Fuzzy Set SEG

Transcription

1 Fuzzy Set SEG

2 Introduction Fuzzy logic has been extremely successful in many applications image processing and control, fuzzy rule-based machine learning classification, clustering, and function approximation SEG

3 The traditional way of representing which objects are members of a set is in terms of a characteristic function where the objects x are elements of some universe X in terms of a functional mapping: Sets to which this applies are called crisp sets, in contrast to fuzzy sets Fuzzy logic 1 µ A ( x) = 0 if if x is an element of x is not an element of µ A (x) : X {0,1} set A set A pays attention to the "excluded middle" and tries to account for the "grays," the partially true and partially false situations which make up most human reasoning in everyday life builds upon the assumption that everything consists of degrees on a sliding scale SEG

4 In fuzzy sets an object may belong partially to a set The degree of membership in a fuzzy set is measured by a generalization of the characteristic function called the membership function or compatibility function defined as: µ A (x) : X [0,1] The membership function maps X into the codomain of real numbers defined in the interval from 0 to I inclusive and symbolized by [0,1] The membership function is a real number: 0 µ A (x) 1 where 0 means no membership and 1 means full membership in the set Using the membership function, real-world situations can be described SEG

5 Fuzzy sets and concepts are commonly used in natural language, such as: John is tall The weather is hot Turn the dial a little higher Most tests are hard If the dough is much too thick, add a lot of water where the words in italics refer to fuzzy sets and quantifiers. A fuzzy proposition may have degrees of truth For example, the fuzzy proposition "John is tall," may be true to some degree: A Little True, Somewhat True, Fairly True, Very True, and so on A fuzzy truth value is called a fuzzy qualifier Fuzzy propositions may have fuzzy quantifiers such as Most, Many, Usually, and so on SEG

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7 This concept of degree of attribute is expressed by the alternate meaning of the membership function as a compatibility The term compatibility means how well one object conforms to some attribute and is really better for describing fuzzy sets Example to illustrate the concept of fuzzy sets John is tall SEG

8 Depending on the application, a membership function may be constructed from one person s opinions or from a group of people's opinions Examples: credit risk for a loan, hostile intent of an unknown aircraft, quality of a product, suitability of a candidate for a job, and so on Intuitively, the membership function for a group of people also may be thought of in terms of an opinion poll A crossover point is where μ= 0.5 For this particular membership function, the crossover point for tall is 6 feet. the opinions are likelihoods because they express a personal belief SEG

9 The S-function is a mathematical function that is often used in fuzzy sets as a membership function: SEG

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11 The S-function can be a valuable tool in defining fuzzy functions such as the word tall all the data may be compactly represented by a formula. In this definition, are parameters that may be adjusted to fit the desired membership data The S-function is flat at a value of 0 for In between the S-function is a quadratic function of x For the TALL membership function of Figure 5.8, SEG

12 The membership function may be expressed as: µ CLOSE (x) With crossover points: x= γ ± β = 1+ 1 x γ β 2 SEG

13 A function that also gives a similar curve but does go to zero at specified points is the following: SEG

14 The β parameter is now the bandwidth or total width at the crossover points. The 11 -function goes to zero at the points: x= γ ± β while the crossover points are at: x= γ ± β/2 The membership function can be a finite set of elements. For example, in the universe of heights defined as: U = { 5, 5.5, 6, 6.5, 7, 7.5, 8 } A fuzzy subset can be defined for a finite Set of elements for TALL as follows: TALL = { 0/5, /5.5, 0.5/6, 0.875/6.5, 1/7, 1/7.5, 1/8 } SEG

15 A finite fuzzy subset of N elements is represented in standard fuzzy notation or SEG

16 The forms of equation (3) are difficult for people to read when numbers are involved. such as in: F = The following notation is better when dealing with numbers: F =.1/ /5 The support of a fuzzy set, F, is a subset of the universe set, X, defined as: The advantage of the support is that a fuzzy set F can be written as: which means that only those fuzzy elements whose membership function is greater than zero contribute to F SEG

17 The TALL set can be written without the 0/5 element is not in the support set: TELL = {0.125/5.5, 0.5/6, 0.875/5.5, 1/7, 1/7.5, 1/8} The reduction in elements can be significant for fuzzy sets with many elements of membership zero The α-cut of a set is a nonfuzzy set of the universe whose elements have a membership function greater than or equal to some value α: Some α-cuts of the TALL set are SEG

18 The height of a fuzzy set is defined as the maximum membership grade of an element For our TALL set, the maximum membership grade is 1 If an element in a fuzzy set attains the maximum possible grade, then the set is called normalized An arbitrary fuzzy subset of the universe over the continuum is written in the form of an integral. using an S-function for TALL SEG

19 Fuzzy set operations Concentration This operation concentrates fuzzy elements by reducing the membership grades more of elements that have smaller membership grades SEG

20 Fuzzy set operations SEG

21 Fuzzy set operations Dilation This operation dilates fuzzy elements by increasing the membership grade more of elements with smaller membership grades. SEG

22 Some commercial applications of fuzzy logic Many commercial applications of fuzzy logic are in everything from cameras to washing machines. Automatic control of dam gates for hydroelectric-powerplants (Tokyo Electric Power). Simplified control of robots (Hirota, Fuji Electric, Toshiba, Omron). Camera aiming for the telecast of sporting events (Omron) Substitution of an expert for the assessment of stock exchange activities (Yamaichi, Hitachi). Preventing unwanted temperature fluctuations in air-conditioning systems (Mitsubishi, Sharp). Efficient and stable control of car-engines (Nissan). Cruise-control for automobiles (Nissan, Subaru) SEG

23 Some commercial applications of fuzzy logic Improved efficiency and optimized function of industrial control applications (Aptronix, Omron, Meiden, Sha, Micom, Mitsubishi, Nisshin-Denki, Oku- Electronics). Positioning of wafer-steppers in the production of semiconductors (Canon). Optimized planning of bus time-tables (Toshiba, Nippon-System, Keihan-Express). Archiving system for documents (Mitsubishi Electric.). Prediction system for early recognition of earthquakes (Inst. of Seismology Bureau of Metrology, Japan). Medicine technology: cancer diagnosis (Kawasaki Medical School). Combination of Fuzzy Logic and Neural Nets (Matsushita). Recognition of handwritten symbols with pocket computers (Sony). SEG

24 Some commercial applications of fuzzy logic Compensation of motion in camcorders (Canon, Minolta) Automatic motor-control for vacuum cleaners with recognition of surface condition and degree of soiling (Matsushita). Back light control for camcorders (Sanyo). Compensation against vibrations in camcorders (Matsushita). Single button control for washing-machines (Matsushita, Hitatchi). Recognition of handwriting, objects, voice (CSK, Hitachi, Hosai Univ., Ricoh). Flight aid for helicopters (Sugeno). Simulation for legal proceedings (Meihi Gakuin Un iv, Nagoy Univ.). Software-design for industrial processes (Aptronix, Harima, Ishikawajima-OC Engeneering). SEG

25 Some commercial applications of fuzzy logic Controlling of machinery speed and temperature for steel-works (Kawasaki Steel, New-Nippon Steel, NKK) Controlling of subway systems in order to improve driving comfort, precision of halting and power economy (Hitachi). Improved fuel-consumption for automobiles (NOK, Nippon Denki Tools). Improved sensitiveness and efficiency for elevator control (Fujitec, Hitachi, Toshiba). Improved safety for nuclear reactors (Hitachi, Bernard, Nuclear Fuel Div.) SEG