Introduction to Fuzzy Logic

1
Introduction to Fuzzy Logic

• Adnan Yazici
• Dept. of Computer Engineering, Middle East
  Technical University, 06531, Ankara/Turkey

2
Introduction

• Mathematics that refers to reality is not
  certain and mathematics that is certain does not
  refer to reality
• Albert Einstein
•  
• While the mathematician constructs a theory in
  terms of perfectobjects, the experimental
  observes objects of which the properties demanded
  by theory are and can, in the very nature of
  measurement, be only approximately true
• Max Black
•  
• What makes society turn is science, and the
  language of science is math, and the structure of
  math is logic, and the bedrock of logic is
  Aristotle, and that is what goes out with fuzzy
  logic
• Bart Kosko

3
Introduction (cont.)

• Uncertainty is produced when a lack of
  information exists.
• The complexity also involves the degree of
  uncertainty.
• It is possible to have a great deal of data
  (facts collected from observations or
  measurements) and at the same time lack of
  information (meaningful interpretation and
  correlation of data that allows one to make
  decisions.)
• Data
  Information
• Database
  Intelligent information systems

? Knowledge Intelligence
? Knowledge base AI
4
Introduction (cont.)

• Knowledge is information at a higher level of
  abstraction.
• Ex Ali is 10 years old (fact)
• Ali is not old (knowledge)
• Our problems are
• Decision
• Management
• Prediction
• Solutions are
• Faster access to more information and of
  increased aid in analysis
• Understanding utilizing information available
• Managing with information not avaliable
• Large amount of information with large amount of
  uncertainty lead to complexity.
• Avareness of knowledge (what we know and what we
  do not know) and complexity goes together.
• Ex Driving a car is complex, driving in an iced
  road is more compex, since more knowledge is
  needed for driving in an iced road.

5
Introduction (cont.)

• Fuzzy logic provides a systematic basis for
  representation of uncertainty, imprecision,
  vagueness, and/or incompletenes.
• Uncertain information Information for which it
  is not possible to determine whether it is true
  or false. Ex a person is possibly 30 years old
• Imprecise information Information which is not
  available as precise as it should be. Ex A
  person is around 30 years old.
• Vague information Information which is
  inherently vague.
• Ex A person is young.
• Inconsistent information Information which
  contains two or more assertions that cannot be
  true at the same time. Ex Two assertions are
  given Ali is 16 and Ali is older than 20
• Incomplete information information for which
  data is missing or data is partially available.
  Ex A persons age is not known or a person is
  between 25 and 32 years old
• Combination of the various types of such
  information may also exist. Ex possibly young,
  possibly around 30, etc.

6
Introduction (cont.)
7
Introduction (cont.)

• Example When uncertainties like heavy traffic,
  unfamiliar roads, unstable wheather conditions,
  etc. increase, the complexity of driving a car
  increases.
• How do we go with the complexity?
• We try to simplify the complexity by making a
  satisfactory trade-off between information
  available to us and the amount of uncertainty we
  allow.
• We increase the amount of uncertainty by
  replacing some of the precise information with
  vague but more useful information.

8
Introduction (cont.)

• Examples
• Travel directions try to do it in mm terms (or
  turn the wheel 23 left, etc.), which is very
  precise and complex but not very useful. So
  replace mm information with city blocks, which is
  not as precise but more meaningful (and/or
  useful) information.
• Parking a car doing it in mm terms, which is
  very precise and complex but difficult and very
  costly and not very useful. So replace mm
  information with approximate terms (between two
  lines), which is not as precise but more
  meaningful (or useful) information and can be
  done in less cost.
• Describing wheather of a day try to do it in
  cloud cover, which is very precise and complex
  but not very useful. So replace cloud
  information with vague terms (very cloudy, sunny
  etc.), which is not as precise but more
  meaningful (or useful) information.

9
Introduction (cont.)

• Fuzzy logic has been used for two different
  senses
• In a narrow sense refers to logical system
  generalizing crisp logic for reasoning
  uncertainty.
• In a broad sense refers to all of the theories
  and technologies that employ fuzzy sets, which
  are classes with imprecise boundaries.
• The broad sense of fuzzy logic includes the
  narrow sense of fuzzy logic as a branch.
• Other areas include fuzzy control, fuzzy pattern
  recongnition, fuzzy arithmetic, fuzzy probability
  theory, fuzzy decision analysis, fuzzy databases,
  fuzzy expert systems, fuzzy computer SW and HW,
  etc.

10
Introduction (cont.)

• With Fuzzy Logic, one can accomplish two things
• Ease of describing human knowledge involving
  vague concepts
• Enhanced ability to develop a cost-effective
  solution to real-world
• In another word, fuzzy logic not only provides a
  cost effective way to model complex systems
  involving numeric variables but also offers a
  quantitative description of the system that is
  easy to comprehend.

11
Introduction (cont.)

• Fuzzy Logic was motivated by two objectives
• First, it aims to alleviate difficulties in
  developing and analyzing complex systems
  encountered by conventional mathematical tools.
  This motivation requires fuzzy logic to work in
  quantitative and numeric domains.
• Second, it is motivated by observing that human
  reasoning can utilize concepts and knowledge that
  do not have well defined, sharp boundaries (i.e.,
  vague concepts). This motivation enables fuzzy
  logic to have a descriptive and qualitative form.
  This is related to AI.

