierarchies is to back up the rational communication of humans past mathematically developing appropriate conceptual structures which can be logically activated.

—Rudolf Wille, [five]

Example [edit]

The data in the example is taken from a semantic field study, where unlike kinds of bodies of water were systematically categorized by their attributes.[6] For the purpose here it has been simplified.

The information table represents a formal context, the line diagram next to it shows its concept lattice. Formal definitions follow below.

Example for a formal context: "bodies of water"
bodies of water attributes
temporary running natural stagnant abiding maritime

objects

 canal Yes Yes
channel Yes Yes
lagoon Yes Yes Yes Yes
lake Yes Yes Yes
maar Yes Yes Yes
puddle Yes Yes Yes
pond Yes Yes Yes
pool Yes Yes Yes
reservoir Yes Yes
river Yes Yes Yes
rivulet Yes Yes Yes
runnel Yes Yes Yes
sea Yes Yes Yes Yes
stream Yes Yes Yes
tarn Yes Yes Yes
torrent Yes Yes Yes
trickle Yes Yes Yes

Line diagram corresponding to the formal context bodies of water on the left

The above line diagram consists of circles, connecting line segments, and labels. Circles represent formal concepts. The lines permit to read off the subconcept-superconcept hierarchy. Each object and aspect name is used equally a label exactly once in the diagram, with objects beneath and attributes above concept circles. This is done in a way that an attribute tin be reached from an object via an ascending path if and only if the object has the attribute.

In the diagram shown, due east.grand. the object reservoir has the attributes stagnant and constant, but not the attributes temporary, running, natural, maritime. Accordingly, puddle has exactly the characteristics temporary, stagnant and natural.

The original formal context can be reconstructed from the labelled diagram, also 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 at that place is an ascending path from that concept circle (in the diagram). In this diagram the concept immediately to the left of the characterization reservoir has the intent stagnant and natural and the extent pool, maar, lake, pond, tarn, pool, lagoon, and body of water.

Formal contexts and concepts [edit]

A formal context is a triple K = (G, Thousand, I), where G is a ready of objects, Thou is a gear up of attributes, and I ⊆ 1000 × Thou is a binary relation called incidence that expresses which objects accept which attributes.[4] For subsets A ⊆ G of objects and subsets B ⊆ M of attributes, one defines two derivation operators as follows:

A′ = {m ∈ M | (g,grand) ∈ I for all g ∈ A}, i.e., a set up of all attributes shared by all objects from A, and dually
B′ = {yard ∈ G | (g,chiliad) ∈ I for all g ∈ B}, i.e., a set of all objects sharing all attributes from B.

Applying either derivation operator and then the other constitutes two closure operators:

A ↦A′′ = (A′)′   for A ⊆ Yard   (extent closure), and
B ↦B′′ = (B′)′   for B ⊆ M   (intent closure).

The derivation operators define a Galois connectedness betwixt sets of objects and of attributes. This is why in French a concept lattice is sometimes called a treillis de Galois (Galois lattice).

With these derivation operators, Wille gave an elegant definition of a formal concept: a pair (A,B) is a formal concept of a context (Thousand, G, I) provided that:

A ⊆ M, B ⊆ One thousand, A′ = B,   andB′ = A.

Equivalently and more intuitively, (A,B) is a formal concept precisely when:

  • every object in A has every attribute in B,
  • for every object in Yard that is not in A, at that place is some aspect in B that the object does non have,
  • for every attribute in M that is not in B, at that place is some object in A that does not have that attribute.

For computing purposes, a formal context may exist naturally represented equally a (0,1)-matrix K in which the rows correspond to the objects, the columns stand for to the attributes, and each entry k i,j equals to 1 if "object i has attribute j." In this matrix representation, each formal concept corresponds to a maximal submatrix (not necessarily contiguous) all of whose elements equal 1. It is all the same misleading to consider a formal context as boolean, because the negated incidence ("object grand does not have attribute m") is not concept forming in the aforementioned way equally defined above. For this reason, the values one and 0 or Truthful and Faux are usually avoided when representing formal contexts, and a symbol like × is used to express incidence.

