Quasi dominance rough set approach

In this section the novel methodology developed by the author (Cyran 2009d), called quasi dominance rough set approach (QDRSA) is presented. QDRSA can be considered as a hybrid of classical rough set approach (CRSA) and dominance rough set approach (DRSA).

After presenting this methodology, the advantages of QDRSA over CRSA and DRSA are illustrated for certain class of problems together with limitations of proposed methodology for other types of problems where CRSA or DRSA are better choice. The analysis of the reasons why QDRSA can produce decision algorithms yielding smaller error rates than DRSA is performed on the real world example, presented in section 4.3.3. This example shows that superiority of QDRSA over CRSA and DRSA in certain types of applications is of practical value.

The DRSA is claimed to have many advantages over CRSA in applications with natural preference-ordered attributes. Not denying this statement in general, it is possible to demonstrate the example of such information system S with preference-ordered attributes, which, when treated as a decision table, can yield better (in the sense of decision error) decision algorithm A than that generated by DRSA (A_DRSA).

The superiority of algorithm A is also true (however in the sense of larger generality level) when the aforementioned algorithm A is compared with the algorithm A_CRSA obtained by application of CRSA. The quasi dominance rough set approach is the framework within which the algorithm A can be derived. That is why the algorithm A will be referred to as AQDRSA.

The QDRSA can be considered as a hybrid of CRSA and DRSA. Like DRSA it is dedicated for problems with preference-ordered attributes, but contrary to DRSA, it does not resign from the classical indiscernibility.

Definition 2.3:28 (Indiscernibility relation in QDRSA, after Cyran 2009d)

For the information system S = (U, Q, Vq, f ) in which Q = C  {d} and for any x, y  U the I_QDRSA is defined as

 

   

I

x _QDRSA C    , ,  , . (2.3:5)

▬

Comparison of formula (1) in Definition 3 with formula (5) in Definition 28 reveals that the eqivalence relations I0 and IQDRSA are identical. Therefore, the notions of lower and upper approximations, as well as particularly important for classification notions of quality of approximation, (relative) cores, (relative) reducts and (relative) value reducts are defined in QDRSA like in CRSA.

In particular, it follows in QDRSA that an attribute q  C is redundant in C, where C  Q, if the indiscernibility relation I_QDRSA(B) is identical to the indiscernibility relation IQDRSA (C – {q}) or, what is equivalent, if the attribute q is functionally dependent of the subset C – {q}, what can be denoted as C – {q}  q. If IQDRSA (C)  IQDRSA (C – {q}) then the attribute q  C is irredundant in C (i.e. it is irremovable from C). Set of attributes C  Q is independent if each attribute q  C is irredundant in C. Otherwise, a set of attributes C  Q is dependent.

Definition 2.3:29 (Reduct, after Pawlak 1995a, adapted to QDRSA)

In QDRSA, similarly like in CRSA, each set Q’  Q is a reduct of the set Q in the information system S = <U, Q, v, f > if Q’ is independent and if I₀(Q’) = I₀(Q).

▬

It follows that reduct Q’ is the smallest (in the sense of sets inclusion) subset of attributes which generates the same classification of the elements in the universe U, as the complete set of attributes Q does. At the same time, the reduct Q‟ is the largest (in the sense of sets inclusion) independent subset of the set Q. So, the attributes not belonging to the reduct Q‟, as being dependent of attributes of this reduct, are redundant for the classification of elements in the universe U. In given information system there could be many reducts, and moreover, it is possible to define not only a reduct of the complete set of attributes Q, but also reducts of some subsets C  Q.

Denote the set of all reducts of the set C in the information system S by RED (C). Then, it follows that (Mrózek 1998)

 



 



RED C

Q C Q

P   ^k  ^k

 , , ' : ' , (2.3:6)

where C ^k P denotes that a set of attributes P  Q depends at the k^th level (0  k  1on the set of attributes C  Q. This latter means that for k^th fraction of all elements of the universe U the values of attributes from P can be reconstructed having values of attributes from C.

Moreover, the following statements are also true (Mrózek 1998)

 

,C RED C C C C

Q

C   

 , (2.3:7)

 

    

     



RED C

Q

C       

 , ' , ', ': p , (2.3:8)

Theorem 2.3:2 (Reduct-based knowledge)

Assuming that Q‟ is a reduct of Q, the knowledge K_Q = Q* contained in the information system S = <U, Q, v, f > considered in QDRSA is identical to the knowledge KQ’ = (Q’)*

contained in the information system S’ = <U, Q’, v, f > derived from S by reducing Q to Q’.

Proof

Any reduct Q’ of the set of attributes Q is such minimum subset of Q, which generates identical set of elementary notions in the information system S. Therefore, knowledge KQ = Q* contained in the information system S = <U, Q, v, f > is based on the same set of elementary notions as knowledge KQ’ = (Q’)* contained in the information system S’ = <U, Q’, v, f >. Hence, based on Lemma 1 valid for CRSA, and using identity of I0 in CRSA with IQDRSA in QDRSA, it follows that knowledge KQ is identical to knowledge KQ’.

■

QDRSA, similarly to CRSA, also uses notion of the core of attributes, which is related to reducts, as described below.

Definition 2.3:30 (Core, after Pawlak 1995a, adapted to QDRSA)

The core of the set of attributes C  Q in the information system, denoted by CORE(C), is a set of all attributes irremovable from C

 





 

  



CORE   :   , (2.3:9)

▬

Hence, the core CORE(Q) of the set of attributes Q in the information system S = <U, Q, v, f > defines the knowledge KCORE(Q) = (CORE(Q))*, which cannot be removed in any reduction process, minimizing the size of the original knowledge KQ = Q*, without loss of classification abilities. Therefore, the knowledge KCORE(Q) is in a sense the most relevant part of the knowledge KQ, and the core itself is the most relevant subset of attributes.

Nevertheless, it is possible that the core CORE(Q) is an empty set and then there is lack of such essential part of knowledge in the information system S = <U, Q, v, f >.

The following relation is true between the notion of core and the reduct of the set of attributes (see Pawlak 1995a)

  

C RED C

C C

CORE Q

C









'

'

: , (2.3:10)

Theorem 2.3:3 (Core and reduct relationship)