12
Introduction (cont.)

• Components of Fuzzy Logic
• Fuzzy Predicates tall, small, kind,
  expensive,...
• Predicates modifiers (hedges) very, quite, more
  or less, extremely,..
• Fuzzy truth values true, very true, fairly
  false,...
• Fuzzy quantifiers most, few, almost, usually, ..
• Fuzzy probabilities likely, very likely, highly
  likely,...

13
Introduction (cont.)

• Applications
• Control If the temperature is very high and
  the presure is decreasing rapidly, then reduce
  the heat significantly.
• Database Retrieve the names of all candidates
  that are fairly young, have a strong background
  in algorithms, and a modest administrative
  experience.
• Medicine Hepatitis is characterized by the
  statement, Total proteins are usually normal,
  albumin is decreased, ?-globulins are slightly
  decreased, ?-globulins are slightly decreased,
  ?-globulins are increased

14
Introduction (cont.)

• Probability theory vs fuzzy set theory
• Probability measures the likelihood of a future
  event, based on something known now. Probability
  is the theory of random events and is not capable
  of capturing uncertainty resulting from vagueness
  of linguistic terms.
• Fuzziness is not the uncertainty of expectation.
  It is the uncertainty resulting from imprecision
  of meaning of a concept expressed by a linguistic
  term in NL, such as tall or warm etc.

15
Introduction (cont.)

• Probability theory vs fuzzy set theory (cont)
• Fuzzy set theory makes statements about one
  concrete object therefore, modeling local
  vagueness, whereas probability theory makes
  statements about a collection of objects from
  which one is selected therefore, modeling global
  uncertainty.
• Fuzzy logic and probability complement each
  other.
• Example highly probable is a concept that
  involves both randomness and fuziness.
• The behaviour of a fuzzy system is completely
  deterministic.
• Fuzzy logic differs from multivalued logic by
  introducing concepts such as linguistic variables
  and hedges to capture human linguistic reasoning.

16
Introduction (cont.)

• Even though the broad sense of fuzzy logic covers
  a wide range of theories and techniques, its core
  technique is based on four basic concepts
• Fuzzy sets sets with smooth boundaries
• Linguistic variables variables whose values are
  both qualitatively and quantitatively described
  by a fuzzy set
• Possibility distribution constraints on the
  value of a linguistic variable imposed by
  assigning it a fuzzy set and
• Fuzzy if-then rules a knowledge representation
  scheme for describing a functional mapping (fuzzy
  mapping rules) or a logical formula that
  generalizes an implication in two-valued logic
  (fuzzy implication rules).
• The first three concepts are fundamental for all
  subareas in fuzzy logic, but the fourth one is
  also important.

17
Fuzzy Sets

• Mathematically speaking, a fuzzy set is
  characterized by mapping from its universe of
  discourse into the interval, 0,1.
• Each fuzzy set is defined in terms of a relevant
  universal set U by a membership function, denoted
  as ?A(u), where u ? U.
• Formally, membership functions are the functions
  of the form
• ?A U --gt 0,1 is called the membership
  function of A.
• The set A(u, ?A(u)) u?U is called a fuzzy
  set in U.
• Given a fuzzy set A, which is a subset of the
  universe set, U, the support of A denoted by Supp
  (A), is an ordinary set defined as the set of
  elements whose degree of membership in A is
  greater than 0.
• Supp (A) u ? U ?A(u) gt 0.

18
Fuzzy Sets (cont.)

• ?A x ?A(u) ? ? is called ?-cut.
• ?1A ? ?2A and ?1A ? ?2A, when ?1? ?2, which
  implies that the set of all distinct ?-cuts (as
  well as strong ?-cuts) is always a nested family
  of crisp sets.
• ?A x ?A(u) gt ? is called strong ?-cut.
• 0A x ?A(u) gt 0 is called support of A.
• 1A x ?A(u) 1 is called core of A.
• When the core of A is not empty, A is called
  normal otherwise, it is called subnormal.
• The largest value of A is called the height of A,
  denoted as hA.
• The set of distinct values of ?A(u),?u? U is
  called the level set of A and denoted as ?A.

19
Fuzzy Sets (cont.)

20
Fuzzy Sets (cont.)

• The significance of ?-cut representation of fuzzy
  sets is that it connects fuzzy sets with crsip
  sets.
• While each crisp set is a collection of a
  colection of objects that are conceived as a
  whole, each fuzzy set is a collection of nested
  crisp sets that are also conceived as a whole.
• Fuzzy sets are thus wholes of a higher category.
• Example A 0.2/x1 0.4/x20.6/x30.8/x41/x5
• Its level set is ?A 0.2,0.4,0.6, 0.8,1, so it
  is associated with only 5-distinct ?-cuts, which
  are defined as follows
• 0.2A 1/x11/x21/x31/x41/x5
• 0.4A 0/x11/x21/x31/x41/x5
• 0.6A 0/x10/x21/x31/x41/x5
• 0.8A 0/x10/x20/x31/x41/x5
• 1A 0/x10/x20/x30/x41/x5

