Structure-Based Attribute Reduction: A Rough Set Approach - Feature Selection for Data and Pattern Recognition - page 111

U

={

u 1 , u 2 ,..., u n } and C

={

c 1 , c 2 ,..., c m } , where n

=|

U

|

and m

=|

C

|

.

Moreover, we define the decision attribute values as V d ={

, ,..., }

.
Remark 1 Decision tables are identical to data sets or data tables for the classifi-
cation problem or the supervised learning in the data mining or machine learning
literature, in which condition attributes are called attributes or independent variables,
the decision attribute is a class attribute or dependent variable, and objects are tuples
or samples. In that literature, each object is given by a tuple of attribute values with
a class label (decision attribute value). However, we use the form of decision tables
in this chapter by two reason. One is that we often deal with subsets of the attributes,
so we prefer to let the symbols of the attributes be explicit. The other is to emphasise
the view that a relation (e.g. the equivalence relation) on the object set is induced
from a structure of the attribute value space (e.g. equivalence of values) through the
attributes (functions).

1 2 p

Example 1 Consider a decision table D = ( U , C ∪{ d } , { V a } ) about car evaluations

in Table 7.1 , where U ={ u 1 , u 2 ,..., u 7 } , C ={ Pr , Ma , Sa } and d = Ev. The

attribute value sets are given by V Pr = V Ma = V Sa ={ low , med , high } , V Ev =

{ . Condition attributes Pr, Ma, and Sa indicate price, maintenance

cost, and safety of a car, respectively, by values high, med (medium), and low.
Decision attribute Ev means evaluation of a car by some customer(s).
The value of u 1 with respect to Pr is Pr

unacc , acc , good }

(

u 1 ) = high, and that of u 2 with respect

to Ev is Ev ( u 2 ) = unacc. The value tuple of u 4 with respect to C ={ Pr , Ma , Sa } is

C ( u 4 ) = ( med , high , low ) .

AT , we define an indiscernibility relation on U
with respect to A , denoted by R A , as follows:

Given an attribute subset A ↆ

u ) ∈ U 2 u ), R A ={ ( u , | a ( u ) = a ( for any a ∈ A } .

R A is the set of the object pairs each of which is indiscernible by the given attributes
A . Obviously, R A is an equivalence relation, which is reflexive, symmetric, and
transitive. From R A , we define the equivalence class of an object u ∈ U , denoted by

R A ( u ) , as follows:

u ∈ u , R A ( u ) ={ U | ( u ) ∈ R A } .

Table 7.1 Decision table of
car evaluations

Car Pr Ma Sa Ev

u 1 High High Low Unacc

u 2 Med Med Med Unacc

u 3 Med Med Med Acc

u 4 Med High Low Acc

u 5 Med Med High Acc

u 6 Med Med High Good

u 7 Low Med Med Good