FUZZY SETS. Precision vs. Relevancy LOOK OUT! A 1500 Kg mass is approaching your head OUT!!

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1 FUZZY SETS Precision vs. Relevancy A 5 Kg mass is approaching your head at at m/sec. m/s. OUT!! LOOK OUT! 4

2 Introduction How to simplify very complex systems? Allow some degree of uncertainty in their description! How to deal mathematically with uncertainty? Using probabilistic theory (stochastic). Using the theory of fuzzy sets (non-stochastic). Proposed in 965 by Lotfi Zadeh (Fuzzy Sets, Information Control, 8, pp ). Imprecision or vagueness in natural language does not imply a loss of accuracy or meaningfulness! 4 Examples Give travel directions in terms of city blocks OR in meters? The day is sunny OR the sky is covered by 5% of clouds? If the sky is covered by % of clouds is still sunny? And 25%? And 5%? Where to draw the line from sunny to not sunny? Member and not member or membership degree? 42 2

3 Probability vs. Possibility Event u: Hans ate X eggs for breakfast. Probability distribution: P X (u) Possibility distribution: X (u) u P X (u)..8. X (u) Applications of fuzzy sets Fuzzy mathematics (measures, relations, topology, etc.) Fuzzy logic and AI (approximate reasoning, expert systems, etc.) Fuzzy systems Fuzzy modeling Fuzzy control, etc. Fuzzy decision making Multi-criteria optimization Optimization techniques 44 3

4 Classical set theory Set: collection of objects with a common property. Examples: Set of basic colors: A = {red, green, blue} Set of positive integers: A{ x x} A line in 3 : A = {(x,y,z) ax + by + cz +d = } 45 Representation of sets Enumeration of elements: A = {x, x 2,..., x n } Definition by property P: A = {x X P(x)} Characteristic function A(x) : X {,}, if x is member of A A( x), if x is not member of A Example: Set of odd numbers: ( x A ) x mod2 46 4

5 Set operations Intersection: C =AB Ccontains elements that belong to A and B Characteristic function: C = min( A, B ) = A B Union: C =AB Ccontains elements that belong to A or to B Characteristic function: C = max( A, B ) Complement: C = C contains elements that do not belong to A Characteristic function: C = A 47 Fuzzy sets Represent uncertain (vague, ambiguous, etc.) knowledge in the form of propositions, rules, etc. Propositions: expensive cars, cloudy sky,... Rules (decisions): Want to buy a big and new house for a low price. If the temperature is low, then increase the heating. 48 5

6 Classical set Example: set of old people A = {age age 7} A age [years] 49 Logic propositions Nick is old... true or false Nick s age: age Nick = 7, A (7) = (true) age Nick = 69.9, A (69.9) = (false) A age [years] 5 6

7 Fuzzy set Graded membership, element belongs to a set to a certain degree. A Membership membership grade age [years] 5 Fuzzy proposition Nick is old... degree of truth age Nick = 7, A (7) =.5 age Nick = 69.9, A (69.9) =.49 age Nick = 9, A (9) = A membership grade age [years] 52 7

8 Context dependent membership grade.5 A tall in China tall in USA tall in NBA h [cm] 53 Typical linguistic values membership grade young middle age old age [years] 54 8

9 Support of a fuzzy set supp(a) = {x X A (x) > } A supp( A) x 55 Core (nucleous, kernel) core(a) = {x X A (x) = } A core( A) x 56 9

10 -cut of a fuzzy set Crisp set: A = { x X A (x) } Strong -cut: A = { x X A (x) > } A A x 57 Resolution principle Every fuzzy set A can be uniquely represented as a collection of -level sets according to ( x) sup [ ( x)] A [,] A Resolution principle implies that fuzzy set theory is a generalization of classical set theory, and that its results can be represented in terms of classical set theory. 58