21
Fuzzy Sets (cont.)

• Theorem (Decomposition theorem of fuzzy sets)
  For any A ? F(X),
• A ???0,1 ?A
• We now convert each of the ?-cuts to a special
  fuzzy set ?A defined for each u?A by the formula
  ?A ?.??A(u). We obtain the following results
• 0.2A 0.2/x10.2/x20.2/x30.2/x40.2/x5
• 0.4A 0/x10.4/x20.4/x30.4/x40.4/x5
• 0.6A 0/x10/x20.6/x30.6/x40.6/x5
• 0.8A 0/x10/x20/x30.8/x40.8/x5
• 1A 0/x10/x20/x30/x41/x5
• The union of these five special fuzzy set is
  exactly the original fuzzy set A, that is, A
  0.2A ? 0.4 A ? 0.6 A ? 0.8 A ? 1A
• A 0.2/x1 0.4/x20.6/x30.8/x41/x5

22
Fuzzy Sets (cont.)

• Any property of fuzzy sets that is derieved from
  classical set theory is called a cutworthy
  property.
• Examples
• A B iff ?A(u) ?B(u), ?x ?X, similarly,
• A B iff ?A ?B, ?? ?0,1
• A ? B iff ?A ? ?B, ?? ?0,1
• The convexity of fuzzy sets A fuzzy set defined
  on the set of real numbers (or more generally, on
  any n-dim Euclidean space) is said to be convex
  iff all of its ?-cuts are convex in the classical
  sense. For a fuzzy set to be convex the graph
  must have just one peak.

convex
non convex
23
Fuzzy Sets (cont.)

• In order to develop computation with fuzzy sets,
  we need to take crisp functions and fuzzify them.
  A principle for fuzzyfying crisp functions is
  called the extension principle.
• f X?Y, where X and Y are crisp sets.
• We say that the function is fuzzified when it is
  extended to act on fuzzy sets defined on X and Y.
  Formally, the fuzzified function, f, has the
  form
• f F(X) ? F(Y), where F(X) and F(Y) denote the
  fuzzy power set (the set of all fuzzy subsets) of
  X and Y, respectively.
• To qualify as a fuzzified version of f, function
  f must conform to f within the extended domain
  F(X) and F(Y). This is guaranteed when a
  principle is employed that is called an extension
  principle. According to this principle,
• B f(A) is determined for any given fuzzy set
  A?F(X) via the formula B(y) max xyf(x) A(x)
  for all y ?Y.
• When the maximum does not exist, it is replaced
  with the supremum.

24
Fuzzy Sets (cont.)

• The inverse function, f-1, is from F(Y) to F(X).
• f-1 F(Y) ? F(X).
• According to the extension principle, for any
  B?F(Y),
• f-1(B) (x) B(f(x)) B(y), for all x ?X,
  where y f(x).
• Example Employees ages and their salaries
• Query What is a young employees salary?
• Answer We use extension principle here. Let us
  have a function f X? Y, where X
  20,25,30,35,40,45,50,55,60,65 and
• Y 2.5, 3, 3.5, 4.0, 4.5, 5.0

Age in years 20 25 30 35 40 45 50 55 60 65
Salary in K 2.5 2.5 3.0 3.5 3.5 4.0 4.0 4.5 4.5 5.0
25
Fuzzy Sets (cont.)

• First step Formulate the meaning of the concept
  young as a fuzzy set A of general form A
  ?A(x) / x for all x ?X. Assume that
• Ayoung 1/20 1/250.8/300.6/350.4/400.2/450
  /500/550/600/65
• Second step Use the fuzzy set A and information
  in the table to determine an appropriate fuzzy
  set B that captures the meaning of the linguistic
  expression young employees salary.
• This fuzzy set is dependent on A via function f
  which for each x in X assigns a particular y
  f(x) in Y. This dependency is expressed by the
  general form
• B(y) max xyf(x) A(x) max xyf(x) ?A(x) /
  f(x)
• B ?A(x) / f(x) 1/f(20) 1/f(25) 0.8/f(30)
  0.6/f(35) 0.4/f(40) 0.2/f(45) 0/f(50)
  0/f(55) 0/f(60) 0/f(65)
• 1/2.5 1/2.50.8/30.6/3.50.4/3.50.2/40/40
  /4.50/4.50/5
• Third step B(y) max xyf(x) A(x) 1/2.5
  0.8/30.6/3.50.2/40/4.50/5, which denotes the
  salary of young employes in the company.

26
Fuzzy Sets (cont.)

• Now let us answer the query Who are employees
  with low salary?
• Answer
• First, assume that Blow 1/2.50.75/30.5/3.50.2
  5/40/4.50/5
• f-1(B) (x) B(f(x))
• B(f(20)/20B(f(25)/25B(f(30)/30B(f(35)/35B(f(4
  0)/40B(f(45)/45 B(f(50)/50B(f(55)/55B(f(60)/60
  B(f(65)/65
• 1/201/250.75/300.5/350.5/400.25/450.25/50
  0/550/600/65
• This fuzzy set is defined on X and represents the
  age of employees with low salaries.

