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
Course Outline Theory Theory of fuzzy sets, from crisp sets to fuzzy sets, basic concepts and definitions, Fuzzy operations, t-norms, t-conorms, aggregation operations Fuzzy arithmetic, fuzzy numbers, linguistic variables Fuzzy relations, fuzzy equivalence, fuzzy relational equations Fuzzy measures, possibility theory, Dempster-Shafer theory of evidence, Fuzzy logic, multi-valued logic, fuzzy qualifiers Uncertainty-based information, uncertainty measures, entropy, nonspecificity,
Course Outline Application Construction of fuzzy sets and operations from experts or datasample Approximate reasoning, fuzzy expert systems, Fuzzy systems, rule-based, data-based, and knowledge based systems Fuzzy control, design of fuzzy controllers Fuzzy modeling, fuzzy regression Fuzzy clustering, fuzzy pattern recognition, cluster validity Fuzzy information retrieval and fuzzy databases Fuzzy decision making, fuzzy ranking, information and fuzzy fusion
References Fuzzy sets and system
Recommended references (Journals) IEEE Trnas. On Fuzzy Systems Fuzzy sets and systems Int. Jou. Of Uncertainty, Fuzziness and Knowledge based systems Jour. Of Intelligent and Fuzzy systems Fuzzy optimization and decision making
Grading Policies Exams 50% Midterm 50% (25% of total) about 15/2/89 Final 50% (25% of total) Final Project (25%) One project (deadline is about 31/6/88) Seminar (15%) Every body present a seminar (select a subject until 15/1/89) Home works (10%) 5 home works
Fuzzy Sets and Systems Class Hours Sunday 10-13 Send your homework to hkhotanloo@yahoo.com with subject Fuzzy
Outline Introduction What is a Fuzzy set? Why Fuzzy? Application Fuzzy sets Introduction to Crisp sets Fuzzy sets theory Definition Membership function
What is a fuzzy set? A set is a collection of its members. The notion of fuzzy sets is an extension of the most fundamental property of sets. Fuzzy sets allows a grading of to what extent an element of a set belongs to that specific set.
What is a fuzzy set? Let us observe a (crisp) reference set X = {1; 2; 3; 4; 5; 6; 7; 8; 9; 10} The (crisp) subset C of X, C = {x 3 < x < 8}, c={4,5,6,7} The set F of big numbers in X F = {10; 9; 8; 7; 6; 5; 4; 3; 2; 1}
Why Fuzzy? Precision is not truth. Henri Matisse So far as the laws of mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality. Albert Einstein As complexity rises, precise statements lose meaning and meaningful statements lose precision. Lotfi Zadeh
Why Fuzzy? Complex, ill-defined processes difficult for description and analysis by exact mathematical techniques Approximate and inexact nature of the real word; vague concepts easily dealt with by humans in daily life Thus, we need other technique, as supplementary to conventional quantitative methods, for manipulation of vague and uncertain information, and to create systems that are much closer in spirit to human thinking. Fuzzy logic is a strong candidate for this purpose.
Fuzzy and Probability, Randomness and Fuzziness Fuzzy is not just another name for probability. The number 10 is not probably big!...and number 2 is not probably not big. Uncertainty is a consequence of non-sharp boundaries between the notions/objects, and not caused by lack of information. Randomness refers to an event that may or may not occur. Randomness: frequency of car accidents. Fuzziness refers to the boundary of a set that is not precise. Fuzziness: seriousness of a car accident.
Example : A fuzzy set of tall man
Another Example: Age groups
Introduction Fuzzy set theory was initiated by Zadeh in the early 1960s L. A. Zadeh, Fuzzy sets. Information and Control, Vol. 8, pp. 338-353. (1965). http://wwwisc.cs.berkeley.edu/zadeh/papers/fuzzy%20s ets-1965.pdf L. A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Transactions on Systems, Man and Cybernetics SMC-3, 28-44, 1973.
Introduction Applications Domain Fuzzy Logic Fuzzy Control o euro - Fuzzy System o Intelligent Control o Hybrid Control Fuzzy Pattern Recognition Fuzzy Modeling
Crisp Set theory
Basic concepts Set: a collection of items To Represent sets List method A={a, b, c} Rule method C = { x P(x) } Family of sets {Ai i I } Universal set X and empty set
A B : x A implies that x B A = B : A B and B A A B : A B and A B
Power set All the possible subsets of a given set X is call the power set of X, denoted by P(X) = {A A X} P(X) = 2^n when X = n X={a, b, c} P(X) = {, a, b, c, {a, b}, {b, c}, {a, c}, X}
Set Operations Union A B = {x x A x B} µ A B (x) = µ A (x) µ B (x) = Max{µ A (x),µ B (x)}
Set Set Operations operations Intersection A B = {x x A x B} µa B(x) = µa(x) µb(x) = Min{µA(x),µB(x)}
Set Operations Set operations Complement A = {x x A x X} µa (x) = 1- µa(x)
Set Operations Difference A-B = {x x A x B} µ A-B (x) = µ A (x)- µ B (x)
Basic properties of set operations
function A function from a set A to a set B is denoted by f: A B Many to one One-to-one
Characteristic function Let A be any subset of X, the characteristic function of A, denoted by χ, is defined by Characteristic function of the set of real numbers from 5 to 10
Real numbers (x- Total ordering: a b Real axis: the set of real number R axis) Interval: [a,b], (a,b), (a,b] One-dimensional Euclidean space
two-dimensional Euclidean space The Cartesian product of two real number R R Plane Cartesian coordinate, x-y axes Cartesian product A B = {<a, b> a A and b B}
Convexity A subset of Euclidean space A is convex, if line segment between all pairs of points in the set A are included in the set.
partition Given a nonempty set A, a family of disjoint subsets of A is called a partition of A, denoted by Π(A), if the union of these subsets yields the set A. Π(A) = {Ai i I, Ai A} Ai Aj = for each pair i j(i,j I) and i IAi=A partition 1, 2, 3 4, 5 1, 2 3, 4, 5 partition 1, 2, 3, 4, 5 partition refinement 1, 2 3 4, 5
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).
Membership functions Assign to each element x of X a number A(x) A: X [0, 1] The degree of membership
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)}
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 1 50 10 2
Fuzzy Sets representations List form Tabular form Rule form Membership form
List representations List representation of very high educated B = 0/0 + 0/1 + 0/2 + 0.1/3 + 0.5/4 +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
Tabular Representation level 0 1 2 3 4 5 6 membership 0 0 0 0.1 0.5 0.8 1
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};
Membership form
MF Terminology MF 1.5 α 0 Core X Crossover points α - cut Support
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.
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.
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.
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.
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
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) = 1+ 1 -a(x-b) e
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
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