Fuzzy set theory provides a means for representing uncertainties

Comments

Posting comment...

Premium member

Presentation Transcript

PowerPoint Presentation:

Fuzzy Logic and Its Applications Textbook: Fuzzy Logic with Engineering Application References: Fuzzy Logic with Engineering Applications, 1995 T.J.Ross, McGraw-Hill Fuzzy Set Teory, 1997 G.Klir et al. Prentice Hall Fuzzy Sets and Fuzzy Logic 1995 G Klir et al. Prentice Hall

PowerPoint Presentation:

Introduction In 1965, Prof. Lofti Zadeh published the first article “Fuzzy Sets”. It becomes billions of dollars business. America Europe Asia Thousands of patents Uncertainty: Incomplete Ambiguity: Imprecise Applications: Air Conditioner Washing Machine Subway System Camera Aerospace Nuclear Submarine Pattern Recognition Control Image Processing Computer Vision

PowerPoint Presentation:

They reflect a recent trend to view fuzzy logic (FL), neurocomputing (NC), genetic computing (GC), Rough Sets (RS) and probabilistic computing (PC) as an association of computing methodologies falling under the rubric of so-called soft computing. Among the basic concepts that underlie human cognition, three stand out in importance: granulation, organization, and causation.

PowerPoint Presentation:

Granulation involves a partitioning of a whole into parts; organization involves an integration of parts into a whole; and causation relates to an association of causes with effects A granule may be viewed as a clump of points (objects) drawn together by indistinguishability, similarity, or functionality. Modes of information granulation (IG) in which granules are crisp play an important role in many theories, methods and techniques, among them interval analysis, quantization, rough set theory, qualitative process theory, and chunking.

PowerPoint Presentation:

In fuzzy logic, fuzzy IG underlies the basic concepts of linguistic variables, fuzzy if-then rules, and fuzzy graphs This perception is reinforced by viewing it in the context of generalization. More specifically, any theory, method, technique, or problem may be fuzzified (or f-generalized) by replacing the concept of a crisp set with that of a fuzzy set. Similarly, any theory, method, technique, or problem can be granulated (g-generalized) by partitioning variables, functions, and relations into granules.

PowerPoint Presentation:

Furthermore, we can combine fuzzification with granulation, which gives rise to fuzzy granulation (f-granulation). Fuzzy granulation, then, provides a basis for what might be called f.g-generalization. The generalization of two-valued logic leads to multivalued logic and parts of fuzzy logic. But fuzzy logic in its wide sense — which is the sense in which it is used today — results from f.g-generalization. This crucial difference between multivalued logic and fuzzy logic explains why fuzzy logic has so many applications, whereas multivalued logic does not.

PowerPoint Presentation:

Introduction Fuzzy set theory provides a means for representing uncertainties. Probablity – random uncertainty But some uncertainty is non-random In fact, a huge amount! Natural Language is vague and imprecise. Fuzzy set theory uses Linguistic variables, rather than quantitative variables to represent imprecise concepts.

PowerPoint Presentation:

Fuzzy Logic Fuzzy Logic is suitable to Very complex models Judgemental Reasoning Perception Decision making Requiring precision – high cost, long time Statistics and random processes Based on Randomness.

PowerPoint Presentation:

Fuzziness Example. Random Errors generally average out over time or space Non-random errors will not generally average out and likely to grow with time. Information World

PowerPoint Presentation:

Information World Crisp set has a unique membership function A (x) = 1 x A 0 x A A (x) {0, 1} Fuzzy Set can have an infinite number of membership functions A [0,1]

PowerPoint Presentation:

Fuzziness Examples: A number is close to 5

PowerPoint Presentation:

Fuzziness Examples: He/she is tall

PowerPoint Presentation:

Fuzziness Randomness versus Fuzziness Drinking Water Problem Classical Sets Fuzzy Sets

PowerPoint Presentation:

Operations on Classical Sets Union: A B = {x | x A or x B} Intersection: A B = {x | x A and x B} Complement: A’ = {x | x A, x X} X – Universal Set Set Difference: A | B = {x | x A and x B} Set difference is also denoted by A - B

PowerPoint Presentation:

Properties of Classical Sets A B = B A A B = B A A (B C) = (A B) C A (B C) = (A B) C A (B C) = (A B) (A C) A (B C) = (A B) (A C) A A = A A A = A A X = X A X = A A = A A =

PowerPoint Presentation:

Properties of Classical Sets If A B C, then A C De Morgan ’ s Law: (A B) ’ = A ’ B ’ (A B) ’ = A ’ B ’ Proof: LHS= {x | x (A and B)}= {x | x A or x B)}= A ’ B ’ = RHS

PowerPoint Presentation:

Can be extended to n sets Generalized De Morgan Law: A A ’ X X Using ( ) to keep original processing order

PowerPoint Presentation:

Generalized Duality Law: X X Using ( ) to keep original processing order

PowerPoint Presentation:

Law of the excluded middle: A A ’ = X Law of the Contradiction: A A ’ = These laws are not true for Fuzzy Sets!