Concept lattice of a formal context [edit]

The concepts (A i , B i ) of a context Yard can exist (partially) ordered past 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 (A 1, B i) and (A 2, B 2) of Grand, we say that (A i, B 1) ≤ (A 2, B 2) precisely when A 1 ⊆ A 2. Equivalently, (A i, B ane) ≤ (A 2, B 2) whenever B 1 ⊇ B ii.

In this club, every set of formal concepts has a greatest common subconcept, or meet. Its extent consists of those objects that are common to all extents of the set. Dually, every prepare of formal concepts has a to the lowest degree common superconcept, the intent of which comprises all attributes which all objects of that set of concepts have.

These run across and join operations satisfy the axioms defining a lattice, in fact a complete lattice. Conversely, it can be shown that every complete lattice is the concept lattice of some formal context (up to isomorphism).

Attribute values and negation [edit]

Real-world data is often given in the form of an object-attribute tabular array, where the attributes accept "values". Formal concept analysis handles such data by transforming them into the basic type of a ("one-valued") formal context. The method is called conceptual scaling.

The negation of an attribute one thousand is an attribute ¬m, the extent of which is just the complement of the extent of m, i.e., with (¬m)′ = G \m′. It is in general not assumed that negated attributes are available for concept formation. But pairs of attributes which are negations of each other often naturally occur, for instance in contexts derived from conceptual scaling.

For possible negations of formal concepts see the department concept algebras beneath.

Implications [edit]

An implication A → B relates two sets A and B of attributes and expresses that every object possessing each aspect from A as well has each attribute from B. When (G,M,I) is a formal context and A, B are subsets of the set M of attributes (i.eastward., A,B ⊆ Thousand), then the implication A → B is valid if A′ ⊆ B′. For each finite formal context, the gear up of all valid implications has a approved basis,[seven] an irredundant set of implications from which all valid implications can be derived by the natural inference (Armstrong rules). This is used in attribute exploration, a noesis conquering method based on implications.[8]

Pointer relations [edit]

Formal concept analysis has elaborate mathematical foundations,[4] making the field versatile. As a basic example nosotros mention the arrow relations, which are unproblematic and like shooting fish in a barrel to compute, but very useful. They are defined equally follows: For g ∈ G and g ∈ M permit

g ↗ m ⇔  (yard, grand) ∉ I and if m⊆north′ and chiliad′ ≠ n′ , so (g, n) ∈ I,

and dually

grand ↙ m ⇔  (thousand, m) ∉ I and if chiliad′⊆h′ and g′ ≠ h′ , then (h, grand) ∈ I.

Since only non-incident object-attribute pairs tin can be related, these relations tin 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 structural information and can exist used for determining, e.one thousand., the congruence relations of the lattice.

Extensions of the theory [edit]