11 Resolution principle A x 59 Other properties Height of a fuzzy set: hgt(a) = sup A (x), x X Fuzzy set is normal(ized) when hgt(a) =. A fuzzy set A is convex iff x,y X and [,]: A ( x + ( ) y) min( A (x), A (y)) A B x 6

12 Other properties (2) Fuzzy singleton: single point x X where A (x) =. Fuzzy number: fuzzy set in that is normal and convex. Two fuzzy sets are equal (A = B) iff: xx, A (x) = B (x) A is a subset of B iff: xx, A (x) B (x) 6 Non-convex fuzzy sets Example: car insurance risk age lifetime [years] 62 2

13 Representation of fuzzy sets Discrete Universe of Discourse: Point-wise as a list of membership/element pairs: A = A (x )/x A (x n )/x n = i A (x i )/x i A = { A (x )/x,..., A (x n )/x n } = { A (x i )/x i x i X} As a list of -level/ -cut pairs: A = { /A,..., n /A n } = { i /A i i [,]} 63 Representation of fuzzy sets Continuous Universe of Discourse: A = X A (x)/x Analytical formula: A( x), x 2 x Various possible notations: A (x), A(x), A, a, etc. 64 3

14 Examples Discrete universe Fuzzy set A = sensible number of children. number of children: X = {,, 2, 3, 4, 5, 6} A =./ +.3/ +.7/2 + /3 +.6/4 +.2/5 +./6 Fuzzy set C = desirable city to live in X = {SF, Boston, LA} (discrete and non-ordered) C = {(SF,.9), (Boston,.8), (LA,.6)} 65 Examples Continuous universe Fuzzy set B = about 5 years old X = + (set of positive real numbers) B = {(x, B (x)) x X} ( x B ) x 5 4 Membership Grades Age 66 4

15 Complement of a fuzzy set c: [,] [,]; A (x c( A (x)) Fundamental axioms. Boundary conditions - c behaves as the ordinary complement c() = ; c() = 2. Monotonic non-increasing a,b [,], if a < b, then c(a) c(b) 67 Complement of a fuzzy set Other axioms: c is a continuous function. c is involutive, which means that c(c(a)) = a, a [,] 68 5

16 Complement of a fuzzy set Equilibrium point c(a) = a = e c, a [,] Each complement has at most one equilibrium. If c is a continuous fuzzy complement, it has one equilibrium point. 69 Examples of fuzzy complements Satisfying only fundamental axioms: Fuzzy complement of threshold type: t=.3, c( a), if a t if a t

17 Examples of fuzzy complements Satisfying fundamental axioms and continuity: Continuous fuzzy complement (non involutive) c( a) cosa Examples of fuzzy complements Sugeno complement: a c( a), ], ] a Sugeno fuzzy complement, lambda = -.9, -.5,, 2,

18 Examples of fuzzy complement Yager complement: c w ( a) w w a, w], ] Yager fuzzy complement, w =.2,.5,, 2, Representation of complement (x) = A (x) A x 74 8

19 Representation of complement x.5 =-.9 = x A (x) - A (x) A (x) w=.2 w=5 A (x) x 75 Intersection of fuzzy sets i: [,] [,] [,]; AB (x i( A (x), B (x)) Fundamental axioms: triangular norm or t-norm. Boundary conditions - i behaves as the classical intersection i(,) = ; i(,) = i(,) = i(,) = 2. Commutativity i(a,b) = i(b,a) 76 9

20 Intersection of fuzzy sets 3. Monotonicity If a a and b b, then i(a,b) i(a,b ) 4. Associativity i(i(a,b),c) = i(a,i(b,c)) Other axioms: i is a continuous function. i(a,a) =a(idempotent). 77 Examples of fuzzy conjunctions Zadeh AB (x) = min( A (x), B (x)) Probabilistic AB (x) = A (x) B (x) Lukaziewicz AB (x) = max(, A (x) B (x) ) 78 2

21 Intersection of fuzzy sets AB (x) = min( A (x), B (x)) A B x 79 Intersection of fuzzy sets A B min(a,b) A*B max(,a+b-) x 8 2

