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be inaccurate in a noisy environment (Krishnapuram and Keller, 1993, 1996). In real
data analysis, noise and outliers are unavoidable. Hence, to reduce this weakness of fuzzy
c-means, and to produce memberships that have a good explanation of the degrees of
belonging for the data, Krishnapuram and Keller (Krishnapuram and Keller, 1993, 1996)
proposed a possibilistic approach to clustering which used a possibilistic type of membership
function to describe the degree of belonging. However, the possibilistic c-means sometimes
generates coincident clusters (Barni, Cappellini, and Mecocci, 1996). Recently, the use of
both fuzzy (probabilistic) and possibilistic memberships in a clustering has been proposed
in (Pal, Pal, Keller, and Bezdek, 2005).
Rough Sets
The theory of rough sets begins with the notion of an approximation space, which is a pair
< U, R >, where U be a non-empty set (the universe of discourse) and R an equivalence
relation on U , i.e., R is reflexive, symmetric, and transitive. The relation R decomposes
the set U into disjoint classes in such a way that two elements x, y are in the same class iff
(x, y)∈R. Let denote by U/R the quotient set of U by the relation R, and
U/R ={X 1 , X 2 ,, X m }
where X i is an equivalence class of R, i = 1, 2,, m. If two elements x, y∈U belong
to the same equivalence class X i ∈U/R, then x and y are called indistinguishable. The
equivalence classes of R and the empty set∅are the elementary sets in the approximation
space < U, R >. Given an arbitrary set X∈2 U , in general it may not be possible to
describe X precisely in < U, R >. One may characterize X by a pair of lower and upper
approximations defined as follows (Pawlak, 1991):
R(X) =
X i ⊆X
R(X) =
X i ∩X
X i ;
X i
That is, the lower approximation R(X) is the union of all the elementary sets which are
subsets of X, and the upper approximation R(X) is the union of all the elementary sets
which have a non-empty intersection with X. The interval [R(X), R(X)] is the representa-
tion of an ordinary set X in the approximation space < U, R > or simply called the rough
set of X. The lower (resp., upper) approximation R(X) (resp., R(X)) is interpreted as the
collection of those elements of U that definitely (resp., possibly) belong to X. Further,
•a set X∈2 U is said to be definable (or exact) in < U, R > iff R(X) = R(X).
•for any X, Y∈2 U , X is said to be roughly included in Y , denoted by X ˜ ⊂Y , iff
R(X)⊆R(Y ) and R(X)⊆R(Y ).
•X and Y is said to be roughly equal, denoted by X≃ R Y , in < U, R > iff
R(X) = R(Y ) and R(X) = R(Y ).
In (Pawlak, 1991), Pawlak discusses two numerical characterizations of imprecision of a
subset X in the approximation space < U, R >: accuracy and roughness. Accuracy of X,
denoted by α R (X), is simply the ratio of the number of objects in its lower approximation
to that in its upper approximation; namely
α R (X) = |R(X)|
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