  • Triadic concept analysis replaces the binary incidence relation between objects and attributes past a ternary relation between objects, attributes, and conditions. An incidence ( thou , m , c ) {\displaystyle (g,thou,c)} then expresses that the object g has the attribute m nether the status c . Although triadic concepts can be divers in analogy to the formal concepts above, the theory of the trilattices formed past them is much less adult than that of concept lattices, and seems to be difficult.[ix] Voutsadakis has studied the northward-ary case.[10]
  • Fuzzy concept analysis: All-encompassing work has been washed on a fuzzy version of formal concept assay.[11]
  • Concept algebras : Modelling negation of formal concepts is somewhat problematic considering the complement (G \ A, Thou \ B) of a formal concept (A, B) is in general non a concept. However, since the concept lattice is complete one can consider the join (A, B)Δ of all concepts (C, D) that satisfy C ⊆ Thousand \ A ; or dually the meet (A, B)𝛁 of all concepts satisfying D ⊆ M \ B . These two operations are known as weak negation and weak opposition, respectively. This tin be expressed in terms of the derivation operators. Weak negation can exist written as (A, B)Δ = ((M \ A)′′, (G \ A)'), and weak opposition can be written as (A, B)𝛁 = ((G \ B)', (M \ B)′′). The concept lattice equipped with the 2 additional operations Δ and 𝛁 is known as the concept algebra of a context. Concept algebras generalize power sets. Weak negation on a concept lattice L is a weak complementation, i.e. an guild-reversing map Δ: L → L which satisfies the axioms x ΔΔ ≤ x and (x⋀y) ⋁ (x⋀y Δ) = ten . 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 weakly dicomplemented lattice. Weakly dicomplemented lattices generalize distributive orthocomplemented lattices, i.due east. Boolean algebras.[12] [thirteen]

Temporal concept analysis [edit]

Temporal concept analysis (TCA) is an extension of Formal Concept Analysis (FCA) aiming at a conceptual description of temporal phenomena. Information technology provides animations in concept lattices obtained from information about changing objects. It offers a general way of agreement alter of physical or abstract objects in continuous, discrete or hybrid space and time. TCA applies conceptual scaling to temporal data bases.[14]

In the simplest case TCA considers objects that change in fourth dimension like a particle in physics, which, at each time, is at exactly ane place. That happens in those temporal data where the attributes 'temporal object' and 'time' together form a key of the data base. Then the country (of a temporal object at a time in a view) is formalized as a certain object concept of the formal context describing the chosen view. In this simple example, a typical visualization of a temporal organization is a line diagram of the concept lattice of the view into which trajectories of temporal objects are embedded. [xv]

TCA generalizes the in a higher place mentioned case by because temporal information bases with an capricious key. That leads to the notion of distributed objects which are at whatsoever given time at peradventure many places, equally for example, a high pressure zone on a weather map. The notions of 'temporal objects', 'time' and 'place' are represented as formal concepts in scales. A land is formalized every bit a gear up of object concepts. That leads to a conceptual estimation of the ideas of particles and waves in physics.[xvi]

Algorithms and tools [edit]

There are a number of unproblematic and fast algorithms for generating formal concepts and for constructing and navigating concept lattices. For a survey, see Kuznetsov and Obiedkov[17] or the book by Ganter and Obiedkov,[eight] where also some pseudo-code tin can be constitute. Since the number of formal concepts may exist exponential in the size of the formal context, the complexity of the algorithms normally is given with respect to the output size. Concept lattices with a few million elements can be handled without problems.

Many FCA software applications are available today.[18] The principal purpose of these tools varies from formal context creation to formal concept mining and generating the concepts lattice of a given formal context and the corresponding implications and clan rules. Most of these tools are academic open-source applications, such every bit:

  • ConExp[19]
  • ToscanaJ[20]
  • Lattice Miner[21]
  • Coron[22]
  • FcaBedrock[23]
  • GALACTIC[24]

[edit]

Bicliques [edit]

A formal context tin can naturally be interpreted as a bipartite graph. The formal concepts and so stand for to the maximal bicliques 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 bipartite dimension (of the complemented bipartite graph) translates[4] to that of Ferrers dimension (of the formal context) and of social club dimension (of the concept lattice) and has applications e.thou. for Boolean matrix factorization.[25]

Biclustering and multidimensional clustering [edit]

Given an object-attribute numerical data-table, the goal of biclustering 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 beliefs for a subset of biological situations (attributes) only: 1 should accordingly produce local patterns to characterize biological processes, the latter should possibly overlap, since a factor 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 nearly the same tastes for a subset of items.[26]

A bicluster in a binary object-attribute data-table is a pair (A,B) consisting of an inclusion-maximal ready of objects A and an inclusion-maximal set of attributes B such that almost all objects from A have almost all attributes from B and vice versa.

