Knowledge Management Institute 707.009 Foundations of Knowledge Management g g „Categorization & Formal Concept Analysis“ Markus Strohmaier Univ. Ass. / Assistant Professor Knowledge Management Institute Graz University of Technology, Austria e-mail: markus.strohmaier@tugraz.at web: http://www.kmi.tugraz.at/staff/markus Markus Strohmaier 2011 1
Knowledge Management Institute Slides in part based on • Gerd Stumme – Course at Otto-von-Guericke Universität Magdeburg / Summer Term 2003 – ECML PKDD Tutorial ECML PKDD Tutorial • Rudolf Wille – „Formal Concept Analysis as Mathematical Theory of Concepts and Concept Hierarchies“, In Formal Concept Analysis, Eds B. Ganter et al., C t Hi hi “ I F l C t A l i Ed B G t t l LNAI 3626, pp1-33, (2005) Further Literature: http://www.aifb.uni-karlsruhe.de/WBS/gst/FBA03/chapter1_2.pdf (Ganter / Stumme) Markus Strohmaier 2011 2
Knowledge Management Institute Overview T d Today‘s Agenda: ‘ A d Categorization & Formal Concept Analysis • Formal Context • Formal Concepts • Formal Concept Lattices • FCA Implications • Constructing Concept Lattices Markus Strohmaier 2011 3
Knowledge Management Institute Categorization Categorization [Mervis Rosch 1981] Intension (Meaning) • The specification of those qualities that a thing must have to be a member of the class Extension (the objects in the class) Extension (the objects in the class) • Things that have those qualities Markus Strohmaier 2011 4
Knowledge Management Institute Categorization Categorization [Mervis Rosch 1981] Six salient problems : Six salient problems : • Arbitrariness of categories . Are there any a priori reasons for dividing objects into categories, or is this division initially arbitrary? • Equivalence of category members . Are all category members equally representative of the category as has often been assumed? • Determinacy of category membership and representation . Are categories specified by necessary and sufficient conditions for membership? Are boundaries of categories well defined? • The nature of abstraction . How much abstraction is required--that is, do we need only memory for individual exemplars to account for categorization? Or, at the other extreme, are higher-order abstractions of general knowledge, beyond the individual categories, necessary? • Decomposability of categories into elements . Does a reasonable explanation of objects consist in their decomposition into elementary qualities? • The nature of attributes . What are the characteristics of these "attributes“ into which categories are to be decomposed? Markus Strohmaier 2011 5
Knowledge Management Institute Formal Concept Analysis Running Example: Taste: Sweet/Sour, Shape: Round/Long/, Color: Red/Yellow/.., Texture: Smooth/Bumpy, Ich möchte nur darauf hinweisen, daß es eine Zeit gab, in der man die Ähnlichkeit der I h ö ht d f hi i d ß i Z it b i d di Äh li hk it d Empfindungen zur Basis der Kategorisierung von Pflanze und Tier gemacht hat. Man denke [...] an die frühen Taxonomien des Ulisse Aldrovandi aus dem 16. Jahrhundert, der die scheußlichen Tiere (die Spinnen, Molche und Schlagen) und die Schönheiten (die scheußlichen Tiere (die Spinnen, Molche und Schlagen) und die Schönheiten (die Leoparden, die Adler usw.) zu eigenen Gruppen [von Lebewesen] zusammenfasste . Heinz von Foerster, Wahrheit ist die Erfindung eines Lügners, Page 22/23 [Mervis Rosch 1981] Markus Strohmaier 2011 6
Knowledge Management Institute Terminology ISO 704: Terminology Work: Principles and methods ISO 704 T i l W k P i i l d th d DIN 2330: Begriffe und Ihre Benennungen Name Definition Representation level Apple: A l C Concept t Taste attribute a Concept level Color attribute b Shape Shape attribute c attribute c Object 1 Object 1 Object 2 Object 2 Object 3 Object 3 property A property A property A Object level property B property B property B property C property C property C property C property C property C Markus Strohmaier 2011 7
Knowledge Management Institute Formal Concept Analysis Formal Concept Analysis [Wille 2005] M d l Models concepts as units of thought . A concept is ht A t i t it f th constituted by its: •Extension: consists of all objects belonging to a Extension: consists of all objects belonging to a concept •Intension: consists of all attributes common to all those •Intension: consists of all attributes common to all those objects Concepts „live“ in relationships with many other concepts where the sub-concept-superconcept-relation concepts where the sub concept superconcept relation plays a prominent role. Markus Strohmaier 2011 8
