Introduction to Fuzzy Systems by mtkhaleeq

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Dr. M. Tahir Khaleeq

        Total Slides 37   1
                    Fuzzy Sets
• Fuzzy logic is an approach to uncertainty that combines
  real values [0,1] and logic operations
• Fuzzy logic is based on the ideas of fuzzy set theory and
  fuzzy set membership.
   Ex: He is very tall  how does this differ from tall?
• In normal sets, membership is binary.
   – An item is either in the set or not in the set.
     Ex: U = { 1,2,3,4,5,6,7,8,9}
          A = {1,3,5,7,9}
          B = {2,4,6,8}.
Fuzzy Sets (continue)
• In fuzzy sets, membership is based on a degree
  between 0 and 1
  – 0 means that the object is not a member of the set
  – 1 means that the object belongs entirely to the set
  – If degree is between 0 and 1, then this degree is the
    degree to which the item is thought to be in the set.
  – Each value of the function is called a membership
     Ex: Jun           is 43, 1.0 in set Hot
            August is 40, 0.7 in set Hot
            September is 35, 0.2 in set Hot              3
Fuzzy Sets (continue)
• The notion of the fuzzy set was introduced by Lotfi Zadeh
  in 1965.
• Fuzzy sets have imprecise boundaries.
• Transition between fuzzy sets is gradual.
• Fuzzy v Crisp:
         Fuzzy (approximate)              Crisp (precise)
    - Elements can belong to         Elements belong to one
        two sets at same time.        set or the other only.
    - cold, warm, hot                 20, 30, 40 (°C)
    - slow, normal, fast               50, 70, 100 (km/h)
    - dry, normal, humid               10, 25, 75 (% R.H.)4
Fuzzy Sets (continue)
• Difference between an ordinary (crisp set) and a fuzzy set
  is shown in the figure
                                • Crisp sets use clear cut
                                  on the boundaries.
      cool  medium
                      Crisp set • Fuzzy sets use grades.

                     Fuzzy set   Ex: values 14.99 and 15.01.
                                   - Belong to the fuzzy set
                                   - Associated with different
                                     crisp sets, cool and
Fuzzy Sets (continue)
Example: Three fuzzy sets , values of a variable height are:
         short, medium, tall.

        Short   Medium          Tall         – The value 170cm belongs
                                              to fuzzy set medium to a
                                              degree of 0.2 and at the

 0.2                                          set tall to a degree of 0.7.

       30                 170          250
                 Height (cm)

    Conceptualizing in Fuzzy terms
• The representation of a problem in fuzzy terms is called
  conceptualization in fuzzy terms.

• Linguistic terms are used in the process of identification
  and specification of a problem and construction of rules.
  Ex: higher, lower, very strong, slowly, much dependent,
        less dependent, good, bad etc.

• Linguistic variable is a variable which takes fuzzy values
  and has a linguistic meaning.
  Ex: Linguistic variable:      Velocity.
       Value:                   low, moderate, or high. 7
Conceptualizing in Fuzzy terms (continue)

 • Linguistic values are also called fuzzy labels, fuzzy
   predicates, or fuzzy concepts.
 • Linguistic values have semantic meaning and can be
   expressed numerically by their membership functions.
 • Linguistic variables can be
    – Quantitative
       Ex: temperature:            low, high
              time:                early, late
    – Qualitative
       Ex: truth, certainty, belief
 • The process of representing a linguistic variable into a set
   of linguistic values is called fuzzy quantization.     8
     Crisp Membership Function
Rule: IF temp >37 THEN day = hot.
– Precise value of set { hot } at 37 °C
– Each temp U belongs to only one set.
      1.0
             35   36     37 38           39   40 U
                       Temp (°C)                     9
     Fuzzy Membership Functions
• Following are the most useful membership functions in
  fuzzy expert systems design:

 1. Single-valued (Singleton)
 2. Triangular
 3. Trapezoidal
 4. S-function (sigmoid function)
 5. Z-function
 6.  function (bell function

Single-valued (Singleton)

• U = b. B is a scalar value
Triangular Function                       b           U
• The triangular functions are uniformly distributed over
  the universe U.