27
Fuzzy Sets (cont.)
28
Fuzzy Sets (cont.)

• Basic operations
• Set union A ? B ? u,?A ? B (u) (u ? A? u ?
  B) ? ? (A ? B) (u) Max (?A(u), ?B(u))
• Set intersection A?B? u,?A ? B (u) (u?A ? u?B)
  ? ? (A ? B) (u) Min (?A(u), ?B(u))
• Set equality A B ? u,?A (u) (u ? A ? u ?
  B) ? ?A(u) ?B(u)

29
Fuzzy Sets (cont.)

• Basic operations
• Set Complement?? u,??A (u) (??A (u)
  (1- ?A(u))
• Set containment A? B ? u ?u (u? A? u? B) ?
  ?A(u) ? ?B(u)
• ConcentrationCON(A)u,?CON(A) (u) (u?A ?
  ? CON(A) (u) (?A(u))2
• DilationDIL(A) u, ?DIL(A) (u) (u ? A ?
• ? DIL(A) (u) (?A(u))1/2

30
Fuzzy Sets (cont.)
 
?very A (u) ? ?A (u) ?
?More-or-Less A (u)
31
Fuzzy Sets (cont.)
?tv(a)
Fairly False
Fairly True
1 0.8 0.45 0.4 0.3 0.2 0
False
Very False
True
Very True
Absolutley False
Absolutley True
0
0.8 (for u) 1
32
Fuzzy Sets (cont.)

• Types of membership functions
• The most commonly used membership functions in
  practice are triangles, trapezoids, bell curves,
  Gaussian, and sigmoid functions.
• Triangular membership function is specified by
  three parameters a,b,cas follows
•   Trapezoidal membership function is specified by
  four parameters a,b,c,d as follows
•   A Gaussian membership function is specified by
  two parameters m,?) as follows
•   Gaussian (xm,?) exp (-(x-m)2/?2)
• where m and ? denote the center and width of the
  function, respectively. We control the shape of
  the function by adjusting the parameter ?. A
  small ? will generate a thin membership
  function, while a big ? will lead to a flat
  membership function.

33
Fuzzy Sets (cont.)

• Designing membership functions
• How do we determine the exact shape of the
  membership function for a fuzzy set? A
  membership function can be designed in three
  ways
• Interview those who are familiar with the
  underlying concepts and later adjust it based on
  a tuning strategy,
• Construct it automatically from data,
• Learn it based on feedback from the system
  performance.

34
Fuzzy Sets (cont.)

• The guidelines for membership function design
• Use parameterizable functions that can be defined
  by a small number of parameters. Parameterizable
  membership functions reduce the system design
  time and facilitate the automated tuning of the
  system.
• The parameterizable membership functions most
  commonly used in practice are the triangular and
  trapezoidal membership functions, because of
  their simplicity.
• If you want to learn the membership function
  using neural network learning techniques, choose
  a differentiable (or even continuous
  differentiable) membership function (e.g.,
  Gaussian).

35
Fuzzy Sets (cont.)

• Designing antecedent membership functions
• The membership functions of an input variables
  fuzzy sets should usually be designed in a way
  that the following two conditions are satisfied
• Unless there is a good reason, use symmetric
  membership functions. This guideline has an
  additional benefit from the viewpoint of
  stability analysis.
• Each membership function overlaps only with the
  closest neighboring membership functions
• Ai ? Aj ? ? j ? i, j1, i-1, where Ai are
  fuzzy sets.
• For any possible input data, its membership
  values in all relevant fuzzy sets should sum to 1
  (or nearly so), ?i ?Ai (x) ? 1

36
Linguistic Variables

• A linguistic variable enables its value to be
  described both qualitatively by a linguistic term
  (i.e., a symbol serving as the name of a fuzzy
  set) and quantitatively by a corresponding
  membership function, (which express the meaning
  of the fuzzy set).
• For example, if TradingQuantity is Heavy, the
  fuzzy set Heavy describes the quantity of the
  stock market trading in one day. The variable
  TradingQuantity demonstrates the linguistic
  variable.

37
Linguistic Variables (cont.)

• A linguistic variable is like a composition of a
  symbolic variable (whose value is a symbol, e.g.,
  Shape is Cylinder)) and a numeric variable (whose
  value is a number, e.g., Height 4)).
• Using the notion of the linguistic variable to
  combine these two kinds of variables into a
  uniform framework is, in fact, one of the main
  reasons that fuzzy logic has been successful in
  offering intelligent approaches in engineering
  and many other areas that deal with continuous
  problem domains.

38
Possibility Distributions

• A possibility distribution, ?, maps a given
  domain of definition into the interval 0,1.
• We can view a possibility distribution as a
  mechanism for interpreting factual statements
  involving fuzzy sets.
• Example the statement, Temperature is High,
  where High is defined as ?High T ? 0,1,
  translates into a possibility distribution, ?(T)
  ?High (T).
• For more complex statement, Temperature is High
  but not too high translates into a possibility
  distribution in terms of conjunction of the terms
  High and Not VeryHigh
• ?(T) min(?High(T),?NotVeryHigh(T))min?High(T),
  (1-?High(T))2. 

39
Possibility Distributions (cont.)

• Fuzzy logic offers an appealing alternative, such
  as assigning the fuzzy set Young to the age of
  the suspect. Thus, we obtain a distribution about
  the possibility degree of the suspects age
  (e.g., the possibility that the suspect is 19 is
  0.7, while the possibility of 21 - 28 is 1.0),
• ?Age(suspect) (x) ?Young (x),
• where ? denotes a possibility distribution of
  the suspects age, and x is a variable
  representing a persons age.
• Nec(A?X) denotes the necessity of the condition
  X is A given the possibility distribution ?X.

