The function ?? maps a concept (X, Y) from the lattice L into a concept of the resulting
lattice L3 by selecting a set O1 of objects:
?? ((X,Y))=(X ??© O1, (X ??© O1)').
Figure 2. Projection of L on two distinct attribute subsets abcd and efghi to produce
L1 and L2
266 Missaoui, Jatteau, Boujenoui, & Naouali
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For example (see Figure 2), the selection on L1 on the set {1, 2, 3, 4} is a lattice that
has the following concept set: {(1234, a), (123, ab), (34, ac), (3, abc), (??…, abcd)}.
When O1 represents the extent of an existing concept c in L, then L3 is nothing but
the principal ideal of c enriched with the partial order.
Operations.on.Data.Mining.Output
In the following, we show how known OLAP operations (drill-down, rollup, slice,
and dice) can be translated in our data mining concept lattice framework as a combination
of operations on lattices such as selection, projection, and assembly (noted
respectively by ??, ? , and ?—). As stated earlier in the section Integrating DM with
DW Techniques, these operations can be assimilated to cubing for exploratory data
mining.
The operations can be defined as follows:
??? Rollup: go up the attribute hierarchy of one or many attributes (e.
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