Of course, formal concepts can be considered as "rigid" biclusters where all objects have all attributes and vice versa. Hence, it is non surprising that some bicluster definitions coming from practice[27] are just definitions of a formal concept.[28]

A bicluster of like values in a numerical object-attribute information-table is usually defined[29] [30] [31] as a pair consisting of an inclusion-maximal set of objects and an inclusion-maximal fix of attributes having similar values for the objects. Such a pair tin be represented every bit an inclusion-maximal rectangle in the numerical table, modulo rows and columns permutations. In[28] it was shown that biclusters of similar values correspond to triconcepts of a triadic context where the third dimension is given past a scale that represents numerical attribute values by binary attributes.

This fact tin be generalized to north-dimensional instance, where northward-dimensional clusters of like values in northward-dimensional information are represented by northward+1-dimensional concepts. This reduction allows one to use standard definitions and algorithms from multidimensional concept assay[31] [10] for computing multidimensional clusters.

Cognition spaces [edit]

In the theory of cognition spaces it is causeless that in any cognition infinite the family of knowledge states is spousal relationship-closed. The complements of cognition states therefore course a closure organisation and may exist represented as the extents of some formal context.

Hands-on experience with formal concept analysis [edit]

The formal concept analysis can be used as a qualitative method for data analysis. Since the early ancestry of FBA in the early 1980s, the FBA research group at TU Darmstadt has gained experience from more than than 200 projects using the FBA (equally of 2005).[32] Including the fields of: medicine and cell biology,[33] [34] genetics,[35] [36] ecology,[37] software applied science,[38] ontology,[39] information and library sciences,[xl] [41] [42] office assistants,[43] police force,[44] [45] linguistics,[46] political science.[47]

Many more examples are eastward.chiliad. described in: Formal Concept Analysis. Foundations and Applications,[32] conference papers at regular conferences such as: International Conference on Formal Concept Analysis (ICFCA),[48] Concept Lattices and their Applications (CLA),[49] or International Conference on Conceptual Structures (ICCS).[50]

See as well [edit]

  • Association dominion learning
  • Cluster analysis
  • Commonsense reasoning
  • Conceptual analysis
  • Conceptual clustering
  • Conceptual space
  • Concept learning
  • Correspondence assay
  • Description logic
  • Factor analysis
  • Graphical model
  • Grounded theory
  • Inductive logic programming
  • Blueprint theory
  • Statistical relational learning
  • Schema (genetic algorithms)

Notes [edit]