22 Yager t-norm Example of week and strong intersections: w w w iw a b a b w (, ) min,, ], ] Yager fuzzy intersection, w =.5, 2, Union of fuzzy sets u: [,] [,] [,]; AB (x u( A (x), B (x)) Fundamental axioms: triangular co-norm or s-norm. Boundary conditions - u behaves as the classical union u(,) = ; u(,) = u(,) = u(,) = 2. Commutativity u(a,b) = u(b,a) 82 22

23 Union of fuzzy sets 3. Monotonicity If a a and b b, then u(a,b) u(a,b ) 4. Associativity u(u(a,b),c) = u(a,u(b,c)) Other axioms: u is a continuous function. u(a,a) =a(idempotent). 83 Examples of fuzzy disjunctions Zadeh AB (x) = max( A (x), B (x)) Probabilistic AB (x) = A (x) B (x) A (x) B (x) Lukasiewicz AB (x) = min(, A (x) B (x)) 84 23

24 Union of fuzzy sets AB (x) = max( A (x), B (x)) A B x 85 Union of fuzzy sets.8.6 A B max(a,b) A+B-A*B min(,a+b) x 86 24

25 Yager s-norm Example of week and strong disjunctions: w w w uw( a, b) min, a b, w], ] Yager fuzzy union, w = 2.5, 5, General aggregation operations Axioms h: [,] n [,]; A (x h( A (x),..., An (x)). Boundary conditions h(,...,) = h(,...,) = 2. Monotonic non-decreasing For any pair a i, b i [,], i If a i b i then h(a i ) h(b i ) 88 25

26 General aggregation operations Other axioms: h is a continuous function. h is a symmetric function in all its arguments: h(a i ) = h(a p(i) ) for any permutation p on 89 Averaging operations When all the four axioms hold: min( a,, a ) h( a,, a ) max( a,, a ) n n n Operator covering this range: Generalized mean h( a,, an) a a n n 9 26

27 Generalized mean Typical cases: Lower bound: Geometric mean: h min( a,, a n ) h aa2a n ( ) n Harmonic mean: h n a a n Arithmetic mean: Upper bound: a a h n h n a a n max(,, ) 9 Fuzzy aggregation operations Parametric t-norms Parametric s-norms Generalized mean i min min max u max Intersection operators Averaging operators Union operators 92 27

28 Membership functions (MF) Triangular MF: xa cx Trxabc ( ;,, ) max min,, ba cb Trapezoidal MF: xa d x Tpxabcd ( ;,,, ) maxmin,,, ba d c Gaussian MF: Gs( x; a, b, c) e 2 xc 2 Generalized bell MF: Bell( xabc ;,, ) xc b 2a 93 Membership functions (a) Triangular MF (b) Trapezoidal MF (c) Gaussian MF (d) Generalized Bell MF

29 Left-right MF cx FL, xc LRxc ( ;,, ) xc FR, xc c= 65 = 6 = Membership Grades c = 25 = = 4 Membership Grades Two-dimensional fuzzy sets A ( xy, ) ( xy, )( xy, ) XY XY A A x y 29

30 2-D membership functions (a) z = min(trap(x), trap(y)) (b) z = m ax(trap(x), trap(y)).5.5 Y - - X Y - - (c) z = min(bell(x), bell(y)) X (d) z = m ax(bell(x), bell(y)).5.5 Y - - X Y - - X 97 Compound fuzzy propositions Small = Short and Light (conjunction) ( h, w) ( h) ( w) Small Short Light Weight(w) LIGHT HEAVY Compact Small Big Thin Height(h) LONG 3

31 Cylindrical extension Cylindrical extension of fuzzy set A in X into Y results in a two-dimensional fuzzy set in X Y, given by cext ( A) ( x)/( xy, ) ( x) ( xy, )( xy, ) XY y A A XY.5 (a) Base Fuzzy Set A (b) Cylindrical Extension of A.5 X y X 99 Projection (a) A Two-dimensional MF (b) Projection onto X (c) Projection onto Y Y X Y ( xy, ) A( x) max R( xy, ) B( y) max R( xy, ) R y X Y x X 3