40
Possibility Distributions

• The possibility and necessity are two related
  measures
• 1a.Total necessity implies total possibility,
  Nec(A?X)1?Pos(A?X) 1
• 1b. No possibility implies no necessity,
• Pos(A?X) 0 ? Nec(A?X) 0
• 2a. A variable is not possible to be NOT A iff
  it is necessarily A
• 1- Pos(?A?X) 1 ? Nec(A?X) 1,
• 2b. Pos(?A?X) 1 ? 1 - Nec(A?X) 1,
• we can review 2b as follows
• 2b. 1- Pos(?A?X) 0 ? Nec(A?X) 0.

41
Possibility Distributions (cont.)

• These observations can provide insights on the
  general relationships between the two measures.
  The relationships 1a and 1b can be generalized to
  Nec (A?X) ? Pos(A?X)
• The relationships 2a and 2b can be generalized
  to
• 1- Pos(?A?X) Nec(A?X).
• Thus, one can automatically derive necessity
  measure using a possibility measure.
• In general, when we assign a fuzzy set A to a
  variable X, the assignment results in a
  possibility distribution of X, which is defined
  by As membership function ?X (x) ?A (x).

42
Possibility Distributions (cont.)

• The possibility measure for a variable X to
  satisfy the condition X is A given a
  possibility distribution ?X is defined to be
• Pos(A?X) sup xi?U (?A ? ?X ),
• where ? denotes a fuzzy intersection (i.e., a
  fuzzy conjunction) operator.
• A common choice of the fuzzy intersection
  operator for calculating the possibility measure
  is the min operator. Thus,
• Pos(A?X) supxi?U min (?A (xi), ?X (xi)).
• It is easy to derive the corresponding formula
  for the necessity measure
• Nec(A?X) infxi?U max (?A (xi), 1-?X (xi)).

43
Possibility Distributions

• Example Let the universe of discourse of a
  persons age be 10,15,20,25,30,35,40,45,50, and
• The age possibility distribution of a suspect
  (denoted J) be
• ?Age (J) 0.2/15 0.5/20 1/25 0.8 /30
• Suppose that the membership function for the
  linguistic term Young is defined as a discrete
  fuzzy set as follows
• Young 1/10 1/15 1/20 0.8 / 25 0.4 /30
  0.2 /35
• Using the equation
• Pos(?A?X) Pos(?Young ?Age (J)) sup
  xi?U (?Young ? ?Age (J)
• Pos(?Young ?Age (J)) max min (?Young ,
  ?Age (J))
• max 0.2?1, 0.5?1, 1?0.8, 0.8?0.4
• max 0.2, 0.5,0.8, 0.4
• Pos(?Young ?Age (J)) 0.8

44
Possibility Distributions (cont.)

• Example (cont.) Let the universe of discourse of
  a persons age be 10,15,20,25,30,35,40,45,50,
  and
• The age possibility distribution of a suspect
  (denoted J) be
• ?Age (J) 0.2/15 0.5/20 1/25 0.8 /30
• Suppose that the membership function for the
  linguistic term Young is defined as a discrete
  fuzzy set as follows
• Young 1/10 1/15 1/20 0.8 / 25 0.4
  /30 0.2 /35
• To calculate the necessity measure, we first
  calculate the complement of the possibility
  distribution of a suspect Js age
• 1-?Age(J) 1/100.8/150.5/200/250.2/301/35
  1/401/451/50
• The necessity measure is obtained by
• Nec(A?X) infxi?U ?A(xi) ? 1-?X (xi)
• Nec(?Young ?Age (J)) infxi?Umax(?Young, 1-
  ?Age (J)
• Nec(?Young?Age(J))min1?1,1?0.8,1?0.5,0.8?0,0.4
  ?0.2,0.2?1,0?1,0?1,0?1
• min 1, 1, 1, 0.8, 0.4, 1, 1,1, 1
  0.4.
• Therefore, the possibility that suspect J is
  young is 0.8, while the necessity that he/she is
  young is 0.4.

45
Fuzzy If-Then Rules

• There are two different kinds of fuzzy rules
  Fuzzy mapping rules and Fuzzy implication rules.
• A fuzzy mapping rule describes an association
  therefore, its fuzzy relation is constructed from
  the Cartesian product of its antecedent fuzzy
  condition and its consequent fuzzy condition.
• A fuzzy implication rule, however, describes a
  generalized logic implication therefore, its
  fuzzy relation needs to be constructed from the
  semantics of a generalization to implication in
  multi-valued logic.
• The difference between the semantics of fuzzy
  mapping rules and fuzzy implication rules can be
  seen from the difference in their inference
  behavior. Even though these two types of rules
  behave the same when their antecedents are
  satisfied, they behave differently when their
  antecedents are not satisfied.
• Example
• Implication rule (logic representation), Mapping
  rule (procedural representation)
• Given x ? 1,3 ? y ? 7,8, Statement
  If x ? 1,3 Then y ? 7,8
• Input x5 Variable value x5
• Infer y is unkown (y ? 0,10 Execution
  result no action

46
Fuzzy Mapping Rules

• The needs to approximate a function of interest
  is often due to one or more of the following
  reasons
• The mathematical structure of the function is not
  precisely known.
• The function is so complex that finding its
  precise mathematical form is either impossible or
  practically infeasible due to its high cost.
• Even if finding the function is not impractical,
  implementing the function in its precise
  mathematical form in a product or service may be
  too costly. This is particularly important for
  low cost high volume products (e.g., automobiles,
  cameras, and many other consumer products).