  1. ^ a b Rudolf Wille, "Restructuring lattice theory: An approach based on hierarchies of concepts". Published in Rival, Ivan, ed. (1982). Ordered Sets. Proceedings of the NATO Advanced Study Institute held at Banff, Canada, August 28 to September 12, 1981. Nato Science Serial C. Vol. 83. Springer Netherlands. pp. 445–470. doi:10.1007/978-94-009-7798-3. ISBN978-94-009-7800-3. , reprinted in Ferré, Sébastien; Rudolph, Sebastian, eds. (12 May 2009). Formal Concept Assay: 7th International Conference, ICFCA 2009 Darmstadt, Germany, May 21–24, 2009 Proceedings. Springer Scientific discipline & Business Media. p. 314. ISBN978-364201814-five.
  2. ^ Hentig, von, Hartmut (1972). Magier oder Magister? Über die Einheit der Wissenschaft im Verständigungsprozeß. Klett (1972), Suhrkamp (1974). ISBN978-3518067079.
  3. ^ a b Johannes Wollbold: Attribute Exploration of Gene Regulatory Processes. PhD thesis, University of Jena 2011, p. 9
  4. ^ a b c d Ganter, Bernhard and Wille, Rudolf: Formal Concept Analysis: Mathematical Foundations. Springer, Berlin, ISBN 3-540-62771-5
  5. ^ Rudolf Wille, "Formal Concept Assay as Mathematical Theory of Concepts and Concept Hierarchies". In Ganter, Bernhard; Stumme, Gerd; Wille, Rudolf, eds. (2005). Formal Concept Analysis. Foundations and Applications. Springer Scientific discipline & Business organization Media. ISBN978-354027891-7.
  6. ^ Peter Rolf Lutzeier (1981), Wort und Feld: wortsemantische Fragestellungen mit besonderer Berücksichtigung des Wortfeldbegriffes: Dissertation, Linguistische Arbeiten 103 (in German), Tübingen: Niemeyer, doi:10.1515/9783111678726.fm, OCLC 8205166
  7. ^ Guigues, J.L. and Duquenne, V. Familles minimales d'implications informatives résultant d'un tableau de données binaires. Mathématiques et Sciences Humaines 95 (1986): five–18.
  8. ^ a b Ganter, Bernhard and Obiedkov, Sergei (2016) Conceptual Exploration. Springer, ISBN 978-3-662-49290-ane
  9. ^ Wille R. "The basic theorem of triadic concept analysis". Order 12, 149–158, 1995
  10. ^ a b Voutsadakis G. "Polyadic Concept Analysis". Club 19(3), 295–304, 2002
  11. ^ "Formal Concept Analysis and Fuzzy Logic" (PDF). Archived from the original (PDF) on 2017-12-09. Retrieved 2017-12-08 .
  12. ^ Wille, Rudolf (2000), "Boolean Concept Logic", in Ganter, B.; Mineau, Yard. Due west. (eds.), ICCS 2000 Conceptual Structures: Logical, Linguistic and Computational Problems, LNAI 1867, Springer, pp. 317–331, ISBN978-3-540-67859-5 .
  13. ^ Kwuida, Léonard (2004), Dicomplemented Lattices. A contextual generalization of Boolean algebras (PDF), Shaker Verlag, ISBN978-3-8322-3350-one
  14. ^ Wolff, Karl Erich (2010), "Temporal Relational Semantic Systems", in Croitoru, Madalina; Ferré, Sébastien; Lukose, Dickson (eds.), Conceptual Structures: From Information to Intelligence. ICCS 2010. LNAI 6208, Lecture Notes in Bogus Intelligence, vol. 6208, Springer-Verlag, pp. 165–180, doi:10.1007/978-3-642-14197-three, ISBN978-3-642-14196-6 .
  15. ^ Wolff, Karl Erich (2019), "Temporal Concept Analysis with SIENA", in Cristea, Diana; Le Ber, Florence; Missaoui, Rokia; Kwuida, Léonard; Sertkaya, Bariş (eds.), Supplementary Proceedings of ICFCA 2019, Conference and Workshops (PDF), Frankfurt, Germany: Springer, pp. 94–99 .
  16. ^ Wolff, Karl Erich (2004), "'Particles' and 'Waves' equally Understood by Temporal Concept Analysis.", in Wolff, Karl Erich; Pfeiffer, Heather D.; Delugach, Harry Southward. (eds.), Conceptual Structures at Work. twelfth International Conference on Conceptual Structures, ICCS 2004. Huntsville, AL, United states of america, July 2004, LNAI 3127. Proceedings, Lecture Notes in Artificial Intelligence, Berlin Heidelberg: Springer-Verlag, pp. 126–141, doi:x.1007/978-3-540-27769-9_8, ISBN978-3-540-22392-iv .
  17. ^ Kuznetsov Southward., Obiedkov S. Comparison Performance of Algorithms for Generating Concept Lattices, fourteen, Journal of Experimental and Theoretical Artificial Intelligence, Taylor & Francis, ISSN 0952-813X (print) ISSN 1362-3079 (online), pp.189–216, 2002
  18. ^ One can discover a not exhaustive listing of FCA tools in the FCA software website: "Formal Concept Analysis Software and Applications". Archived from the original on 2010-04-16. Retrieved 2010-06-10 .
  19. ^ "The Concept Explorer". Conexp.sourceforge.internet . Retrieved 27 December 2018.
  20. ^ "ToscanaJ: Welcome". Toscanaj.sourceforge.net . Retrieved 27 December 2018.
  21. ^ Boumedjout Lahcen and Leonard Kwuida. "Lattice Miner: A Tool for Concept Lattice Construction and Exploration". In: Supplementary Proceeding of International Conference on Formal concept assay (ICFCA'10), 2010
