chap 2: imprecise categories, approximations and rough sets

1. chap 2:imprecise categories, approximations and rough sets Presented by Farzana Forhad & Xiaqing he

2. OUTLINE • Rough Sets • Approximations of Set • Properties of Approximations • Approximations and Membership Relation • Numerical Characterization of Imprecision The is presented by Farzana Forhad And Xiaqing He will present by follows: • Topological Characterization of Imprecision • Approximation of Classification • Rough Equality of Sets

3. Rough Sets: Let, X is a subset of U. X is R-definable X’ is the union of some R-basic categories; otherwise Xis R-undefinable..means rough sets.

4. Approximation of set

5. Proposition 2.1 X is R-definable if and only if b. X is rough with respect to R if and only if

6. Proposition 2.2

7. Approximation and Membership relation Proposition 2.3 1) Implies implies 2) Implies ( implies and implies ) 3) if and only if and 4) If and only if and 5) Or implies 6) Implies and 7) If and only if non 8) If and only if non

8. Numerical Characterization of Imprecision

9. Topological Characterization of Imprecision • Approximation of set

10. Topological Characterization of Imprecision • If R X ≠ and ≠ U, then we say that X is roughlyR-definable • If R X = and ≠ U, then we say that X is internallyR-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.

11. Topological Characterization of Imprecision Proposition 1 • Set X is R-definable (rough R-definable, totally R-undefinable) if and only if so is –X • Set X is externally (internally) R-undefinable, if and only if, -X is internally (externally) R-undefinable

12. approximation of classifications • Extention of definition of approximation of set • Classification : a family of non empty sets • Example: F = { X1, X2, …, Xn} R-upper approximation of the family F: R-upper approximation of the family F: R F = { RX1, RX2, …, RXn}

13. approximation of classifications • The 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.

14. approximation of classifications Proposition 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.)

15. approximation of classifications Proposition 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 = Uthen for each j ≠ i and j є {1,2,…,n} R Xj = 0. (The opposite is not true.)

16. approximation of classifications Proposition 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.)

17. approximation of classifications Proposition 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.)

18. Rough Equality of Sets Formal 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