47
Fuzzy Mapping Rules

• Fuzzy rule-based function approximation is a
  partition-based technique.
• The partition-based approximation techniques
  approximate a function by partitioning the input
  space of the function and approximate the
  function in each partitioned region separately
  (e.g., piecewise linear approximation).
• Because each fuzzy rule approximates a small
  segment of the function, the entire function is
  approximated by a set of fuzzy mapping rules. We
  refer to such a collection of fuzzy mapping rules
  as fuzzy rule-based models or simply fuzzy models
  (describing a mapping (i.e., function) from a set
  of input variables to a set of output variables.)
  
• Example a fuzzy model of the stock market can be
  used to predict future changes of the IMKB
  average. A fuzzy control model of a petrochemical
  process can be used to predict the future state
  of the process.

48
Fuzzy Mapping Rules

• A fuzzy model can be defined as a model that is
  obtained by fusing multiple local models that are
  associated with fuzzy subspaces of the given
  input space.
• The result of fusing multiple local models is
  usually a fuzzy conclusion, which is converted to
  a crisp final output through a defuzzification
  process.
• The main difference between fuzzy and nonfuzzy
  rules for function approximation lies in their
  interpolative reasoning capability, which
  allows the output of multiple fuzzy rules to be
  fused for a given input.

49
Fuzzy Mapping Rules

• The four major concepts in fuzzy rule-based
  models thus are as follows
• 1.       Fuzzy partition,
• 2.       Mapping of fuzzy subregion to local
  models,
• 3.       Fusion of multiple local models,
• 4.       Defuzzification.

50
Fuzzy partition

• A fuzzy partition of a space is a collection of
  fuzzy subspaces whose boundaries partially
  overlap and whose union is the entire space.
• Formally, a fuzzy partition of a space as a
  collection of fuzzy subspace Ai of S that
  satisfies the following condition
• ? ?Ai(x) 1, ?x ? S.
• That is, for any element of the space, its
  membership degree in all subspaces always adds up
  to 1.
• We call a collection of fuzzy subspaces Ai of S a
  weak fuzzy partition of S if and only if it
  satisfies the following condition
• 0lt ? ?Ai(x) ? 1, ?x ? S.
• The greater than 0 condition requires each
  element in the space S to be covered by at least
  one fuzzy subspace in the partition. The sum to
  1 condition of a fuzzy partition can be relaxed
  to the sum to less or equal to 1 condition
  because the interpolative reasoning of fuzzy
  models includes a normalization step.
• Research Note It has been shown that ? ?Ai(x)
  1 is a desirable property in a framework for
  analyzing the stability of fuzzy logic
  controllers.

51
Mapping a Fuzzy Subspace to a Local Model

• A local model for a subspace of the entire input
  space describes the systems input-output mapping
  relationship in the small subspace. In contrast,
  a global model for an input space describes the
  systems input-output relationship for the entire
  input space.
• Because the scope of the local model is smaller
  than that of a global model, it is usually easier
  to develop a local model.
• In particular, a nonlinear global model (i.e.,
  whose input-output mapping function is not
  linear) can often be approximated by a set of
  linear local models. This can be understood by
  remembering the well-known approximation
  technique called piecewise linear approximation,
  which approximates an arbitrary nonlinear
  function using segments of lines.
• The following figure shows such an approximation
  technique, where dotted line indicates the
  function being approximated. 

52
Mapping a Fuzzy Subspace to a Local Model

• Piecewise linear approximation has two major
  components 
• 1.       Partitioning the input space to crisp
  regions
• 2.       Mapping each partitioned region to a
  linear local model.
• The main difference between fuzzy modeling and
  piecewise linear approximation is that the
  transition from one local subregion to a
  neighboring one is gradual rather than abrupt.
• Generally, the mapping from a fuzzy subspace to a
  local model is represented as a fuzzy if-then
  rule in the form of
• If ?x is in FSi Then yj LMi (x) 
• where ?x and yj denote the vector of input
  variables and output variable, respectively, FSi
  and LMi denote ith fuzzy subspace and the
  corresponding local model, respectively.

53
Mapping a Fuzzy Subspace to a Local Model

• The local model can be of four different types
• 1.  Crisp constant This type of local model is
  simply a crisp (nonvisual) constant. For example
  
• If xi is Small Then y 4.5
• 2. Fuzzy constant A local model that is a fuzzy
  constant (e.g., Small) belong to this type. For
  example
• If xi is Small Then y is Medium
• 3. Linear Model this describes the output as a
  linear function of the input variables, such as
  
• If x1 is Small And x2 is Large Then y 2x1
  5x2 3.