  22. ^ "The Coron System". Coron.loria.fr . Retrieved 27 December 2018.
  23. ^ "FcaBedrock Formal Context Creator". SourceForge.net . Retrieved 27 December 2018.
  24. ^ "GALACTIC GAlois LAttices, Concept Theory, Implicational system and Closures". galactic.univ-lr.fr . Retrieved two February 2021.
  25. ^ Belohlavek, Radim, and Vychodil, Vilem. "Discovery of optimal factors in binary data via a novel method of matrix decomposition". Journal of Computer and System Sciences 76.i (2010): 3–20.
  26. ^ Adomavicius C., Tuzhilin A. "Toward the side by side generation of recommender systems: a survey of the state-of-the-art and possible extensions". IEEE Transactions on Knowledge and Data Engineering, 17(6): 734–749, 2005.
  27. ^ Prelic, S. Bleuler, P. Zimmermann, A. Wille, P. Buhlmann, W. Gruissem, L. Hennig, L. Thiele, and E. Zitzler. "A Systematic Comparison and Evaluation of Biclustering Methods for Gene Expression Information". Bioinformatics, 22(9):1122–1129, 2006
  28. ^ a b Kaytoue M., Kuznetsov Southward., Macko J., Wagner Meira Jr., Napoli A. "Mining Biclusters of Similar Values with Triadic Concept Assay". CLA : 175–190, 2011
  29. ^ R. Thousand. Pensa, C. Leschi, J. Besson, J.-F. Boulicaut. "Assessment of discretization techniques for relevant pattern discovery from gene expression data". In G. J. Zaki, Southward. Morishita, and I. Rigoutsos, editors, Proceedings of the 4th ACM SIGKDD Workshop on Information Mining in Bioinformatics (BIOKDD 2004), 24–30, 2004.
  30. ^ Besson J., Robardet C. Raedt L.D., Boulicaut, J.-F. "Mining bi-sets in numerical data". In S. Dzeroski and J. Struyf, editors, KDID, LNCS 4747, p.11–23. Springer, 2007.
  31. ^ a b Cerf 50., Besson J., Robardet C., Boulicaut J.-F. "Closed patterns encounter n-ary relations". TKDD, 3(one), 2009
  32. ^ a b Bernhard Ganter; Gerd Stumme; Rudolf Wille, eds. (2005), Formal Concept Analysis. Foundations and Applications, Lecture Notes in Reckoner Scientific discipline, vol. 3626, Berlin Heidelberg: Springer Science & Business Media, doi:10.1007/978-3-540-31881-1, ISBN3-540-27891-v , retrieved 2015-11-14
  33. ^ Susanne Motameny; Beatrix Versmold; Rita Schmutzler (2008), Raoul Medina; Sergei Obiedkov (eds.), "Formal Concept Analysis for the Identification of Combinatorial Biomarkers in Breast Cancer", Icfca 2008, LNAI, Berlin Heidelberg: Springer, vol. 4933, pp. 229–240, ISBN978-3-540-78136-3 , retrieved 2016-01-29
  34. ^ Dominik Endres; Ruth Adam; Martin A. Giese; Uta Noppeney (2012), Florent Domenach; Dmitry I. Ignatov; Jonas Poelmans (eds.), "Agreement the Semantic Structure of Human fMRI Brain Recordings with Formal Concept Analysis", Icfca 2012, LNCS, Berlin Heidelberg: Springer, vol. 7278, pp. 96–111, doi:10.1007/978-3-642-29892-9, ISBN978-three-642-29891-2, ISSN 0302-9743, S2CID 6256292
  35. ^ Denis Ponomaryov; Nadezhda Omelianchuk; Victoria Mironova; Eugene Zalevsky; Nikolay Podkolodny; Eric Mjolsness; Nikolay Kolchanov (2011), Karl Erich Wolff; Dmitry Eastward. Palchunov; Nikolay Thousand. Zagoruiko; Urs Andelfinger (eds.), "From Published Expression and Phenotype Information to Structured Knowledge: The Arabidopsis Gene Internet Supplementary Database and Its Applications", Kont 2007, KPP 2007, LNCS, Heidelberg New York: Springer, vol. 6581, pp. 101–120, doi:x.1007/978-3-642-22140-8, ISBN978-3-642-22139-2, ISSN 0302-9743
  36. ^ Mehdi Kaytoue; Sergei Kuznetsov; Amedeo Napoli; Sébastien Duplessis (2011), "Mining factor expression data with blueprint structures in formal concept analysis" (PDF), Information Sciences, Elsevier, vol. 181, no. x, pp. 1989–2001, CiteSeerXten.1.1.457.8879, doi:10.1016/j.ins.2010.07.007, retrieved 2016-02-13
  37. ^ Aurélie Bertaux; Florence Le Ber; Agnès Braud; Michèle Trémolières (2009), Sébastien Ferré; Sebastian Rudolph (eds.), "Identifying Ecological Traits: A Concrete FCA-Based Approach", Icfca 2009, LNAI, Berlin Heidelberg: Springer-Verlag, vol. 5548, pp. 224–236, doi:10.1007/978-3-642-01815-2, ISBN978-3-642-01814-5, S2CID 26304023
  38. ^ Gregor Snelting; Frank Tip (1998), "Reengineering class hierarchies using concept assay", Proceeding. SIGSOFT '98/FSE-6, New York: ACM, vol. 23, no. vi, pp. 99–110, doi:10.1145/291252.288273, ISBN1-58113-108-9 , retrieved 2016-02-04
  39. ^ Gerd Stumme; Alexander Maedche (2001), Universität Leipzig (ed.), "FCA-Merge: Bottom-up merging of ontologies" (PDF), IJCAI, Leipzig, pp. 225–230, archived from the original (PDF) on 2016-02-13, retrieved 2016-02-thirteen
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References [edit]

