13,094
edits
Changes
mLine 1:
Line 1: − + − A '''Concept lattice'''' or formal concept analysis is a principled way of automatically deriving an [[ontology]] from a collection of objects and their properties. The term was introduced by Rudolf Wille in 1984, and builds on applied [[lattice theory|lattice]] and order theory that was developed by [[Garrett Birkhoff|Birkhoff]] and others in the 1930's.+ − + + + + + + + + + + + + + + + + + + + + + + + + + Line 11:
Line 35: − + Line 19:
Line 43: − +
Text replacement - "http://" to "https://"
[[Image:lighterstill.jpg]]
[[Image:lighterstill.jpg]][[Image:Star-Lattice_2.jpg|right|frame]]
'''Lattice''' work is an ornamental, lattice framework consisting of a criss-crossed pattern of strips of building material, usually wood or metal, but it can be made of any building [[material]]. The [[design]] is created by crossing the strips to form a decorative network. Latticework can also be used to support a [[structure]], such as lattice girder bridge supports.
==Intuitive description==
*Lattice is defined as "a structure of crossed strips arranged to form a regular pattern of open spaces".
*In India the house of a rich or noble person may be built with a baramdah or verandah surrounding every level leading to the living area. The upper floors often have balconies overlooking the street that are shielded by screens jaalis carved in stone latticework, allowing privacy and coolness.
==Mathematics==
[[Image:Lattice_of_partitions_of_an_order_4_set2.jpg|right|frame|<center>The name "lattice" is suggested by the form of the "Hasse diagram" depicting it.</center>]]
In [[mathematics]], a '''lattice''' is a partially ordered set (also called a ''poset'') in which subsets of any ''two elements'' have a unique supremum (the elements' least upper bound; called their '''|join''') and an infimum (greatest lower bound; called their '''meet'''). Lattices can also be characterized as algebraic [[structure]]s satisfying certain axiomatic |identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras. These "lattice-like" structures all admit order-theoretic as well as algebraic descriptions.[https://en.wikipedia.org/wiki/Lattice_(order)]
===Concept lattice===
Concept lattice or formal concept analysis is a principled way of automatically deriving an [[ontology]] from a collection of objects and their properties. The term was introduced by Rudolf Wille in 1984, and builds on applied [[lattice theory|lattice]] and order theory that was developed by [[Garrett Birkhoff|Birkhoff]] and others in the 1930's.
===Intuitive description===
Formal [[concept]] [[analysis]] refers to both an unsupervised machine learning technique and, more broadly, a method of [[data analysis]]. The approach 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
Formal [[concept]] [[analysis]] refers to both an unsupervised machine learning technique and, more broadly, a method of [[data analysis]]. The approach 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" ''object'' cluster is the set of all objects that share a common subset of attributes, and
Note the strong parallel between "natural" property clusters and definitions in terms of individually necessary and jointly sufficient conditions, on one hand, and between "natural" object clusters and the extensions of such definitions, on the other. Provided the input objects and input concepts provide a complete description of the world (never true in practice, but perhaps a reasonable approximation), then the set of attributes in each concept can be interpreted as a set of singly necessary and jointly sufficient conditions for defining the set of objects in the concept. Conversely, if a set of attributes is ''not'' identified as a concept in this framework, then those attributes are not singly necessary and jointly sufficient for defining ''any'' non-empty subset of objects in the world.
Note the strong parallel between "natural" property clusters and definitions in terms of individually necessary and jointly sufficient conditions, on one hand, and between "natural" object clusters and the extensions of such definitions, on the other. Provided the input objects and input concepts provide a complete description of the world (never true in practice, but perhaps a reasonable approximation), then the set of attributes in each concept can be interpreted as a set of singly necessary and jointly sufficient conditions for defining the set of objects in the concept. Conversely, if a set of attributes is ''not'' identified as a concept in this framework, then those attributes are not singly necessary and jointly sufficient for defining ''any'' non-empty subset of objects in the world.
== Example ==
=== Example ===
[[Image:Concept_lattice2.jpg|right|frame|<center>A concept lattice for objects consisting of the integers from 1 to 10, and attributes composite (c), square (s), even (e) odd (o) and prime (p).</center><center>The lattice is drawn as a Hasse diagram</center>]]
[[Image:Concept_lattice2.jpg|right|frame|<center>A concept lattice for objects consisting of the integers from 1 to 10, and attributes composite (c), square (s), even (e) odd (o) and prime (p).</center><center>The lattice is drawn as a Hasse diagram</center>]]
It can readily be seen that both of these example concepts satisfy the formal definitions below
It can readily be seen that both of these example concepts satisfy the formal definitions below
The full set of concepts for these objects and attributes is shown in the illustration. It includes a concept for each of the original attributes: the composite numbers, square numbers, even numbers, odd numbers, and prime numbers. Additionally it includes concepts for the even composite numbers, composite square numbers (that is, all square numbers except 1), even composite squares, odd squares, odd composite squares, even primes, and odd primes.[http://en.wikipedia.org/wiki/Concept_lattice]
The full set of concepts for these objects and attributes is shown in the illustration. It includes a concept for each of the original attributes: the composite numbers, square numbers, even numbers, odd numbers, and prime numbers. Additionally it includes concepts for the even composite numbers, composite square numbers (that is, all square numbers except 1), even composite squares, odd squares, odd composite squares, even primes, and odd primes.[https://en.wikipedia.org/wiki/Concept_lattice]
[[Category: Mathematics]]
[[Category: Mathematics]]