Fuzzy Sets and Systems. Lecture 2 (Fuzzy Sets) Bu- Ali Sina University Computer Engineering Dep. Spring 2010

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1 Fuzzy Sets and Systems Lecture 2 (Fuzzy Sets) Bu- Ali Sina University Computer Engineering Dep. Spring 2010

2 Fuzzy Sets Formal definition: A fuzzy set A in X (universal set) is expressed as a set of ordered pairs: A = {( x, µ ( x )) x X } A Fuzzy set Membership function (MF) Universe or universe of discourse A fuzzy set is totally characterized by a membership function (MF).

3 Membership functions Assign to each element x of X a number A(x) A: X [0, 1] The degree of membership

4 Discrete Fuzzy Sets Fuzzy set C = desirable city to live in X = {SF, Boston, LA} (discrete and nonordered) C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)} Fuzzy set A = sensible number of children X = {0, 1, 2, 3, 4, 5, 6} (discrete universe) A = {(0,.1), (1,.3), (2,.7), (3, 1), (4,.6), (5,.2), (6,.1)}

5 Continuous Fuzzy Sets Fuzzy set B = about 50 years old X = Set of positive real numbers (continuous) B = {(x, µb(x)) x in X} µ B ( x ) = 1 + x

6 Fuzzy Sets representations List form Tabular form Rule form Membership form

7 List representations List representation of very high educated B = 0/0 + 0/1 + 0/ / /4 +0.8/5 + 1/6 Or B={<0,0>,<1,0>,<2,0>,<3,0.1>,<4,0.5>,<5,0.8>,<6, 1>} OR General notation A= A(x)/x

8 Tabular Representation level membership

9 Rule form M = {x ɛ X x meets some conditions}; where the symbol denotes the phrase "such that". M = {x ɛ X x is old man};

10 Membership form

11 MF Terminology MF 1.5 α 0 Core X Crossover points α - cut Support

12 Basic Concepts The support of a fuzzy set A in the universal set X is a crisp set that contains all the elements of X that have nonzero membership values in A, that is, A fuzzy singleton is a fuzzy set whose support is a single point in X.

13 Basic concepts A crossover point of a fuzzy set is a point in X whose membership value to A is equal to 0:5. The height, h(a) of a fuzzy set A is the largest membership value attained by any point. If the height of a fuzzy set is equal to one, it is called a normal fuzzy set, otherwise it is subnormal. α A An α- cut of a fuzzy set A is a crisp set that contains all the elements in X that have membership value in A greater than or equal to α.

14 Basic Concepts A strong α-cut of a fuzzy set A is a crisp set α+a that contains all the elements in X that have membership value in A strictly greater than α. We observe that the strong α-cut 0+A is equivalent to the support supp(a). The 1-cut 1A is often called the core of A. h ( a) Note! Sometimes the highest non-empty α-cut is Acalled the core of A. (in the case of subnormal fuzzy sets, this is different). The word kernel is also used for both of the above definitions.

15 Basic concepts The set of all levels α ɛ [0, 1] that represent distinct α-cuts of a given fuzzy set A is called a level set of A.

16 MF functions Main types of membership functions (MF): (a) Triangular MF is specified by 3 parameters {a,b,c}: trn(x : a,b,c) = 0, (x - a) (b - a), (c - x) (c - b), if x < a if a x b if b x c 0, if x > c 0, if x < a (b) Trapezoidal MF is specified by 4 parameters {a,b,c,d}: (x - a) (b - a), if a x b trn(x : a,b,c) = (c - x) (c0, - b), if b x cif x < a 0, (x - a) (b - a), if x > if c a x < b trp(x : a,b,c,d) = 1, if b x < c (d - x) (d - c), if c x < d 0, if x d

17 MF functions (c) Gaussian MF is specified by 2 parameters {a,δ}: gsn(x : a, δ) = - (x - a) exp 2 δ 2 (d) Bell-shaped MF is specified by 3 parameters {a,b,δ}: bll(x : a,b, δ) = 1 x - a (e) Sigmoidal MF is specified by 2 parameters {a,b}: + 1 δ 2b sgm(x : a,b) = a(x-b) e

18 MF Formulation Sigmoidal MF: sigm f ( x ; a, b, c ) 1 = 1 ( ) + e a x c Extensions: Abs. difference of two sig. MF Product of two sig. MF disp_sig.m

19 MF Formulation L-R MF: L R ( x ; c, α, β ) = F F L R c x, x < α x c, x β c c Example: F x = 0 1 x 2 L ( ) max(, ) F ( R x ) = exp( 3 x ) c=65 a=60 b=10 c=25 a=10 b=40 difflr.m