PowerPoint Presentation:

Fuzzy Sets Characteristic function X, indicating the belongingness of x to the set A X(x) = 1 x A 0 x A or called membership Hence, A B X A B (x) = X A (x) X B (x) = max(X A (x),X B (x)) Note: Some books use + for , but still it is not ordinary addition! Some more explanations follow…

PowerPoint Presentation:

Fuzzy Sets A B X A B (x) = X A (x) X B (x) = min( X A (x), X B (x)) A’ X A’ (x) = 1 – X A (x) A B X A (x) X B (x) A’’ = A

PowerPoint Presentation:

Fuzzy Sets Note (x) [0,1] not {0,1} like Crisp set A = { A (x1) / x1 + A (x2) / x2 + … } = { A (xi) / xi } Note: ‘+’ add ‘/ ’ divide Only for representing element and its membership. Also some books use (x) for Crisp Sets too.

PowerPoint Presentation:

Fuzzy Set Operations A B (x) = A (x) B (x) = max( A (x), B (x)) A B (x) = A (x) B (x) = min( A (x), B (x)) A’ (x) = 1 - A (x) De Morgan’s Law also holds: (A B)’ = A’ B’ (A B)’ = A’ B’ But, in general A A’ A A’

PowerPoint Presentation:

Properties of Fuzzy Sets A B = B A A B = B A A (B C) = (A B) C A (B C) = (A B) C A (B C) = (A B) (A C) A (B C) = (A B) (A C) A A = A A A = A A X = X A X = A A = A A = If A B C, then A C A’’ = A

PowerPoint Presentation:

Sets as Points in Hypercubes Explore to n-dimension Classical Relations Fuzzy relations Logic, Approximate reasoning, Rule-based learning systems, Nonlinear Simulation, Classification, Pattern Recognition, etc.

PowerPoint Presentation:

Cartesian Product A = {a,b} B = {0,1} A x B = { (a,0) (a,1) (b,0) (b,1) } Ordered Pairs Consider A x A or A x B x C if C is given Based on the above, Crisp Relations are discussed next…

PowerPoint Presentation:

Crisp Relations A subset of a Cartesian Product A1 x A2 x … x Ar is called an r-ary relation over A1,A2,…,Ar If r = 2, the relation is a subset of A1 x A2 Binary relation from A1 into A2 The strength of a relation: Characteristic Function X(x,y) = 1 (x,y) X x Y 0 (x,y) X x Y For Classical relations, the value is 1 or 0 If the universes or sets are finite, we can use relational matrix to represent it.

PowerPoint Presentation:

Crisp Relations Example: If X = {1,2,3} Y = {a,b,c} R = { (1 a),(1 c),(2 a),(2 b),(3 b),(3 c) } a b c 1 1 0 1 R = 2 1 1 0 3 0 1 1 Using a diagram to represent the relation

PowerPoint Presentation:

Crisp Relations Relations can also be defined for continuous universes R = { (x,y) | y 2x, x X, y Y } X = 1 y 2x 0 otherwise

PowerPoint Presentation:

Crisp Relations Cardinality: N: # of elements in X M: # of elements in y Cardinality of R n X x Y = n X • n Y = M • N Cardinality of the Power set of this relation n P(X x Y) = 2 M N

Operations on Crisp Relations Union R S X R S (x,y) X R S (x,y) = max{ X R (x,y),X S (x,y) } Intersection R S X R S (x,y) X R S (x,y) = min{ X R (x,y),X S (x,y) } Complement R’ X R’ (x,y) X R’ (x,y) = 1 – X R (x,y) Containment R S X R (x,y) X S (x,y) Identity 0 X E

PowerPoint Presentation:

Properties of Crisp Relations Commutativity Associativity Distributivity Idempotency All hold De Morgan Law Excluded middle Law Etc.