• Membership function (for a<b<c):
                                       x -a c -x
        mtri ( x; a , b, c) = max{min{      ,     }, 0 },
                                       b - a c -b
 Example:        If X= 32.5 then
                                        32.5 - 25       35 - 32.5
  µtri (32.5; 25,30,35) = max { min {               ,             },0}
                                         30 - 25         35 - 30
                      = max { min { 1.5, 0.5},0}
                      = max {0.5,0} = 0.5

Trapezoidal Function
• Member ship function (for a  b < c  d )
                                          x -a     d -x
      m trap ( x; a , b, c,d ) = max{min{      ,1,      }, 0 },
                                          b -a     d -c

- Fuzzy membership for u between 36 and 38 °C as uU
- About half of persons would call the day “hot” when the
  temp is 37 °C. m(T) represents the fraction of people,
  who would assign the term “hot” to the day.          13
S-Function (sigmoid function)
• Member function for a<b:
           msigm ( x; a , b) =
                                 1  e - a ( x -b )

- Fuzzy membership for u between 36 and 38 °C as uU

                   Fuzzy Logic
• Just as fuzzy sets are an extension to sets, fuzzy logic is
  an extension to classical logic.
• Fuzzy logic is a multi-valued logic while the classical
  logic is a binary logic.
• Classical logic holds that every thing can be expressed in
  binary terms: 0 or 1, black or white, yes or ne, in terms of
  Boolean algebra, every thing is in one set or another but
  not in both.
• Fuzzy logic allows for values between 0 and 1, shades of
  gray and may be partial membership in a set.       15
Fuzzy Logic (continue)
• When the approximate reasoning of fuzzy logic is used
  with an expert system, logical inferences can be drawn
  from imprecise relationships.
• EX: To optimize automatically the wash cycle of a
  washing machine by sensing
   – the load size
   – fabric mix, and
   – quantity of detergent.
• The most distinguishing property of fuzzy logic is that
  deal with fuzzy propositions, which contain fuzzy
  variables and fuzzy values.
                   Fuzzy Rules
• Several types of fuzzy rules have been used for fuzzy
  knowledge engineering.
• IF x is A THEN y is B
   – Where (x is A) and (y is B) are two fuzzy
      • x and y are fuzzy variables defined over universe
        of discourse U and V respectively; and
      • A and B are fuzzy sets defined by their fuzzy
        membership functions
         µA : U  [0,1],      µB : V  [0,1]
Fuzzy Rules (continue)
 • May contain AND, OR, NOT and other operators
 • The conclusion is computed by applying these fuzzy
   operators to the fuzzified inputs
 • Implication operation will be applied to rule output
 Example:       IF day is hot THEN drink lots of water

Fuzzy Rules (continue)
• The rule would recommend to drink 6 - 10 glasses of
• The implication of the rule will be the minimum of the
  intersection of the 0.7 membership line with the “lots
  of” implication weight function.
        mA B ( x, y) = min[mA ( x), mB ( y)]

• When the input data are crisp then the fuzzification is
  applied over fuzzy rules of the type
  IF x1 is A1 AND x2 is A2 THEN y is B.

• Fuzzification is the process of finding the membership
  degrees µA1(x1’) and µA2(x2’) to which input data x1’
  and x2’ belong to the fuzzy sets A1 and A2 in the
  antecedent part of a fuzzy rule.

Fuzzification (continue)
– Crisp input x is input to fuzzy membership function m(x)
  Example: Temperature = 37.4 degC
– Result is fuzzy degree of membership
  Example: Membership in “hot” is 0.7

• Rule evaluation takes place after the fuzzification
• It deals with single values of membership degrees
  mA(x) and mA(y) and produces output membership
• There are two major methods which can be applied:
  1. Minimum inference:
        mA B ( x, y) = min[mA ( x), mB ( y)]
   2. Product inference:

             mA B ( x, y) = mA ( x)mB ( y)
                                                       22
• If the output is crisp then the defuzzification is applied
  over fuzzy rules.
• Defuzzification is the process of calculating a single-
  output numerical value for a fuzzy output variable on
  the basis of the inferred resulting membership function
  for this variable.

• Following two methods are widely used
  1. The Center-of-Gravity method (COG)
  2. The Mean-of-Maxima method (MOM)
The Center-of-Gravity Method
– This method finds the geometical centre y’ in the
  universe V of an output variable y, which is center
  balance the inferred membership function B as a fuzzy
  value for y.
                         v . mB(v)
                           mB(v)
The Mean-of-Maxima Method
– This method finds the value y’ for output variable y
  which has maximum membership degree according to
  the fuzzy membership function B.
– If values have maximum values then find mean of them.



