1. Fuzzy Set
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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
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3. Fuzzy sets and natural language
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
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
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4. Fuzzy sets and natural language
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:
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:
where 0 means no membership and 1 means full membership in the set
Using the membership function, real-world situations can be described
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5. Fuzzy sets and natural language
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
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6. Fuzzy sets and natural language
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7. Fuzzy sets and natural language
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”
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8. Fuzzy sets and natural language
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
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9. Fuzzy sets and natural language
The S-function is a mathematical function that is often used in fuzzy sets as a membership function:
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10. Fuzzy sets and natural language
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11. Fuzzy sets and natural language
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,
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12. Fuzzy sets and natural language
The membership function may be expressed as:
With crossover points: x= γ ± β
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13. Fuzzy sets and natural language
A function that also gives a similar curve but does go to zero at specified points is the following:
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14. Fuzzy sets and natural language
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 }
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15. Fuzzy sets and natural language
A finite fuzzy subset of N elements is represented in standard fuzzy notation or
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16. Fuzzy sets and natural language
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
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17. Fuzzy sets and natural language
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
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18. Fuzzy sets and natural language
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
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19. Fuzzy set operations Concentration
This operation concentrates fuzzy elements by reducing the membership grades more of elements that have smaller membership grades
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20. Fuzzy set operations
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21. Fuzzy set operations Dilation
This operation dilates fuzzy elements by increasing the membership grade more of elements with smaller membership grades.
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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)
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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).
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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).
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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.)
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