1. Rough Set Theory and Databases Senior Lecturer: Laurie Webster II, M.S.S.E.,M.S.E.E., M.S.BME, Ph.D., P.E. Lecture 28 A First Course in Database Systems

2. Rough Set Theory and Databases Rough Set Theory and Databases: U Q AD a1a2 d1 x1V a1-x1 V a2-x2 d1-x1 x2V a1-x2 V a2-x2 d1-x2. xmV a1-xm V a2-xm d1-xm Information System => entity-attribute database

3. Rough Set Theory and Databases Rough Set Theory in databases: Let U be a nonempty set called the universe Let R be an equivalence relation on U, called an indiscernibility relation An ordered pair A = (U, R) => called an approximation space

4. Rough Set Theory and Databases Rough Set Theory Databases: For any element (object) x of U, the equivalence class of R containing x will be denoted by [ x ] R Equivalence classes of R [ x ] R are called elementary sets in A We assume that the empty set  is also elementary

5. Rough Set Theory in Databases Rough Set Theory: Any finite union of elementary sets in A is called a definable set in A Let X be a subset of U, i.e. assuming U consists of a number of classes of objects, then X is one of these classes We define X in terms of definable sets in A

6. Rough Set Theory in Databases Rough Set Theory: - Lower Approximation of X in A denoted RX RX = {x  U | [ x ] R  X } - Upper Approximation of X in A denoted R’X R’X = {x  U | [ x ] R  X   } - A Rough Set in A is the family of all subsets of U having the same lower and upper approximations in A

7. Rough Set Theory in Databases Rough Set Theory: Let x be in U We say that x is certainly in X iff x  RX We say that x is possibly in X iff x  R’X RX  X  R’X

8. Rough Set Theory in Databases Rough Set Theory: The difference between the entity-attribute database and and the relational database is that the entities of the information system do not need to be distinguished by their attributes or by their relationship to entities of another type In the information system entities are called objects The main goal of the information system is the basis for knowledge acquisition, to help discover new rules from examples

9. Rough Set Theory in Databases Rough Set Theory: Objects of the information system are described by values of conditions. Classifications of experts are represented by values of decisions Classifications of experts => Physician Diagonses

10. Rough Set Theory in Databases Rough Set Theory: Some objects in the information system are characterized by the same condition values, when this is true, the objects are recognizable up to an indiscernibility relation determined by the set of conditions Two objects are indiscernible whenever they have the same values for all conditions

11. Rough Set Theory in Databases Rough Set Theory: If the real-world phenomena (Universe U) is a hospital, the objects x1,……….,xm are patients, the conditions are test, and the decisions are diseases (conditions C = attributes A) Let S represent the Information System S = ( U, Q, V,  )

12. Rough Set Theory in Databases Rough Set Theory: Let S represent the Information System S = ( U, Q, V,  ) U = Universe => a nonempty finite set The elements x 1, ….., x m of U are called objects of S Q = C  D = A  D is a set of attributes where C is a nonempty finite set whose elements are conditions of S D is a nonempty finite set of decisions of S

13. Rough Set Theory in Databases Rough Set Theory: C  D =  (empty) V =  q  Q V q is a nonempty finite set whose elements are called values of attributes V q is the set of values of attributes q, called the domain of q

14. Rough Set Theory in Databases Rough Set Theory:  is a function of U  Q on to V, called the description function of S such that  (x,q)  V q for all x  U and q  Q Let R be a nonempty subset of Q and x,y be members(objects) of U

15. Rough Set Theory and databases Rough set theory: x and y are indiscernible by R in S, denoted by x ~ R y iff for each a  R,  (x,a)   (y,a) ~ R is an equivalence relation in U Thus, R defines a partition on U; such a partition is a set of all equivalence classes of ~ R

16. Rough Set Theory and Databases Rough Set Information System: R defines a partition on U; such a partition is a set of all equivalence classes of ~ R also called a classification of U generated by R in S or briefly a classification generated by R

17. Rough Set Theory and Databases Rough Set Theory - Information System Example: U Q C c1c2 D d X10L0 x20H0 x30H0 x41L0 x51L1 x61H1 x71H1 x82L0 x92L1 x102H1

18. Rough Set Information System: Rough Set Information System-Example: x1,…,xm => patients Conditions c1 and c2 => medical tests d => disease c1 = body temp of patient = normal, subfebrile, high = 0, 1, 2 c2 = L for ‘low level’ and H for ‘high level’ d = 0 for healthy and 1 for sick

19. Rough Set Information System Rough Set Theory-Information System Example : {{x1}, {x2, x3}, {x4, x5}, {x6,x7}, {x8,x9}, {x10}} ==> classification, generated by the set C of conditions

20. Deductive Databases Next Lecture Rough Set Theory and Database