          0           1           2    3        4
                      (0 × 0) + (1 × 1) + (1 × 2) + (0.7 × 3)
          y'(COG) =                                            1.9
                                   1 + 1 + 0.7
          y'(MOM) =       = 1.5

           Fuzzy Expert System
• A fuzzy expert system is like an ordinary expert system
  but methods of fuzzy logic are applied.
• Fuzzy expert systems use:
   – Fuzzy data (fuzzy input and output variables)
   – Fuzzy rules
   – Fuzzy inference
   – Other components of the ordinary expert system

Block diagram of a fuzzy expert system

        Fuzzy Rule Base      Learning Fuzzy Rules

        Fuzzy Inference       Data Base (Fuzzy)


Membership                      Defuzzyfication

         User Interface
                  Fuzzy data/Exact data
                  Fuzzy queries/Exact queries

1. Fuzzy Rule-Base
– The fuzzy rules and the membership functions make up
  the system knowledge base.
– Some systems use production rules extended with fuzzy
  variables and confidence factors.
– Different types of production rules can be used:
           antecedent part  consequent part
                   Crisp  Crisp (CF)
                   Crisp  Fuzzy (CF)
                   Fuzzy  Crisp (CF)
                   Fuzzy  Fuzzy (CF)
2. Data-Base
– Data can be exact or fuzzy data with certainty factors.
  Ex: economic situation good CF = 0.95
3. Fuzzy Inference Machine
– A fuzzy inference machine is built on the theoretical
  basis of fuzzy inference methods
– A fuzzy inference machine activates all the satisfied
  rules at every cycle.
– A characteristic of fuzzy expert system is the realization
  of partial match between exact or fuzzy facts.
– A rule is fired only if the matching degree of the left
  hand side of the rule is greater than a predefined
  threshold.                                              29
– A measure of the degree of matching is calculated for
  every case.
  Ex:       fuzzy fact --- fuzzy condition
            Crisp fact --- fuzzy condition
            fuzzy fact --- exact condition
            crisp fact --- exact condition

4. Fuzzification and Defuzzification
– These may be used according to the type of inference
  machine implemented in the fuzzy expert system.

5. User Interface
– The interface unit communicates with the user or the
  environment, or both for collecting input data and
  reporting output results.
– Fuzzy queries might be possible when the user input
  information in fuzzy terms.
  Ex: high temperature, severe headache etc.
6. Learning Fuzzy Rules
– This is optional module.
– Learning can take place either before the inference
  machine starts the reasoning process, or during the fuzzy
  inference process.
– If learning takes place before the inference machine
  starts the learning module uses AI machine learning
  methods or neural networks.
– If the learning takes place during the fuzzy inference
  process the fuzzy neural networks can be used for

7. Explanation
– The explanation module explains the way the expert
  system is functioning during the inference process or
  explains how the final solution has been reached.
– The system may use fuzzy terms for explanation as well as
  exact terms and values.

            Fuzzy System Design
• The following are the main steps of the fuzzy system
1. Identification the Problem
– Identify the problem and choosing the type of fuzzy
  system, which best suits the problem requirements.
– A modular system can be designed consisting of several
  modules linked together.
– The modular approach simply the design of the whole
  system, reduce the complexity and make the system more
2. Defining the Input and Output Variables:
– Define the input and output variables, their fuzzy values
  and their membership functions.
3. Articulating the Set of Fuzzy Rules

4. Choosing the fuzzy inference method, fuzzification and
   defuzzification methods if necessary.

5. Experiment and Validate the System
– Experimenting with the fuzzy system prototype,
– drawing the goal function between input and output fuzzy
– changing the membership functions and fuzzy rules if
– tuning the fuzzy system,
– validation of the results.                          34
Methods for Obtaining Fuzzy Rules
• The main problem in building fuzzy expert systems is that
   of articulating the fuzzy rules and membership functions
   for the fuzzy terms.
• Some methods for obtaining fuzzy rules are as follows:
1. Interview an Expert
– Sometimes, communication between expert and
   interviewer can be difficult because of a lake of common
– The shape of membership functions, the number of labels,
   and so forth should be defined by the expert. But,
   sometimes the human expert is unfamiliar with fuzzy sets
   or fuzzy logic and the knowledge engineer is unfamiliar
   with the domain area.                                35
2. Imagine the Behavior of the System
– The system designer has to be particularly experienced
  with the system in order to imagine physical behavior of
  the system and think about physical meaning in natural
  and technical languages.

3. Using Learning Methods
– Use the methods of machine-learning, neural networks,
  and genetic algorithms to learns fuzzy rules from data
  and to learn membership functions if they are not given
  in advance.

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