The notion of concept is then nothing but a cluster since it
consists of objects grouped together by proximity or similarity (here, according to
a common intent). In the closed itemset mining framework (Pasquier et al., 2005;
Valtchev et al., 2002b; Zaki et al., 2002), X and Y correspond to the notion of closed
tidset and closed itemset respectively.
Table 2. Context K = (O={1, 2,...,8}, A={a, b,..., h, i}, R)
A
A1 A2
Tid a b c d e f g h i
1 X X X
2 X X X X
3 X X X X X
4 X X X X X
5 X X X X
6 X X X X X
7 X X X X
8 X X X X
260 Missaoui, Jatteau, Boujenoui, & Naouali
Copyright ?© 2007, Idea Group Inc. Copying or distributing in print or electronic forms without written permission of
Idea Group Inc. is prohibited.
The set G of all concepts extracted from the context K is partially ordered by intent/
extent inclusion:
(X1, Y1) ?‰¤ (X2, Y2) ?‡” X1 ??† X2, Y2 ??† Y1.
A concept (Galois) lattice associated with context K is then a complete lattice L=B(O,
A, R) = ??© G, ?‰¤???, where concepts in G are linked according to a partial order.
The Hasse diagram of the lattice L drawn from Table 2 is shown on Figure 1 where
tidsets and itemsets are indicated on both sides of nodes. For example, node #11
represents the concept (678, acd) which means that the itemset acd is a closed itemset
with three supporting objects: 6, 7, and 8.
A subset B of concepts in G is an order filter (order ideal respectively) if ???a ??? G, b
??? B, b ?‰¤ a ?‡’ a???B (a ?‰¤ b ?‡’ a ??? B respectively).
Pages:
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487