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approximations, called singleton, subset, and concept approximations [7]. The
singleton lower and upper approximations were studied in [14,15,23-25]. Sim-
ilar ideas were studied in [2, 22, 26-28]. In this chapter we further discuss
applications to data mining of all three kinds of approximations: singleton,
subset and concept. As it was observed in [7], singleton lower and upper ap-
proximations are not applicable in data mining.
The next topic of this chapter is demonstrating how certain and possible
rules may be computed from incomplete decision tables. An extension of the
well-known LEM2 (Learning from Examples Module, version 2) rule induction
algorithm [1, 5], called MLEM2, was introduced in [6]. LEM2 is a component
of the LERS (Learning from Examples based on Rough Sets) data mining
system. Originally, MLEM2 induced certain rules from incomplete decision
tables with numerical attributes and with missing attribute values interpreted
as lost. Using the idea of lower and upper approximations for incomplete
decision tables, MLEM2 was further extended to induce both certain and
possible rules from a decision table with some numerical attributes and with
some attribute values being lost and some attribute values being “do not care”
conditions.
A preliminary version of this chapter was presented at the Workshop on
Foundation of Data Mining, associated with the Fourth IEEE International
Conference on Data Mining, Brighton, UK, November 1-4, 2004 [10].
2 Complete Data: Elementary Sets and Indiscernibility
Relation
An example of a decision table, taken from [9], is presented in Table 1.
Rows of the decision table represent
cases
, while columns are labeled by
variables
. The set of all cases will be denoted by
U
. In Table 1,
U
=
{
1
,
2
,...,
8
}
.
Table 1.
A complete decision table
Attributes
Decision
Case
Temperature
Headache
Nausea
Flu
1
High
Yes
No
Yes
2
Very high
Yes
Yes
Yes
3
High
No
No
No
4
High
Yes
Yes
Yes
5
High
Yes
Yes
No
6
Normal
Yes
No
No
7
Normal
No
Yes
No
8
Normal
Yes
No
Yes