Knowledge Management Institute Formal Concept Analysis Formal Concept Analysis [Wille 2005] Formal context: Formal context: A Formal Context is a tripel (G, M, I) for which G and M are sets while I is a binary relation between G and M. a binary relation between G and M Formal Concept: A formal concept of a formal context K := ( G,M, I ) is defined as a pair ( A,B ) with and A = B´ , and B = A´ ; A and B are called the extent and the intent of the formal concept ( A,B ), respectively. Markus Strohmaier 2011 9
Knowledge Management Institute Formal Concept Analysis Def.: A formal context is a tripel ( G,M,I ), where Taste: Sweet/Sour, Shape: Round/Long/, Color: Red/Yellow/ Color: Red/Yellow/.., Texture: Smooth/Bumpy, Texture: Smooth/Bumpy • G is a set of objects, • M is a set of attributes • and I is a relation between G and M . ( g,m ) I is read as „object g has attribute m “. Markus Strohmaier 2011 10
Knowledge Management Institute Formal Concept Analysis Derivation Operators Taste: Sweet/Sour, Shape: Round/Long/, Color: Red/Yellow/ Color: Red/Yellow/.., Texture: Smooth/Bumpy, Texture: Smooth/Bumpy • A G, B M (A…Extent, B…Intent) • all attributes shared by all objects of A • all objects having all attributes of B all objects having all attributes of B A 1 ´ X X X A formal concept is A 1 defined as a pair ( A,B ) X X X A = B´ , and B = A´ X X X Markus Strohmaier 2011 11
Knowledge Management Institute Formal Concept Analysis Def.: A formal concept is a pair ( A,B ), with Taste: Sweet/Sour, Shape: Round/Long/, Color: Red/Yellow/ Color: Red/Yellow/.., Texture: Smooth/Bumpy, Texture: Smooth/Bumpy • A G, B M all attributes shared by all objects of A all objects having all attributes of B j g Intent • A´=B and B´= A Set A is called the extent Set A is called the extent t Extent (a set of objects) Set B is called the intent (a set of attributes) Of the formal concept (A,B) Markus Strohmaier 2011 12
Knowledge Management Institute Formal Concept Analysis Sub/Superconcept Relation Taste: Sweet/Sour, Shape: Round/Long/, Color: Red/Yellow/.., Texture: Smooth/Bumpy, Color: Red/Yellow/ Texture: Smooth/Bumpy • A G, B M • all attributes shared by all objects of A • all objects having all attributes of B j g B 1 ↔ A 1 ´ B 2 ↔ A 2 ´ X X X A 2 A A 1 X X X X X X The orange concept is a subconcept of the blue concept, since its extent is contained in the blue one (equivalent to the blue intent is contained in the orange one) blue one. (equivalent to the blue intent is contained in the orange one) Markus Strohmaier 2011 13
Knowledge Management Institute Formal Concept Analysis Formal Concept Analysis Concept Lattices Concept Lattices (cf. Galois Lattices) Taste: Sweet/Sour, Shape: Round/Long/, Color: Red/Yellow/.., Texture: Smooth/Bumpy, Color: Red/Yellow/ Texture: Smooth/Bumpy Where is C1 & C2 located? A 2 ´ A 1 ´ X X X X X X A 1 1 A 2 X X X X X X X X X C1 Extent Intent Formal Concept C1 (A1, A1´) C2 The set of objects that are „yellow“, „sweet“ and „smooth“ „sweet and „smooth Markus Strohmaier 2011 15
Knowledge Management Institute Formal Concept Analysis Formal Concept Analysis Concept Lattices Concept Lattices (cf. Galois Lattices) Taste: Sweet/Sour, Shape: Round/Long/, Color: Red/Yellow/ Color: Red/Yellow/.., Texture: Smooth/Bumpy, Texture: Smooth/Bumpy A 1 ´ X X X X X X ) (Attributes) A 1 1 X X X X X X X X X Intent A1 cts) xtent (Objec Ex Markus Strohmaier 2011 16
Knowledge Management Institute Formal Concept Analysis Formal Concept Analysis Concept Lattices Def.: The concept lattice of a formal context ( G,M,I ) is Th f f l t t ( G M I ) i D f t l tti the set of all formal concepts of ( G,M,I ), together with the partial order The concept lattice is denoted by B ( G,M,I ) . Theorem: The concept lattice is a lattice, i.e. for two concepts ( A 1 , B 1 ) and ( A B ) there is always and ( A 2 , B 2 ), there is always • a greatest common subconcept • and a least common superconcept p p Markus Strohmaier 2011 17
Knowledge Management Institute Formal Concept Analysis Formal Concept Analysis Greatest Common Subconcept Which objects share the attributes „smooth“ and „red“ and „sour“? A: Grapes, Apples greatest common subconcept (infimum) • a greatest common subconcept • and a least common superconcept p p Markus Strohmaier 2011 18
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