20 2D MF Projection Two-dimensional MF Projection onto X Projection onto Y µ R ( x, y) µ A( x) = µ B( y) = max µ ( x, y) max µ ( x, y) project.m y R x R

21 Fuzzy Partition Fuzzy partitions formed by the linguistic values young, middle aged, and old : lingmf.m

22 Convexity of Fuzzy Sets A fuzzy set A is convex if for any λ in [0, 1], µ ( λx + ( 1 λ ) x ) min( µ ( x ), µ ( x )) A 1 2 A 1 A 2 Alternatively, A is convex if all its α-cuts are convex.

23 Set-Theoretic Operations Subset: A B µ A µ B Complement: Standard union: C = A B µ ( x ) = max( µ ( x ), µ ( x )) = µ ( x ) µ ( x ) c A B A B Standard Intersection: A = X A µ ( x ) = 1 µ ( x ) C = A B µ ( x ) = min( µ ( x ), µ ( x )) = µ ( x ) µ ( x ) c A B A B A A

24 Set-Theoretic Operations subset.m fuzsetop.m

25 µ ( x) = min( µ ( x), µ ( x)) A. B A B

26 µ ( x) max( µ ( x), µ A + B = A B ( x))

27 µ ( x) = 1 ( x) µ A A µ A (x) µ A (x) µ A (x) µ A (x)

28 Discrete Fuzzy sets µ ( x) = min( µ ( x), µ ( x)) A. B A B Let A and B be fuzzy subsets of the universe X={-3, -2, -1, 0, 1, 2, 3, 4} A= 0.6/ / / / / / / /4 B= 0.2/ / / / / / / /4 µa B = 0.2/ / / / / / / /4

29 Discrete Fuzzy Sets µ ( x) max( µ ( x), µ A + B = A B ( x)) Let A and B be fuzzy subsets of the universe X={-3, -2, -1, 0, 1, 2, 3, 4} A= 0.6/ / / / / / / /4 B= 0.2/ / / / / / / /4 µa B = 0.6/ / / / / / / /4

30 Fuzzy Union- S-norm or t-conorm General notation: µ µ A B A B (x) (x) = = s [ µ t [ µ A A (x), µ (x), µ B B (x)] (x)] (fuzzy disjunction operator) (fuzzy conjunction operator) Operator s is called an s-norm if it satisfies to the following axioms for any x, y, z and w [0,1]: [Ax.1] [Ax.2] [Ax.3] [Ax.4] s[1,1] = 1, s[0, x] = s[x,0] = x s[x, y] s[z, w] if x z and y s[x, y] = s[y,x] s[x, s[y,z] ] = s[ s[x,y], z] w (zero - identity) (monotonicity) (commutativity) (associativity)

31 S-norm (S1) Drastic sum: s D [x,y] = max(x, y), 1, if if x > 0 min(x, y) = and y > 0 0 (S2) Hamacher sum: s[x, y] = x + y - 2x y 1- x y (S3) Dubois-Prade class: s γ [x,y] x + = γ [0,1] y - x y - min(1- γ, x, y) max( γ, 1- x, 1- y), (S4) Yager class: s σ [x,y] 1 σ σ = min1, (x + y ) σ, σ [0, [ Note: for arbitrary fuzzy sets A and B membership values x and y stand for µ A (x) and µ B (x), correspondingly

32 Fuzzy Intersection- t-norm Operator t is called an t-norm (triangular norm) if it satisfies to the following axioms for any x, y, z and w [0,1]: [Ax.1] t[0,0] = 0, t[1,x] = t[x,1] = x (one - identity) [Ax.2] t[x, y] t[z, w] if x z and y w (monotonicity) [Ax.3] t[x, y] = t[y,x] (commutativity) [Ax.4] t[x, t[y,z] ] = t[ t[x,y], z] (associativity) Some of the operators (t-norms) that model (extend) fuzzy intersection: (T1) Drastic product: t D [x,y] = min(x,y), 0, if x if max(x, y) = 1 < 1 and y < 1 (T2) Hamacher product: t[x, y] = x x y + y - x y

33 T-norm (T3) Dubois-Prade class: t γ [x,y] = x max( γ y, x, y), γ [0,1] (T4) Yager class: t σ [x, y] 1 - min 1, σ [0, [ [ σ σ ] (1 - x) + (1 - y) 1 = σ,