54
Fusion of local models through interpolative
reasoning

• Fuzzy models use interpolative reasoning to fuse
  multiple local models into a global model.
• The basic idea behind interpolative reasoning is
  analogous to drawing a conclusion from a panel of
  experts, each of whom is specialized in a subarea
  of the entire problem.
• Each experts opinion is associated with a
  weight, which reflects the degree to which the
  current situation is in the experts specialized
  area.
• These weighted opinions are combined to form an
  overall opinion.
• In this analogy, an expert corresponds to a fuzzy
  if-then rule, the specialized subarea of the
  expert corresponds to the fuzzy subspace
  associated with the if-part of the rule.
• The weight of an experts opinion is determined
  by the degree to which the current situation
  belongs to the subspace.

55
Defuzzification

• We may interpret a possibility distribution
  either through linguistic approximation, or
  through defuzzification.
• The former gives a qualitative interpretation,
  while the latter gives a quantitative summary and
  is more commonly used in fuzzy logic
  applications, i.e., industrial applications.
• Given a possibility distribution of a fuzzy
  models output, defuzzification amounts to
  selecting a single representative value that
  captures the essential meaning of the given
  distribution. There are three common
  defuzzification techniques mean of maximum,
  center of area, and height.
• Mean of Maximum (MOM) This calculates the
  average of those output values that have the
  highest possibility degrees. Suppose y is A is
  a fuzzy conclusion to be fuzzified. We can
  express the MOM defuzzification method using the
  following formula MOM (A) ?y?P y / P
• Where P is the set of output values y with
  highest possibility degree in A. If P is an
  interval, the result of MOM defuzzification is
  obviously the midpoint in that interval. This
  technique does not take into account the overall
  shape of the possibility distribution.

56
Defuzzification

• Center of Area (COA) This method (also referred
  to as the center-of-gravity, or centroid method)
  is the most popular defuzzification technique.
  Unlike MOM, the COA method takes into account the
  entire possibility distribution in calculating
  its representative point. This method is similar
  to the formula for calculating the center of
  gravity in physics, if we view ?A(x) as the
  density of mass at x. If x is discrete, the
  fuzzification result of A is
• COA(A) ?x ?A(x) x / ?x ?A(x).
•   The main disadvantage of the COA method is its
  high computational cost. However, the calculation
  can be simplified for some fuzzy models.

57
Defuzzification

• The Height Method This method can be viewed as a
  two step procedure. First we convert the
  consequent membership function Ci into crisp
  consequent y ci where ci is the center of
  gravity of Ci. The centroid defuzzification is
  then applied to the rules with crisp consequents
  with the following formula
• y ?Mi1 wici / ?Mi1 wi
• where wi is the degree to which ith rule matches
  the input data. This method reduces the
  computation cost and facilitates the application
  of neural networks learning to fuzzy systems
  hence, many well-known neuro-fuzzy models use
  this type of defuzzification method. The main
  disadvantage of this method is that it is not
  well justified and is often considered an
  approximation to the centroid defuzzification.

58
A Theoretical Foundation of Fuzzy Mapping Rules

• A mathematical representation of fuzzy mapping
  rules A fuzzy mapping rule imposes an elastic
  constraint on possible associations between input
  and output variables.
• It is elastic because a fuzzy tule can describe
  input-output associations that are somewhat
  possible (i.e., the gray area between totally
  possible and totally impossible).
• The degree of possibility of an input-output
  association imposed by a rule R can be expressed
  as a possibility distribution, denoted by ?R.
• Since a fuzzy relation is a general way for
  describing a possibility distribution, it is
  natural to use it to represent the possibility
  distribution imposed by a fuzzy rule.
• How do you construct the fuzzy relation that
  represent fuzzy mapping rules?
• The answer is to use the concept of Cartesian
  product.
• A fuzzy mapping rule is represented
  mathematically as fuzzy relations formed by the
  Cartesian product of the variables referred to in
  the rules if-part and then-part. For example,
  the mapping rule is IF x is A, THEN y is
  B, which is mathematically represented as a fuzzy
  relation R defined as
• ?R(x,y)?A?B(x,y)min?A(x), ?B(y).

59
A Theoretical Foundation of Fuzzy Mapping Rules

• Example Let us consider the following fuzzy
  mapping rule from X to Y where X
  2,3,4,5,6,7,8,9 and Y 1,2,3,4,5,6
• If x is Medium, Then y is Small
• where Medium and Small are fuzzy subsets of X and
  Y characterized by the following membership
  functions
• Medium ? 0.1/2 0.3/3 0.7/4 1/5 1/6
  0.7/7 0.5/8 0.2/9
• Small ? 1/1 ½ 0.9/3 0.6/4 0.3/5 0.1/6
• The fuzzy relation R representing the rule is the
  Cartesian product of Medium and Small. If we use
  the min operator to construct the Cartesian
  product, we have ?R(x,y) min?Medium(x),
  ?Small(y).
• The resulting fuzzy relation representing the
  rule is

60
A Theoretical Foundation of Fuzzy Mapping Rules

• The theoretical foundation of fuzzy mapping rules
  is a fuzzy graph and a compositional rule of
  inference. A fuzzy graph can be conveniently
  described by fuzzy rules in the form of
• If x is A Then y is B
• Such a statement (or rule) generalizes the
  dependency relationship between variables in a
  lookup table such as
• If x is 5 Then y is 10
• If x is 10 Then y is 14
• A set of such dependencies form a functional
  mapping from x to y. Generalizing point-to-point
  mappings to a mapping from fuzzy sets to fuzzy
  sets introduces two benefits. 
• We can reduce the total number of point-to-point
  rules required for approximating a function
• Using words in fuzzy rules makes it easier to
  capture, understand, and communicate the
  underlying human knowledge. 
• Let f be a fuzzy graph described by a set of
  fuzzy mapping rules in the form of
• If x is Aj Then y is Bj. 

