Dependency Analysis and Attribute Reduction in the Probabilistic Approach to Rough Sets - Feature Selection for Data and Pattern Recognition - page 98

6.4.2 Decision Tables

A knowledge representation system [ 11 ] is a pair , where U is a universe and

A is a nonempty and finite set of attributes defined on U . In the context of rough
set approach, decision tables are constructed in terms of knowledge representation
systems as follows.
Let C

( U , A )

A be two disjoint subsets of attributes, called condition and deci-
sion attributes, respectively. The condition attributes generate the partitioning of the
universe U into classes of objects having identical values of attributes belonging
to C , thus forming the structure of approximation space on U . The corresponding
collection of elementary sets of this approximation space is denoted by U/C . Simi-
larly, the decision attributes D induce a structure of approximation space on U , with
U/D denoting its elementary sets. The knowledge representation systemwith defined
condition and decision attributes is called a decision table [ 11 ]. Decision tables fall
into two broad groups: deterministic decision tables and non-deterministic decision
tables .
Deterministic decision tables describe the functional relation between a set of
observations (inputs, conditions) and the corresponding decisions (outcomes). In
practice, deterministic decision knowledge is not always available. When only some,
but not all, decisions can uniquely be determined by combinations of attribute values,
the decision table is called non-deterministic. In a non-deterministic decision table,
the relationship between conditions and decisions is only partially functional.
Compared to the previous two types of decision tables, which are based on the
original rough set theory, a probabilistic decision table is developed within the frame-
work of the variable precision rough set theory. It contains some built-in probabilistic
measures to help in the process of decision making or prediction in non-deterministic
cases.
When defining the probabilistic decision tables, we focus on elementary sets (our
target sets) of the decision attribute D , X

, D ↂ

∈ U/D , of the partition generated by the

decision attributes.
For a given target set X , the probabilistic decision table can be defined as a
mapping associating each combination of condition attribute values, corresponding
to an elementary set E ∈ U/C , with a triple of values representing:

1. the unique designation of the rough approximation region (positive, negative, or
boundary region),
2. the respective values of the elementary set probability P

( E ) , and

3. the conditional probability P ( X | E ) .

In practice, when deriving a probabilistic decision table, the measures of P ( E )

and P are usually computed based on available data. An example probabilistic

decision table is shown in Table 6.1 . It should be noted at this point, that while prob-
abilistic decision tables are containing information about set approximation regions
of the variable precision rough set model, and consequently depend on the settings of
the parameters l and u , similar decision tables can be constructed based on Bayesian

( X | E )