34 Fuzzy complement Operator c is called a fuzzy complement if it satisfies to the following axioms for any x and y [0,1]: [Ax.1] [Ax.2] [Ax.3] c[0] = 1 and c[1] = 0 c[x] > c[y] if x < y c[ c[x] ] = x (boundary) (monotonicity) (involution) Some of the operators that model (extend) fuzzy complement: (C1) Sugeno s complement: c γ [x] = 1- x 1+ γ x, γ ( 1, [ (C2) Yager s complement: c σ [x] σ 1 = (1- x ) σ, σ (0, [

36 Generalized DeMorgan s Law T-norms and T-conorms are duals which support the generalization of DeMorgan s law: T(a, b) = C(S(C(a), C(b))) S(a, b) = C(T(C(a), C(b))) Tm(a, b) Ta(a, b) Tb(a, b) Td(a, b) Sm(a, b) Sa(a, b) Sb(a, b) Sd(a, b)

37 Cartesian Product Assume X and Y are two arbitrary classical sets. The Cartesian product of sets X and Y is a set of all ordered pairs (x i,y j ), x i X, y j Y; that is X Y = { (xi,yj) xi X, y j Y, i = 1,n, j = 1,m } Suppose X = { x 1, x 2, x 3, x 4 }, Y = { y 1, y 2, y 3 }; the set X x Y consists of 12 ordered pairs (x i,y j ), i=1,2,3,4, j=1,2,3- in this case, a graphical representation (as nodes of a grid) is convenient Generalization of Cartesian product to n arbitrary classical sets X 1, X 2,, X n : i j 2 X1 X2... Xn = { ( x1, x,..., x ) x X, x X,..., x X k n i 1 1 j 2 2 k n n } card(x 1 ) = ξ1, card(x 2) = ξ2,..., card(xn) = ξ n

38 Cartesian Product Y y 3 12 pairs elements of the set XxY y 2 y 1 Y x1 x2 x3 x4 X If the cardinality of the set X is n(x) and the cardinality of the set Y is n(y), then the cardinality of the Cartesian product (set of elements) is n(x x Y) = n(x)*n(y) y 2 y 1 X =[x 1,x 2 ], Y=[y 1,y 2 ] Set XxY (Cartesian product) x1 x2 X

39 Classical Relations An ordered sequence of n elements (x 1, x 2,, x n ) is called an ordered n- n tuple A subset of the Cartesian product X 1 X 2... X n is called an n- ary relation built over domains (sets) X 1,X 2,, X n - if n=2, the binary relation on X 1 and X 2 (from X 1 to X 2 ) can be formally defined as a set of ordered pairs in X 1 xx 2 ; that is where P(x 1,x 2 ) is a property to which each pair (x 1,x 2 ) λ satisfies Example λ = { (x, x ) P(x, x ), x X, x } X2 Suppose that both X 1 and X 2 are sets of real numbers [5,20], i.e. X 1 = X 2 = R [5,20]. The binary relation λ λ(x 1,X 2 ) «less than» has

Classical Relations the following formal analytical representation: λ = {, x ) x < x, x X, x X } (x1 2 1 2 1 1 2 2 Graphical form of the binary relation λ: 20 X 2 5 Set λ is built on the Cartesian

40 Classical Relations the following formal analytical representation: λ = {, x ) x < x, x X, x X } (x Graphical form of the binary relation λ: 20 X 2 5 Set λ is built on the Cartesian product X 1 xx 2 Points located on the diagonal (x 1 = x 2 ) of the square do not belong to the set. The area is defined as follows x 1 5 x 2 20 x 1 < x X 1 A binary relation λ can be also represented by:

41 Classical Relations means of membership function: µ λ = µ λ (X1,X2,..., Xn) = 1, 0, if if (x (x 1 1,x,x 2 2,...,,..., x x n n ) λ ) λ (arbitrary n-ary relation is a mapping: λ(x 1,X 2,, X n ) : X 1x X 2x x X n {0,1} ) If a set X 1x X 2 is finite, then the values of function µ λ can be collected into a relational matrix Relations are intimately involved in logic, approximate reasoning, rule-based systems, etc. A rule «IF x is A THEN y is B» describes a relation between the variables x and y - as implication A B, rule expresses a mapping (subset of Cartesian product) between input and output domains

42 Binary relation representation The representations of binary relations List Coordinate Matrix Mappings Directed graph

43 Coordinate Rep. R ={<eggs, hens>, <milk,cows>, milks,goats>}

44 Matrix Repr. 0 if the relation does not exist between the two individuals and 1 if it does.

45 Mapping Rep. Functions: binary relations in which no element of the first set is mapped to more than one element of the second set.

