chap 2: imprecise categories, approximations and rough sets

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 ,