61
A Theoretical Foundation of Fuzzy Mapping Rules

• The fuzzy graph can be expressed mathematically
  as
• f ?j A j ? Bj
• where A and B are two fuzzy subsets of X and Y
  respectively. A fuzzy graph f from X to Y is
  union of Cartesian products involving linguistic
  input-output associations (iei., pairs if x is
  Ai and y is Bi). The resulting fuzzy graph is
  basically a fuzzy relation. 
• The inference (i.e., interpolative reasoning) of
  such a fuzzy rule-based model is based on the
  compositional rule of inference. The net effect
  is a possibility distribution over the domain of
  definition of the output variable.
• In particular, B A o f
• where f represents the fuzzy graph of a given
  fuzzy model, A is an input which can be fuzzy or
  crisp, and B is the inferred output value before
  defuzzification.
• Using the definion of a compositional rule of
  inference, we express this as
• A o f ProjY (cyl-ext(A) ? f) ProjY
  cyl-ext(A) ? (?i Ai?Bi)
• ?x ?X cyl-ext(A) ? (?i Ai?Bi)
• where X and Y are the universe of discourse of x
  and y resepectively, and cyl-ext(A) is the
  cylindirical extension of A to X ?Y.

62
A Theoretical Foundation of Fuzzy Mapping Rules

• Example Consider the following rule (again)
• If x is Medium Then y is Small
• Input data is X is Small, where Small for x is
  defined as
• Small ? 1/2 0.9/3 0.6/4 0.3/5 0.1/6
• To find out the possible values of y, we compose
  the possible values of x with the fuzzy relation
  T using the sup-min composition

0.6 0.6 0.6 0.6 0.3 0.1, y0.6/10.6/20.6/
30.6/40.3/50.1/6 as the result of the
inference.
In this example, we consider only one rule.
However, a fuzzy model for function approximation
is usually formed by a set of fuzzy mapping
rules. In such a case, the fuzzy relation of the
entire model (denoted FM) is constructed by
forming the union of fuzzy relations of
individual rules ?FM ?R1 ? ?R2 ? ??Rn
63
Types of Fuzzy Rule-Based Models

• There are three types of fuzzy rule-based models
  for function approximation
• 1.       The Mamdani model
• 2.       The Takagi-Sugeno-Kang (TSK) model,
• 3.       Koskos additive model (SAM) 
• The inference scheme of SAM is similar to that of
  TSK model. Both of them use an inference
  analogous to the weighted sum to aggregate the
  conclusion of multiple rules into a final
  conclusion. Therefore, we refer to these rule
  models as additive rule models.
• One of the main advantages of the TSK model is
  that it can approximate a function using fewer
  rules.
• In contrast, the Mamdani model combines inference
  results of rules using superimposition, not
  addition. Hence nonadditive rule model.
• The Mamdani and SAM use rules whose consequent
  part is a fuzzy set (uses a fuzzy constant as its
  rules local model).
• The TSK model uses a rule whose then part is a
  linear model (uses a linear local model). The
  fundamental difference between the Mamdani and
  SAM lies in the choice of composition,
  conjunction, and disjunction operators in their
  reasoning (inference mechanism).

64
Types of Fuzzy Rule-Based Models

•  The Mamdani Model
• One of the most widely used fuzzy models in
  practice is the Mamdani model, which consists of
  the following linguistic rules that describe a
  mapping from U1 ? U2 ? ? Ur to W.
• Ri If x1 is Ai1 and and x r is Air Then y is
  Ci
•   where xj is (j 1,2,..r) are the input
  variables, y is the output variable, and Aij and
  Ci are fuzzy sets for xj and y respectively.
• Given inputs of the form x1 is A1 , x2 is A2
  x r is Ar where A1 ,A2 Ar are fuzzy subsets
  of U1, U2, ,Ur (e.g., fuzzy numbers), the
  contribution of rule Ri to a Mamdani models
  output is a fuzzy set whose membership function
  is computed by
• ?Ci (y) (?i1 ? ?i2 ? ? ?ir ) ? ?Ci (y)
• where ?Ci (y) is the matching degree of rule
  Ri, and where ?ij is the matching degree between
  xj and Ris condition about xj.
• ?ij sup xj (?Aj (xj) ? ?Aij (xj) )

65
Types of Fuzzy Rule-Based Models

•   and ? denotes the min operator. This is the
  clipping inference method.
• The final output of the model is the aggregation
  of outputs from all rules using the max operator.
• ?C (y) max (?C1(y), ?C2(y),..., ?Cm(y))
• Notice that the output C is a fuzzy set. This
  output can be defuzzified into a crisp output
  using one of the defuzzification techniques.
• The Mamdani model can be derived from the
  following operators
• Sup-min composition
• Min for Cartesian product
• Min for conjunctive conditions in rules
• Max for aggregating multiple rules

66
Fuzzy If-Then Rules