Formal Concept Analysis
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This document discusses formal concept analysis (FCA) as a method for analyzing data. FCA takes a data set consisting of objects and their attributes and identifies natural clusters of objects that share attributes and natural clusters of attributes shared by objects. These clusters form a concept lattice that represents the conceptual relationships in the data. FCA can be used to gain insight from data, foresee patterns and relationships, and structure data for delivery and presentation. The document provides examples and explanations of key FCA terms and processes like formal contexts, derivation, concepts and implications.
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Formal Concept Analysis
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- 6. Formal Concept Analysis? Lets start with understanding a concept? “Orangutan” Orangutan Mamorset Baboon … Has black fur Has tail Has two legs … objects related to attributes Objects, attributes and a relation form a formal concept
- 7. The Universe of Discourse A repertoire of objects and attributes (which might or might not be related) constitutes the „context“ of our considerations Orangutan Mamorset Baboon … Has black fur Has tail Has two legs … Object_1 Object_2 Object_3 Attribute_1 Attribute_2 Attribute_3 relation objects attributes Attribute_4
- 8. Formal Concept Analysis? • Formal Concept Analysis is a method used for investigating and processing explicitly given information, in order to allow for meaningful and comprehensive interpretation • An analysis of data • Structures of formal abstractions of concepts of human thought • Formal emphasizes that the concepts are mathematical objects, rather than concepts of mind
- 9. Formal Concept Analysis? • Formal Concept Analysis takes as input a matrix specifying a set of objects and the properties thereof, called attributes, and finds both all the “natural” clusters of attributes and all the “natural” clusters of objects in the input data, where • a “natural” object cluster is the set of all objects that share a common subset of attributes, and • a “natural” property cluster is the set of all attributes shared by one of the natural object clusters
- 10. Formal Concept Analysis? • Natural property clusters correspond one-for-one with natural object clusters, and a concept is a pair containing both a natural property cluster and its corresponding natural object cluster • The family of these concepts obeys the mathematical axioms defining a lattice, and is called a concept lattice
- 11. FCA: Formal Context? • Context: A triple (G, M, I) is a (formal) context if • G is a set of objects (Gegenstand) • M is a set of attributes (Merkmal) • I is a binary relation between G and M called incidence • Incidence relation: I ⊆ G x M • if gG, mM in (g,m)I, then we know that “object g has attribute m„ and we write gIm • Derivation operators: • For A ⊆ G, A‘={mM | (g,m)I for all gA} • For B ⊆ M, B‘={gG | (g,m)I for all mB}
- 12. FCA: Formal Context? • A pair (A,B) is a formal concept of (G,M,I) if and only if • A ⊆ G • B ⊆ M • A‘ = B, and A = B‘ • Note that at this point the incidence relationship is closed; i.e. all objects of the concept carry all its attributes and that there is no other object in G carrying all attributes of the concept • A is called the extent (Umfang) of the concept (A,B) • B is called the intent (Inhalt) of the concept (A,B)
- 13. FCA: Generating a Formal Context • Using the derivation operators we can derive formal concepts from our formal context with the following routine: 1.Pick a set of objects A 2.Derive the attributes A' 3.Derive (A')' 4.(A'',A') is a formal concept • A dual approach can be taken starting with an attribute
- 14. Example 1.Pick any set of objects A, e.g. A={orangutan}. 2.Derive the attributes A'={big, two legs, black fur, long tail, swim} 3.Derive (A')'={big, two legs, black fur, long tail, swim}'={orangutan, spider monkey} 4.(A'',A')=({orangutan, spider monkey},{big, two legs, black fur, long tail, swim}) is a formal concept.
- 15. FCA: Concept Lattice? • The concepts of a given context are naturally ordered by a subconcept-superconcept relation: • (A1,B1) ≤ (A2,B2) :⇔ A1⊆A2 (⇔ B2⊆B1) • The ordered set of all formal concepts in (G,M,I) is denoted by B(G,M,I) and is called concept lattice (Begriffsverband) • A concept lattice consists of the set of concepts of a formal context and the subconcept-superconcept relation between the concepts
- 16. FCA: Example
- 17. FCA: Extent and Intent in a Lattice • The extent of a formal concept is given by all formal objects on the paths which lead down from the given concept node • The extent of an arbitrary concept is then found in the principle ideal generated by that concept • The intent of a formal concept is given by all the formal attributes on the paths which lead up from the given concept node • The intent of an arbitrary concept is then found in the principle filter generated by that concept
- 18. FCA: Subconcepts in the Concept Lattice • The Concept B is a subconcept of Concept A because • The extent of Concept B is a subset of the extent of Concept A • The intent of Concept B is a superset of the intent of Concept A • All edges in the line diagram of a concept lattice represent this subconcept-superconcept relationship Intent: Two legs, long tail, swim, small Extent: squirrel monkey Intent: brown fur, two legs, long tail, swim, small Extent: mamorset Concept “A” Concept “B”
- 19. FCA: Implication • An implication A → B (between sets A,BM of attributes) holds in a formal context if and only if B⊆A‘‘ • i.e. if every object that has all attributes in A also has all attributes in B • e.g. if X has feather and has beak then is a bird • The implication determines the concept lattice up to isomorphism and therefore offers an additional interpretation of the lattice structure • Implications can be used for a step-wise construction of conceputal knowledge
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- 23. What is DKAN? • DKAN is a Drupal (7) based Open Data Publishing suite that is compatible with CKAN.
- 24. Why Drupal:DKAN as Data > Content Delivery System? • DKAN is built with Drupal making it a secured Open Data Publishing Portal and at the same time a robust enterprise grade content management system • Easy to publish dataset • Model Data presentation standards • Line Graphs • Pie – share Graph • Geographic Information System • Open Data and Open Government Compliant • Share Data Presentations via API
- 25. Data Farm > Middleware > DKAN Data Farm Analytics Platform Middleware Publish Data – form matching Model Data Delivery Standard Share Publish Data Presentation Mobile App Social Media Partner Website Community Board
- 26. Thank You!!