PowerPoint Presentation:

Properties of Crisp Relations Composition Let R be a relation representing a mapping from X to Y X Y University sets Let S be a relation, a mapping from Y to Z Can we find T from R to S?

PowerPoint Presentation:

Properties of Crisp Relations T: mapping from X to Z T = R S Two ways to compute X T (xz) X T (xz) = (X R (xy) X s (yz)) = max(min{X R (xy),X S (yz)}) Max-min composition X T (xz) = (X R (xy) X s (yz)) Max-product composition multiplication y Y y Y y Y

PowerPoint Presentation:

Properties of Crisp Relations Using Matrix representation: y1 y2 y3 y4 x1 1 0 1 0 R = x2 0 0 0 1 x3 0 0 0 0 z1 z2 y1 0 1 z1 z2 y2 0 0 x1 0 0 S = y3 0 1 T = x2 0 0 y4 0 0 x3 0 0 T (x1,z1) = max[min(1,0) min(0,0) min(1,0) min(0,0)] = max[0,0,0,0] = 0 Similar, but not the same as matrix multiplication!

PowerPoint Presentation:

Fuzzy Relations Cardinality of Fuzzy Relations Since the cardinality of fuzzy sets on any universe is infinity, the cardinality of a fuzzy relation is also infinity. Note: other books have different discussions!

PowerPoint Presentation:

Operations on Fuzzy Relations Union: R S = max{ R (x,y), S (x,y) } Intersection: R S = min{ R (x,y), S (x,y) } Complement: R ’ (x,y) = 1 - R (x,y) Containment: R S R (x,y) S (x,y)

PowerPoint Presentation:

Properties of Fuzzy Relations Commutativity Associativity Distributivity Idempotency All hold De Morgan Law Excluded middle Law Etc. Note: R R’ E R R’ 0 In general.

PowerPoint Presentation:

Properties of Fuzzy Relations Fuzzy Cartesian Product and Composition R (x y) = A x B (x y) = min( A (x), B (y)) Example: A = 0.2/x1 + 0.5/x2 + 1/x3 B = 0.3/y1 + 0.9/y2 y1 y2 0.2 x1 0.2 0.2 A x B = 0.5 0.3 0.9 = x2 0.3 0.5 1 x3 0.3 0.9

PowerPoint Presentation:

Properties of Fuzzy Relations Vector Outer Product If R is a fuzzy relation on the space X x Y S is a fuzzy relation on the space Y x Z Then, fuzzy composition is T = R S Fuzzy max-min composition T (xz) = ( R (xy) s (yz)) 2. Fuzzy max-production composition T (xz) = ( R (xy) s (yz)) Note: R S S R y Y y Y

PowerPoint Presentation:

Properties of Fuzzy Relations Example: y1 y2 z1 z2 z3 R = x1 0.7 0.5 S = y1 0.9 0.6 0.2 x2 0.8 0.4 y2 0.1 0.7 0.5 z1 z2 z3 Using max-min, T = x1 0.7 0.6 0.5 x2 0.8 0.6 0.4 z1 z2 z3 Using max-product, T = x1 0.63 0.42 0.25 x2 0.72 0.48 0.20 Note: Set, Relation, Composition How to find new membership from the given ones!

PowerPoint Presentation:

Tolerance and Equivalence Relation Crisp Equivalence Relation R X x X Relation has the following properties: Reflexivity (xi xi) R or X R (xi xi) = 1 Symmetry (xi xj) R (xj xi) R or X R (xi xj) = X R (xj xi) Transitivity (xi xj) R and (xj xk) R (xi xk) R or X R (xi xj) = 1 and X R (xj xk) = 1 X R (xi xk) = 1

PowerPoint Presentation:

Tolerance and Equivalence Relation Graph representation:

PowerPoint Presentation:

Crisp Tolerance Relation (or proximity relation) Only has reflexivity and symmetry A tolerance relation, R1 can become an Equivalence Relation by at most (n-1) compositions ( n-1), n is the cardinal member of X. R 1 n-1 = R1 R1 … R1 = R

Fuzzy Tolerance and Equivalence Relation A fuzzy relation R has: 1. Reflexivity R (xi xi) = 1 2. Symmetry R (xi xj) = R (xj xi) 3. Transitivity R (xi xj) = 1 R (xj xk) = 2 R (xi xk) = where min{1, 2} Fuzzy tolerance relation R1 has reflexivity, symmetry. It can be transformed into a fuzzy equivalence relation by at most (n-1) ( n-1) compositions. R 1 n-1 = R1 R1 … R1 = R