32 Cartesian product Cartesian product of fuzzy sets A and B is a fuzzy set in the product space X Y with membership ( x, y) min( ( x), ( y)) AB A B Cartesian co-product of fuzzy sets A and B is a fuzzy set in the product space X Y with membership ( x, y) max( ( x), ( y)) AB A B Cartesian product (a) trap(x) (b) trap(y) (X).5 (Y).5 - X (c) z = min(trap(x), trap(y)) - Y (d) z = max(trap(x), trap(y)) Y - X - Y - X 2 32

33 Classical relations Classical relation R(X, X 2,..., X n ) is a subset of the Cartesian product: R Characteristic function: R X, X2,, Xn XX2Xn x, x,, x 2 n, iff x, x2,, x, otherwise n R 3 Example X = {English, French} Y = {dollar, pound, euro} Z = {USA, France, Canada, Britain, Germany} R(X, Y, Z) = {(English, dollar, USA), (French, euro, France), (English, dollar,canada), (French, dollar, Canada), (English, pound,britain)} 4 33

34 Matrix representation USA Fra Can Brit Ger Dollar Pound Euro English USA Fra Can Brit Ger Dollar Pound Euro French 5 Fuzzy relation Fuzzy relation: R: X X 2... X n [,] Each tuple (x, x 2,..., x n ) has a degree of membership. Fuzzy relation can be represented by an n-dimensional membership function (continuous space) or a matrix (discrete space). Examples: x is close to y x and y are similar x and y are related (dependent) 6 34

35 Discrete examples Relation R very far between X = {New York, Lisbon} and Y = {New York, Beijing, London}: R(x,y) = /(NY,NY) + /(NY,Beijing) +.6/(NY,London) +.5/(Lisbon,NY) +.8/(Lisbon,Beijing) +./(Lisbon,London) Relation: is an important trade partner of Holland Germany USA Japan Holland,9,5,2 Germany,3,4,2 USA,3,4,7 Japan,6,8,9 7 Continuous example R: x y ( x is approximately equal to y ) ( x, y) e 2 ( xy) y x

36 Composition of relations R(X,Z) = P(X,Y) Q(Y,Z) Conditions: (x,z) R iff exists y Y such that (x,y) P and (y,z) Q. Max-min composition ( xz, ) max min ( xy, ), ( yz, ) PQ P Q yy 9 Properties Associativity: R( ST) ( RS) T Distributivity over union: R( ST) ( RS) ( RT) Weak distributivity over intersection: R( ST) ( RS) ( RT) Monotonicity: ST( RS) ( RT) 36

37 Other compositions Max-prod composition ( xz, ) max ( xy, ) ( yz, ) PQ P Q yy Max-t composition ( xz, ) max t ( xy, ), ( yz, ) PQ P Q yy Example a b c d P Y Q Z

38 Example Composition R = P Q x z R (x,z) a.6 a 2.7 b.6 b 2.8 c 2 d 2.4 Composition R = P Q? 3 Matrix notation examples

39 Relations on the same universe Let R be a relation defined on U U, then it is called: Reflexive, if u U, the pair (u,u) R Anti-reflexive, if u U, (u,u) R Symmetric, if u,v U, if (u,v) R, then (v,u) R too Anti-symmetric, if u,v U, if (u,v) and(v,u)r, then u = v Transitive, if u,v,w U, if (u,v) and(v,w)r, then (u,w)r too. 5 Examples R is an equivalence relation if it is reflexive, symmetric and transitive. R is a partial order relation if it is reflexive, antisymmetric and transitive. R is a total order relation if R is a partial order relation, and u, v U, either (u,v) or (v,u) R. Examples: The subset relation on sets () is a partial order relation. The relation on is a total order relation. 6 39