46 Directed Graph Rep. Properties of directed graph Each element of the set X is represented by a node in the diagram Directed connections between nodes indicate pairs of elements that are included in the relation

47 Binary Relation Operations Equivalence and compatibility relation Equivalence: Any binary relation that satisfies reflexive, symmetric and transitive properties. R is Reflexive <x, x> R for each x R R is symmetric both <x, y> R and <y, x> R or both not. R is transitive for any three elements x, y, z in X, <x, z> R whenever <x, y> R and <y, z> R Compatibility: satisfy reflexive and symmetric

48 Operations Partial ordering Relations that satisfy reflexive, transitive and anti-symmetric, denoted by x y R on X is anti-symmetric for any x and y in X, if <x,y> R and <y, x> R then x=y.

49 Fuzzy Relation A fuzzy relation generalizes the concept of classical (crisp) relation introducing a degree of membership for each ordered n-tuple (x 1, x 2,, x n ) in X 1 X2... Xn For 2D case it can be defined as follows: ~ λ = { ( (x,x ), µ ~ (x, x ) ) (x,x ) X X } λ 2 Examples ~ λ denotes a fuzzy relation a) x 1 is close to x 2 (both x 1 and x 2 are numbers) b) if x 1 is medium, then x 2 is high (x 1 is an observed state, whereas x 2 is a result state or action) c) x 1 is similar to x 2 (x 1 and x 2 can be objects, human beings, properties)

50 Fuzzy Relations Formally, fuzzy relation ~ λ in X 1 X2... Xn can be defined as a fuzzy set ~ λ = {((x,x,..., x ), µ ~ (x,x,...,x )) (x,x,..., x ) X X... X } where 1 2 n 1 2 n 1 2 n 1 2 λ µ λ ~ is a mapping: X1 X2... Xn [0,1] R R ~ In different sources notations and can be used for crisp and fuzzy relations, correspondingly µ ~ λ ( x1,x2,..., xn) means «membership degree of the ordered n-tuple in the ~ fuzzy relation λ(x1,x2,..., Xn), where xi Xi, i = 1,n», or «a degree to which fuzzy relation λ ~ holds true for objects x,x,...,x )» ( 1 2 n n

51 Example U={France, Iran} V={Germany,Pakistan} U u 1 u 2 V Distance France Iran v 1 Germany.1.8 v 2 Pakistan.9.2

52 Fuzzy Relations Suppose that X 1 and X 2 are line segments [0,50] and [20,40], respectively, on the set of real numbers R. A ~ ~ fuzzy relation λ1 λ 1 (X1,X2 ) «x 1 is approximately equal to x 2» may be defined by the membership 2 function -(x1 x2 ) µ ~ = e, x (X,X ) 1 X1, x2 λ X Fuzzy relations enhance our capability to deal with relational concepts expressed in a natural language! X 1 X 2 X 2 X 1

53 Fuzzy Relations A fuzzy relation «x 1 is much larger than x 2» may be defined by the membership function µ = ~ λ2(x1,x2 ) -0.5 (x1 x2 ) 1+ e 1 Membership values; line segment [0,1] Fuzzy relations are also fuzzy sets, and fundamental properties of fuzzy sets hold for fuzzy relations as well

54 Fuzzy relations Fuzzy set operations (complements, unions, intersections) are applicable to fuzzy relations too Sugeno s complement (parameter γ = 2) Standard complement

55 Fuzzy Relation Operations. Union R S = {U U R U S} µr S(U) = µr(u) µs(u) = Max{µR(U),µS(U)} )} Intersection R S = {U U R U S} µr S(U) = µr(u) µs(u) = Min{µR(U),µS(U)} )} Complement µr (U) = 1- µr(u) Subset Subset R S µr(u) µs(u)

56 Fuzzy Relations Operations on fuzzy sets defined on different universal domains produce a multidimensional fuzzy set (the following shows graphically the result of operation ( A I B) X Y ) Membership values 1 Fuzzy set A Fuzzy set B Domain X Domain Y Cartesian product XxY result of intersection A B

57 Representations of fuzzy relation List of ordered pairs with their membership grades Matrices Mappings Directed graphs

58 Matrices Fuzzy relation R on X Y X={x 1, x 2,, x n }, Y={y 1, y 2,, y m } r ij =R(x i, y j ) is the membership degree of pair (x i, y j )

