- 2. FuzzyFuzzy SetsSets • Dr. Lotfi Zadeh propose this approach. • In his approach an element can belong to a degree k (0 <= k <= 1). • In classical set theory an element must belong or not belong to a set. • Fuzzy membership function can be presented as: µX(x) є (0,1) where, X is a set and x is an element. 2
- 3. Fuzzy Sets (Continue)Fuzzy Sets (Continue) • Fuzzy membership function has the following properties. • µU -x(x) = 1 - µX(x) for any x є U • µxUy(x) = max(µX(x), µy(x)) for any x є U • µx ∩y(x) = min(µX(x), µy(x)) for any x є U 3
- 4. Rough SetsRough Sets • Rough set theory is another approach to handle vagueness. • Imprecision in this approach is expressed by a boundary region of a set, and not by a partial membership, like in fuzzy set theory. • Rough set concept can be defined by topological operation interior and closure called approximations. 4
- 5. Rough Sets (Continue)Rough Sets (Continue) • Suppose we are given a set of objects U called the universe and an indiscernibility relation R ⊆ U × U, representing our lack of knowledge about elements of U. • For simplicity we assume that R is an equivalence relation. • Let X be a subset of U. • We want to characterize the set X with respect to R. To do this we will need the basic concepts of rough set theory given in next slide. 5
- 6. Rough Sets (Continue)Rough Sets (Continue) • The lower approximation of a set X with respect to R is the set of all objects, which can be for certain(sure) classified as X with respect to R (are certainly X with respect to R). • The upper approximation of a set X with respect to R is the set of all objects which can be possibly(maybe) classified as X with respect to R (are possibly X in view of R). • The boundary region of a set X with respect to R is the set of all objects, which can be classified neither as X nor as not-X with respect to R. 6
- 7. Rough Sets (Continue)Rough Sets (Continue) • So that, • Set X is crisp (Exact with respect to R), if the boundary region of X is empty. • Set X is rough (Inexact with respect to R), if the boundary region of X is nonempty. 7
- 8. Rough ApproximationRough Approximation • Formal definitions of approximations and the boundary region are as follows: • R-lower approximation of X R*(x) = U {R(x): R(x) ⊆ X} • R- upper approximation of X R* (x) = U {R(x): R(x) ∩ X ≠ ɸ} • R-boundary approximation of X RNR (X) = R* (X) - R* (X) 8
- 9. Rough ApproximationRough Approximation • As we can see from the definition, approximations are expressed in terms of granules of knowledge. • The lower approximation of a set is union of all granules which are entirely included in the set. • The upper approximation − is union of all granules which have non-empty intersection with the set. • The boundary region of set is the difference between the upper and the lower approximation. 9
- 10. Rough Membership functionRough Membership function • Rough sets can be also defined as rough membership function. µx R : U (0,1) Where µx R (x) = |X∩ R(x)| / |R(x)| And |X| denotes the cardinality of X. 10
- 11. Rough MembershipRough Membership function(Conti)function(Conti) • The rough membership function can be expresses conditional probability. • That x belongs to X given R. • And can be interpreted as a degree that x belongs to X in view of information about x expressed by R. • The meaning of rough membership function can be defined as shown in fig 1. 11
- 12. Rough MembershipRough Membership function(Conti)function(Conti) X X X R(x) R(x) R(x) x x x µx R (x) = 0 0 < µx R (x) < 1 µx R (x) = 1 12 Fig 1
- 13. Rough MembershipRough Membership function(Conti)function(Conti) • The rough membership function can be used to define approximations and the boundary region of a set, as shown below: R* (x) = {xєU : µx R (x) = 1} R* (x) = {xєU : µx R (x) > 0} RNR (X) = {xєU : 0 < µx R (x) < 1 } 13
- 14. Rough MembershipRough Membership function(Conti)function(Conti) • The rough membership function has the following properties: • µx R (x) = 1 iff x є R* (x) • µx R (x) = 0 iff x є U - R* (x) • 0 < µx R (x) < 1 iff x є RNR (X) • µR U-x (x) = 1 - µx R (x) for any x є U • µxUy(x) => max(µR X(x), µR y(x)) for any x є U • µx ∩y(x) <= min (µR X(x), µR y(x)) for any x є U 14
- 15. IndiscernibilityIndiscernibility • A decision system (i.e. a decision table) express all the model. • This table may be unnecessarily large. • The same or indiscernible objects may be represented several times. 15
- 16. Indiscernibility (Conti)Indiscernibility (Conti) Element Age LEMS Walk X1 16-30 50 Yes X2 16-30 0 No X3 31-45 1-25 No X4 31-45 1-25 Yes X5 46-60 26-49 No X6 16-30 26-49 No X7 46-60 26-49 Yes LEMS = Lower Extremity(boundary) Motor Score Table 1 16
- 17. IndiscernibilityIndiscernibility • A binary relation R ⊆ X x X which is reflexive (i.e. an object is in relation with itself xRx), symmetric (if xRy then yRx) and transitive (if xRy and yRz then xRz) is called an equivalence relation. • The equivalence class of an element x є X consists of all objects y є X such that xRy. • Let A = (U, A) be an information system then with any B ⊆ A there is associated an equivalence relation INDA (B) : INDA (B) = {(x,x’) є U2 | a є B a(x) = a(x’)} 17
- 18. IndiscernibilityIndiscernibility • INDA (B) is called the B-indiscernibility relation. If (x,x') є INDA (B) then x and x' are indiscernible from each other by attributes from B. • The equivalence classes of the B-indiscernibility relation are denoted [x]B. Example: • Let us illustrate how a decision table such as Table 1 defines an indiscernibility relation. • The non-empty subsets of the conditional attributes are {Age}, {LEMS} and {Age, LEMS}. 18
- 19. IndiscernibilityIndiscernibility • If we consider, for instance, {LEMS}, objects X3 and X4 belong to the same equivalence class and are indiscernible. • The relation IND defines three partitions of the universe. IND ({Age}) = {{x1,x2,x6},{x3,x4},{x5,x7}} IND ({LEMS}) = {{x1},{x2},{x6},{x3,x4},{x5,x6,x7}} IND ({Age,LEMS}) = {{x1},{x2},x6},{x3,x4},{x5,x7},{x6}} 19