Rb (x, y) = Rc (x, y) =

1.0 0.0 0.568 1.0 1.0 0.0 1.0 0.0 0.036 0.0 0.0 0.0

0.0 1.0 0.0 0.0 0.0 0.137 0.0 1.0 0.036 0.518 0.518 0.518

0.568 0.0 1.0 0.568 0.568 0.0 0.036 0.036 1.0 0.0 0.0 0.0

1.0 0.0 0.568 1.0 1.0 0.0 0.0 0.518 0.0 1.0 1.0 1.0

1.0 0.0 0.568 1.0 1.0 0.0 0.0 0.518 0.0 1.0 1.0 1.0

0.0 0.137 0.0 0.0 0.0 1.0 0.0 0.518 0.0 1.0 1.0 1.0

Again, the rst step is to compute the lower approximations of each concept for each feature. Consider feature a and the decision concept {1, 3, 6} in the example dataset: Ra {1,3,6} (x) = inf I ( Ra (x, y), {1,3,6} (y))

For object 3, the lower approximation is Ra {1,3,6} (3) = inf I ( Ra (3, y), {1,3,6} (y))

= inf{I (0.699, 1), I (0.699, 0), I (1, 1), I (0, 0), I (0, 0), I (0, 1)} = 0.301 For the remaining objects, the lower approximations are Ra {1,3,6} (1) = 0.0 Ra {1,3,6} (2) = 0.0 Ra {1,3,6} (4) = 0.0 Ra {1,3,6} (5) = 0.0 Ra {1,3,6} (6) = 0.0 For concept {2, 4, 5}, the lower approximations are Ra {2,4,5} (1) = 0.0 Ra {2,4,5} (2) = 0.0 Ra {2,4,5} (3) = 0.0 Ra {2,4,5} (4) = 0.301

Ra {2,4,5} (5) = 0.0 Ra {2,4,5} (6) = 0.0 Hence the positive regions for each object are POS Ra (Q) (1) = 0.0 POS Ra (Q) (2) = 0.0 POS Ra (Q) (3) = 0.301 POS Ra (Q) (4) = 0.301 POS Ra (Q) (5) = 0.0 POS Ra (Q) (6) = 0.0 The resulting degree of dependency is therefore {a} (Q) =

x U POS Ra (Q) (x)

0.602 = 6 = 0.1003 Calculating the dependency degrees for the remaining features results in {b} (Q) = 0.3597 {c} (Q) = 0.4078 As feature c results in the largest increase in dependency degree, this feature is selected and added to the reduct candidate. The algorithm then evaluates the addition of all remaining features to this candidate. Fuzzy similarity relations are combined using (9.3). This produces the following evaluations: {a,c} (Q) = 0.5501 {b,c} (Q) = 1.0 Feature subset {b, c} produces the maximum dependency value for this dataset, and the algorithm terminates. The dataset can now be reduced to these features only. The complexity of the algorithm is the same as that of FRFS in terms of the number of dependency evaluations. However, the explosive growth of the number of considered fuzzy equivalence classes is avoided through the use of fuzzy similarity relations and (9.3). This ensures that for one subset, only one fuzzy similarity relation is used to compute the fuzzy lower approximation.

Most approaches to crisp rough set FS and all approaches to fuzzy-rough FS use only the lower approximation for the evaluation of feature subsets. The lower approximation contains information regarding the extent of certainty of object membership to a given concept. However, the upper approximation contains information regarding the degree of uncertainty of objects, and hence this information can be used to discriminate between subsets. For example, two subsets may result in the same lower approximation but one subset may produce a smaller upper approximation. This subset will be more useful as there is less uncertainty concerning objects within the boundary region (the difference between upper and lower approximations). The fuzzy-rough boundary region for a fuzzy concept X may thus be de ned: BND RP (X) (x) = RP X (x) RP X (x) (9.10)

The fuzzy-rough negative region for all decision concepts can be de ned as follows: NEG RP (x) = N ( sup RP X (x))