FUZZY SETS. Precision vs. Relevancy LOOK OUT! A 1500 Kg mass is approaching your head OUT!! João M. C. Sousa 38

Transcription

1 FUZZY SETS Precision vs. Relevancy A 5 Kg mass is approaching your head at at m/sec. m/s. OUT!! LOOK OUT! João M. C. Sousa 38

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! João M. C. Sousa 39 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? João M. C. Sousa 4 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) João M. C. Sousa 4 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 João M. C. Sousa 42 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 = } João M. C. Sousa 43 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 João M. C. Sousa 44 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 João M. C. Sousa 45 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. João M. C. Sousa 46 5

6 Classical set Example: set of old people A = {age age 7} A age [years] João M. C. Sousa 47 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] João M. C. Sousa 48 6

7 Fuzzy set Graded membership, element belongs to a set to a certain degree. A Membership membership grade age [years] João M. C. Sousa 49 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] João M. C. Sousa 5 7

8 Context dependent membership grade.5 A tall in China tall in USA tall in NBA h [cm] João M. C. Sousa 5 Typical linguistic values membership grade young middle age old age [years] João M. C. Sousa 52 8

9 Support of a fuzzy set supp(a) = {x X A (x) > } A supp( A) x João M. C. Sousa 53 Core (nucleus, kernel) core(a) = {x X A (x) = } A core( A) x João M. C. Sousa 54 9

10 -cut of a fuzzy set Crisp set: A = { x X A (x) } Strong -cut: A = { x X A (x) > } A A x João M. C. Sousa 55 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. João M. C. Sousa 56

11 Resolution principle A x João M. C. Sousa 57 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 João M. C. Sousa 58

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) João M. C. Sousa 59 Non-convex fuzzy sets Example: car insurance risk age lifetime [years] João M. C. Sousa 6 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 [,]} João M. C. Sousa 6 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. João M. C. Sousa 62 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)} João M. C. Sousa 63 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 João M. C. Sousa 64 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) João M. C. Sousa 65 Complement of a fuzzy set Other axioms: c is a continuous function. c is involutive, which means that c(c(a)) = a, a [,] João M. C. Sousa 66 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. João M. C. Sousa 67 Examples of fuzzy complements Satisfying only fundamental axioms: Fuzzy complement of threshold type: t=.3, c( a), if a t if a t João M. C. Sousa 68 6

17 Examples of fuzzy complements Satisfying fundamental axioms and continuity: Continuous fuzzy complement (non involutive) c( a) cosa João M. C. Sousa 69 Examples of fuzzy complements Sugeno complement: a c( a), ], ] a Sugeno fuzzy complement, lambda = -.9, -.5,, 2, João M. C. Sousa 7 7

18 Examples of fuzzy complement Yager complement: c w ( a) w w a, w], ] Yager fuzzy complement, w =.2,.5,, 2, João M. C. Sousa 7 Representation of complement (x) = A (x) A x João M. C. Sousa 72 8

19 Representation of complement x.5 =-.9 = x x A (x) - A (x) A (x) w=.2 w=5 A (x) João M. C. Sousa 73 Intersection of fuzzy sets i: [,] [,] [,]; AB (xi( 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) João M. C. Sousa 74 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). João M. C. Sousa 75 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) ) João M. C. Sousa 76 2

21 Intersection of fuzzy sets AB (x) = min( A (x), B (x)) A B x João M. C. Sousa 77 Intersection of fuzzy sets A B min(a,b) A*B max(,a+b-) x João M. C. Sousa 78 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, João M. C. Sousa 79 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) João M. C. Sousa 8 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). João M. C. Sousa 8 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)) João M. C. Sousa 82 23

24 Union of fuzzy sets AB (x) = max( A (x), B (x)) A B x João M. C. Sousa 83 Union of fuzzy sets.8.6 A B max(a,b) A+B-A*B min(,a+b) x João M. C. Sousa 84 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, João M. C. Sousa 85 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 ) João M. C. Sousa 86 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 João M. C. Sousa 87 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 João M. C. Sousa 88 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(,, ) João M. C. Sousa 89 Fuzzy aggregation operations Parametric t-norms Parametric s-norms Generalized mean i min min max u max Intersection operators Averaging operators Union operators João M. C. Sousa 9 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 João M. C. Sousa 9 Membership functions (a) Triangular MF (b) Trapezoidal MF (c) Gaussian MF (d) Generalized Bell MF João M. C. Sousa 92 28

29 Left-right MF cx FL, xc LRxc ( ;,, ) xc FR, xc c= 65 = 6 = Membership Grades c = 25 = = 4 Membership Grades João M. C. Sousa 93 Two-dimensional fuzzy sets A ( xy, ) ( xy, )( xy, ) XY XY A A x y João Miguel da Costa Sousa 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 João M. C. Sousa 95 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 João Miguel da Costa Sousa 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 João M. C. Sousa 97 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 João M. C. Sousa 98 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 João M. C. Sousa 99 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 João M. C. Sousa 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 João M. C. Sousa 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)} João M. C. Sousa 2 33

34 Matrix representation USA Fra Can Brit Ger Dollar Pound Euro English USA Fra Can Brit Ger Dollar Pound Euro French João M. C. Sousa 3 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) João M. C. Sousa 4 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 João M. C. Sousa 5 Continuous example R: x y ( x is approximately equal to y ) ( x, y) e 2 ( xy) y x.5 João M. C. Sousa 6 35

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 João M. C. Sousa 7 Properties Associativity: R( ST) ( RS) T Distributivity over union: R( ST) ( RS) ( RT) Week distributivity over intersection: R( ST) ( RS) ( RT) Monotonicity: ST( RS) ( RT) João M. C. Sousa 8 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 João M. C. Sousa 9 Example a b c d P Y Q Z 2 João M. C. Sousa 37

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? João M. C. Sousa Matrix notation examples João M. C. Sousa 2 38

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. João M. C. Sousa 3 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. João M. C. Sousa 4 39