Toward Integrating Data Warehousing with Data Mining Techniques 2
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Concept Lattices
Let K = (O, A, R) be a formal context (see Table 2), where O, A, and R are a set of
objects (e.g., transactions), a set of attributes or properties (e.g., items in a transaction
database), and a binary relation between O and A, respectively. Two functions f and
g summarize the links between subsets of objects and subsets of attributes induced
by R. Function f maps a set of objects into a set of common attributes, whereas g
is the dual for attribute sets:
??? f:P (O)?†’ P (A), f(X)=X??™={a ??? A|???o ??? X, oRa}, where P (O) is the power set
of O.
??? g: P (A)?†’ P (O), g(Y)=Y??™={o ??? O|???a ??? Y, oRa}.
Table 2 shows, for example, that f({2, 6}) = {a, b} and g({a, c, d}) = {6, 7, 8}1.
Furthermore, the compound operators g?°f(X) and f?°g(Y) (denoted by '') are closure
operators over P (O) and P (A) respectively. This means, in particular, that Z ??† Z''
and (Z'')''=Z'' for any Z??? P (A) or Z ??? P (O).
A formal concept c is a pair of sets (X, Y) where X ??? P (O), Y ??? P (A), X=Y??™ and
Y=X??™. X is called the extent of c (denoted by Extent(c)) and Y represents its intent
(denoted by Intent(c)).
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