Fuzzy Tolerance and Equivalence Relation Value Assignment How to find the membership values for the relation? Cartesian Production Note: you have to know the membership value for the sets! Will discuss in chapter 4. 2. Y = f(x) X – input vector Y – output vector 3. Look up table y1 y2 y3 x1 x2 x3

PowerPoint Presentation:

Fuzzy Tolerance and Equivalence Relation Value Assignment 4. Linguistic rule of knowledge – chapters 7 – 9 5. Classification – chapter 11 6. Similarity methods in data manipulation The more robust a data set, the more accurate the relation entries!

PowerPoint Presentation:

Cosine Amplitude X = {x 1 ,x 2 , … ,x n } each element is also a vector X i = {x i1 ,x i2 , … ,x im } ij = R (x i ,x j ) It will be n x n symmetric,reflexive … i.e. a tolerance relation! Note: this relates to the vector dot product for cosine function

Other Similarity Methods Absolute Exponential: Exponential Similarity Coefficient: Where, S k = any general measure for all the data i.e. (S k ) 2 ≥ 0

PowerPoint Presentation:

Other Similarity Methods Other methods produce scalar quantities which are similar to the cosine amplitude, such as the following: Geometric average minimum: Scalar Product: Where:

PowerPoint Presentation:

Other Similarity Methods Some methods are analogous to popular statistical quantities, such as: Correlation Coefficient: Where: and Arithmetic Average Minimum:

PowerPoint Presentation:

Other Similarity Methods Some methods are based on the inverse relationships, for example: Absolute Reciprocal: Where M is selected to make 0 ≤ r ij ≤ 1 Absolute subtrahend: Where c is selected to make 0 ≤ r ij ≤ 1

PowerPoint Presentation:

Other Similarity Methods Other methods are nonparametric, such as: Nonparametric: where x ’ ik = x ik – x i and x ’ jk – x j n + = number of elements > 0 in {x ’ i1 x ’ j1 ,x ’ i2 x ’ j2 , … ,x ’ im x ’ jm } n - = number of elements < 0 in {x ’ i1 x ’ j1 ,x ’ i2 ,x ’ j2 , … ,x ’ im ,x ’ jm } In the above equations, terms such as x ’ i1 x ’ j1 are products of data elements.

PowerPoint Presentation:

Membership Function Membership Functions characterize the fuzziness of fuzzy sets. There are an infinite # of ways to characterize fuzzy infinite ways to define fuzzy membership functions. Membership function essentially embodies all fuzziness for a particular fuzzy set, its description is essential to fuzzy property or operation.

PowerPoint Presentation:

Features of Membership Function Core: comprises of elements x of the universe, such that A (x) = 1 Support: comprises of elements x of universe, such that A (x) > 0 Boundaries: comprise the elements x of the universe 0 < A (x) < 1 A normal fuzzy set has at least one element with membership 1 For fuzzy set, if one and only one element has a membership = 1, this element is called as the prototype of set. A subnormal fuzzy set has no element with membership=1.

PowerPoint Presentation:

Features of Membership Function Graphically,

PowerPoint Presentation:

Features of Membership Function A convex fuzzy set has a membership whose value is: 1. strictly monotonically increasing, or 2. strictly monotonically decreasing, or 3. strictly monotonically increasing, then strictly monotonically decreasing Or another way to describe: (y) ≥ min[(x), (z)], if x < y < z If A and B are convex sets, then A B is also a convex set Crossover points have membership 0.5 Height of a Fuzzy set is the maximum value of the membership: max{ A (x)}

PowerPoint Presentation:

Features of Membership Function If height < 1, the fuzzy set is subnormal. Fuzzy number: like a number is close to 5. It has to have the properties: 1. A must be a normal fuzzy set. 2. A must be closed for all (0,1]. 3. The support, 0A must be bounded.

You do not have the permission to view this presentation. In order to view it, please
contact the author of the presentation.

Send to Blogs and Networks

Processing ....

Premium member

Use HTTPs

HTTPS (Hypertext Transfer Protocol Secure) is a protocol used by Web servers to transfer and display Web content securely. Most web browsers block content or generate a “mixed content” warning when users access web pages via HTTPS that contain embedded content loaded via HTTP. To prevent users from facing this, Use HTTPS option.