59 Example (very far from) Example: X={Beijing, Chicago, London, Moscow, New York, Paris, Sydney, Tokyo} X Very far from

60 Mappings The visual representations On finite Cartesian products Document D = {d 1, d 2,, d 5 } Key terms T = {t 1, t 2, t 3, t 4 }

61 Directed graphs

62 Inverse operation Given fuzzy binary relation R X Y Inverse R -1 Y X R -1 (y,x) = R(x,y) (R -1 ) -1 = R The transpose of R

63 Composition of Crips Relations Let R be a relation that relates elements from universe X To universe Y, and let S be a relation that relates elements From Universe Y to universe Z. A useful question we seek to answer is whether we can find a relation, T, that relates the same elements in universe X that R contains to the same elements in universe Z that S contains. T = R o S R X S Y Z χ T ( x, z) = ( χ ( x, y) χ ( y, z) y Y R S T = R o S

64 Composition of Crips Relations S R T o = ), ( ), ( ( ), ( z y y x z x S R Y y T χ χ χ = x 1 Y X x 2 x 3 y 1 y 2 y 3 y 4 z 1 z 2 Z y y y y x x x R = z z y y y y S = z z x x x T = 0 max[min(1,1), min(0,0), min(1,1), min(0,0)] ), ( 0 max[min(1,0), min(0,0), min(1,0), min(0,0)] ), ( = = = = z x z x T T χ χ o =

65 The composition of fuzzy relations P and Q Given fuzzy relations P X Y, Q Y Z Composition on P and Q = P Q = R X Z The membership degree of a chain <x, y, z> is determined by the degree of the weaker of the two links, <x, y> and <y, z>. R(x, z) = (P Q )(x, z) = max y Y min [ P(x, y ), Q(y, z )]

66 Example (composition of fuzzy relations) X = {a,b,c} Y = {1,2,3,4} Z = {A,B,C}

67 Example (matrix composition) P X Y Q Y Z R X Z

68 Fuzzy equivalence relations & compatibility relations Equivalence: Any fuzzy relation R that satisfies reflexive, symmetric and transitive properties. R is Reflexive R(x, x) = 1 for all x X. R is symmetric R(x, y) = R(y, x) for all x, y X. R is transitive R(x, z) max y Y min [ R(x, y ), R(y, z )] all x, z X. Compatibility: satisfy reflexive and symmetric

69 Example (fuzzy equivalence)

70 Example(fuzzy compatibility) R(1,4)<max {min[r(1,2),r(2,4)], min[r(1,5),r(5,4)]}

71 Fuzzy Partial ordering Fuzzy relations that satisfy reflexive, transitive and anti-symmetric, denoted by x y R on X is anti-symmetric R(x,y)>0 and R(y, x)>0 imply that x=y for any x,y X.

72 Example (fuzzy partial ordering)

73 Projections Given an n-dimensional fuzzy relation R on X =X 1 X 2 X n and any subset P(any chosen dimensions) of X, the projection R P of R on P for each p P is Where R P ( p) = max p P R( p, p) P is the remaining dimensions

74 Projections example S ={s 1, s 2, s 3 } D ={d 1, d 2, d 3, d 4, d 5 } Projection on S symptoms diseases Projection on D Q1(s) = Q( s, d) max d D Q2(s) = Q( s, d) max s S

75 Projections Two-dimensional MF Projection onto X Projection onto Y µ R ( x, y) µ A( x) = µ B( y) = max µ ( x, y) max µ ( x, y) project.m y R x R

76 Cylindric extension Given an n-dimensional fuzzy relation R on X =X 1 X 2 X n, any (n+k)-dimensional relation whose projection into the n dimensions of R yields R is called an extension of R. An extension of R with respect to Y =Y 1 Y 2 Y h is call the cylindric extension EY R of R into Y For all x X and y Y. EY R(x,y) = R(x)

77 Example (Cylindric extension)

78 Cylindric extension Base set A Cylindrical Ext. of A

79 Assignment 3-3, 3-4,3-6,3-7,3-8,3-14, ,2-4,2-11, ,1-11,

Fuzzy Sets and Systems. Lecture 1 (Introduction) Bu- Ali Sina University Computer Engineering Dep. Spring 2010

Fuzzy Sets and Systems. Lecture 1 (Introduction) Bu- Ali Sina University Computer Engineering Dep. Spring 2010 Fuzzy Sets and Systems Lecture 1 (Introduction) Bu- Ali Sina University Computer Engineering Dep. Spring 2010 Fuzzy sets and system Introduction and syllabus References Grading Fuzzy sets and system Syllabus