Presented byFarzana Forhad & Xiaqing he
OUTLINERough SetsApproximations of SetProperties of ApproximationsApproximations and Membership RelationNumerical Characterization of ImprecisionThe is presented by Farzana Forhad
And Xiaqing He will present by follows:Topological Characterization of ImprecisionApproximation of ClassificationRough Equality of Sets
Rough Sets:
Let,X is a subset of U.X is R-definableX’ is the union of some R-basic categories;
otherwise Xis R-undefinable..means rough sets.
Approximation of set
}:/{ XYRUYXR
}:/{ XYRUYXR
Proposition 2.1 a. X is R-definable if and only if
b. X is rough with respect to R if and only if
XR XR
XR XR
Proposition 2.2
Approximation and Membership relationProposition 2.31) Implies implies2) Implies ( implies and
implies )3) if and only if and4) If and only if and 5) Or implies 6) Implies and 7) If and only if non 8) If and only if non
Xx Xx XxYX Xx Yx Xx
Yx)(XUYx Xx Yx)(XUYx
)( Xx
Xx YxXx Yx )(XUYx
)( YXx Xx Yx)( Xx Xx
Xx
Numerical Characterization of Imprecision
Topological Characterization of Imprecision
Approximation of set
}:/{ XYRUYXR
}:/{ XYRUYXR
Topological Characterization of Imprecision
If R X ≠ and ≠ U, then we say that X is roughly R-definableIf R X = and ≠ U, then we say that X is internally R-undefinable R X ≠ and = U, then we say that X is externally R-undefinable If R X = and = U, then we say that X is totally R-undefinable.
XR
XR
XR
XR
Topological Characterization of Imprecision
Proposition 1
I. Set X is R-definable (rough R-definable, totally R-undefinable) if and only if so is –X
II. Set X is externally (internally) R-undefinable, if and only if, -X is internally (externally) R-undefinable
approximation of classificationsExtention of definition of approximation of setClassification : a family of non empty setsExample: F = { X1, X2, …, Xn} R-upper approximation of the family F:
R-upper approximation of the family F:
R F = { RX1, RX2, …, RXn}
}...,{ ,2,1 nXRXRXRFR
approximation of classificationsThe accuracy of classification expresses the
percentage of possible correct decisions when classifying objects employing the knowledge R.
The quality of classification expresses the percentage of objects which can be correctly classified to classes of F employing knowledge R.
I
i
R
XRcard
XRcardF )(
cardU
XRcardF
i
R)(
approximation of classificationsProposition 2
Let F = {X1, X2,…, Xn}, where n ≥1 be a classification of U and let R be an equivalence relation. If there exists i є {1,2,…,n} such that R Xi ≠ 0, then for each j ≠ I and j є {1,2,…,n} Xj ≠ U. (The opposite is not true.)
R
approximation of classificationsProposition 3
Let F = {X1, X2,…, Xn}, where n ≥1 be a classification of U and let R be an equivalence relation. If there exists i є {1,2,…,n} such that, Xi = U then for each j ≠ i and j є {1,2,…,n} R Xj = 0. (The opposite is not true.)
R
approximation of classificationsProposition 4
Let F = {X1, X2,…, Xn}, where n ≥1 be a classification of U and let R be an equivalence relation. If for each i є {1,2,…,n} R Xi ≠ 0 holds, then Xi ≠ U for each i є {1,2,…,n}. (The opposite is not true.)
R
approximation of classificationsProposition 5
Let F = {X1, X2,…, Xn}, where n ≥1 be a classification of U and let R be an equivalence relation. If for each i є {1,2,…,n} Xi = U holds, then R Xi = 0 for each i є {1,2,…,n}. (The opposite is not true.)
R
Rough Equality of SetsFormal definitions of approximate (rough) equality of sets
Let K= (U,R) be a knowledge base, and R є IND(K)Sets X and Y are bottomR - equal (X ≈R Y )if RX =
RY means that positive examples of the sets X and Y are the same
Sets X and Y are topR – equal (X ≈R Y )if X = Y means the negative examples of sets X and Y are the same
Sets X and Y are R – equal (X ≈R Y )if X ≈R Y and X ≈R Y means both positive and negative examples of sets X and Y are the
same
R R
UYX ,