  • Ganter, Bernhard; Stumme, Gerd; Wille, Rudolf, eds. (2005), Formal Concept Analysis: Foundations and Applications, Lecture Notes in Bogus Intelligence, no. 3626, Springer-Verlag, ISBN3-540-27891-5
  • Ganter, Bernhard; Wille, Rudolf (1998), Formal Concept Analysis: Mathematical Foundations, translated by C. Franzke, Springer-Verlag, Berlin, ISBNthree-540-62771-five
  • Carpineto, Claudio; Romano, Giovanni (2004), Concept Information Analysis: Theory and Applications, Wiley, ISBN978-0-470-85055-8
  • Wolff, Karl Erich (1994), F. Faulbaum in StatSoft 1993 (ed.), A first course in Formal Concept Assay (PDF), Gustav Fischer Verlag, pp. 429–438, archived from the original (PDF) on 2006-03-23
  • Davey, B.A.; Priestley, H. A. (2002), "Chapter 3. Formal Concept Analysis", Introduction to Lattices and Society, Cambridge University Press, ISBN978-0-521-78451-1

External links [edit]

  • A Formal Concept Analysis Homepage
  • Demo
  • 11th International Conference on Formal Concept Analysis. ICFCA 2013 – Dresden, Germany – May 21–24, 2013

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Source: https://en.wikipedia.org/wiki/Formal_concept_analysis

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