introduction to fuzzy systems

8/7/2019 Introduction to Fuzzy Systems

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Total Slides 37 1

INTRODUCTION TO

FUZZY SYSTEMS

By

Dr. M. Tahir Khaleeq


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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 tallp 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}.


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Fuzzy Sets (continue)

In fuzzy sets, membership is based on a degreebetween 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

degree.

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


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The notion of the fuzzy set was introduced by Lotfi Zadehin 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.)


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Difference between an ordinary (crisp set) and a fuzzy setis shown in the figure

cool medium

Fuzzy setmedium

Crisp set

Crisp sets use clear cut

on the boundaries.

Fuzzy sets use grades.

Ex: values 14.99 and 15.01.

- Belong to the fuzzy set

medium.- Associated with different

crisp sets, cool and

medium.


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Example: Three fuzzy sets , values of a variable height are:short, medium, tall.

Medium TallShort

Height (cm)

25017030

0.2

0.7

1 The value 170cm belongs

to fuzzy set medium to a

degree of 0.2 and at the

set tall to a degree of 0.7.


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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.


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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.


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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

0.8

0.6

0.4

0.2

0

35 36 37 38 39 40

hot

Temp (C)

U

Q


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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

6.4 function (bell function


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Single-valued (Singleton)

U = b. B is a scalar value

UbTriangular Function

The triangular functions are uniformly distributed over

the universe U.


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Membership function (for a


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Trapezoidal Function

Member ship function (for a e b < c e d )

m x a b c,dx a

b a

d x

d ctrap( ; , , ) max{min{ , },!

1, 0 },

- Fuzzy membership foru between 36 and 38 C as uU

- 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.


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S-Function (sigmoid function)

Member function for a


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Fuzzy Logic

Just as fuzzy sets are an extension to sets, fuzzy logic isan 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 butnot in both.

Fuzzy logic allows for values between 0 and 1, shades of

gray and may be partial membership in a set.


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Fuzzy Logic (continue)

When the approximate reasoning of fuzzy logic is usedwith 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.


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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 fuzzypropositions:

x and y are fuzzy variables defined overuniverse

of discourse U and V respectively; and

A and B are fuzzy sets defined by their fuzzy

membership functions

A : U p [0,1], B : V p [0,1]


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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 outputExample: IF day is hot THEN drink lots of water

Fuzzy Rules (continue)


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The rule would recommend to drink 6 - 10 glasses ofwater.

The implication of the rule will be the minimum of theintersection of the 0.7 membership line with the lotsof implication weight function.

Fuzzy Rules (continue)

m x y m x m yA B A Bp !( , ) min[ ( ), ( )]


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Fuzzification

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 A1(x1) and A2(x2) to which input data x1and x2 belong to the fuzzy sets A1 and A2 in the

antecedent part of a fuzzy rule.


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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

Fuzzification (continue)


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RULE EVALUATION

Rule evaluation takes place after the fuzzificationprocedure.

It deals with single values of membership degreesmA(x) and mA(y) and produces output membership

function. There are two major methods which can be applied:

1. Minimum inference:

2. Product inference:m x y m x m yp !( , ) min[ ( ), ( )]

m x y m x m yA B A Bp !( , ) ( ) ( )


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DEFUZZIFICATION

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 widelyused

1. The Center-of-Gravity method (COG)

2. The Mean-of-Maxima method (MOM)


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The Center-of-Gravity Method

This method finds the geometical centre y in theuniverse V of an output variable y, which is center

balance the inferred membership function B as a fuzzy

value fory.

The Mean-of-Maxima Method

This method finds the value y for output variable ywhich has maximum membership degree according to

the fuzzy membership function B.

If values have maximum values then find mean of them.

y =7

v . mB(v)7 mB(v)


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0.7

1 2 3 40

1

1.51.9

y'(MOM)

y'(COG)

y'(COG) =(0 0) + (1 1) + (1 2) + (0.7 3)

1 + 1 + 0.7} 1.9

y'(MOM) = = 1.51 + 2

2

y

Example


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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


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Fuzzy Rule Base Learning Fuzzy Rules

Fuzzy Inference

MachineData Base (Fuzzy)

Fuzzification

DefuzzyficationMembership

Function

User Interface

Fuzzy data/Exact data

Fuzzy queries/Exact queries

Block diagram of a fuzzy expert system


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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 p consequent part

Crispp Crisp (CF)

Crispp Fuzzy (CF)Fuzzyp Crisp (CF)

Fuzzyp Fuzzy (CF)


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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 realizationof 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.


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A measure of the degree of matching is calculated for

every case.

Ex: f uzzy 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.


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5. User Interface

The interface unit communicates with the user or the

environment, or both for collecting input data andreporting output results.

Fuzzy queries might be possible when the user inputinformation 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 fuzzyinference process.

If learning takes place before the inference machinestarts the learning module uses AI machine learning

methods or neural networks.


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If the learning takes place during the fuzzy inference

process the fuzzy neural networks can be used forlearning.

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 mayuse fuzzy terms for explanation as well as

exact terms and values.


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Fuzzy System Design

The following are the main steps of the fuzzy systemdesign:

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 morecomprehensible.

2. Defining the Input and Output Variables:

Define the input and output variables, their fuzzy values

and their membership functions.


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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 fuzzyvariables,

changing the membership functions and fuzzy rules ifnecessary,

tuning the fuzzy system,

validation of the results.


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Methods for Obtaining Fuzzy Rules

The main problem in building fuzzy expert systems is thatof articulating the fuzzy rules and membership functionsfor 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 commonunderstanding.

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 setsor fuzzy logic and the knowledge engineer is unfamiliar

with the domain area.


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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 dataand to learn membership functions if they are not given

in advance.


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END


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All lectures are available on

www.geocities.com/mtkhaleeq/AI.htm

introduction to fuzzy systems

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