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FUZZY SYSTEMS

A Brief Introduction

FUZZY $($EXPERT$)$ SYSTEMS

$$ Fuzzy logic

$$ Fuzzy control

$$ Fuzzy if $-$ then rules

$$ Examples

$$ Reasoning methodsFUZZY $($EXPERT$)$ SYSTEMS

$$ Fuzzy logic

$$ Fuzzy control

$$ Fuzzy if $-$ then rules

$$ Examples

$$ Reasoning methods

FUZZY LOGIC

$$ Examples

expensive cars, intelligent men, pile of Sand,

high temperature, nice weather

there are no sharp boundaries in concepts

$$ Basic Idea

Generalize

true$/$false$,$ black$/$white$,$ all$/$none dichotomy

traditional logic $/$ set concept : $\{0,1\}$

To grades or degrees $($of truth or validity$)$

fuzzy logic $/$ set concept : $[0,1]$

FUZZY LOGIC

$$ Advantages

$$ A more general theory:

traditional theory becomes limiting case

$$ More realistic modeling of real world:

allows varying degrees of validity

$$ Easier problem solving:

start from a general $($qualitative$)$ description,

arrive at a specific $($quantitative$)$ conclusion

FUZZY CONTROL

$$ Most successful application of fuzzy logic

$$ Basis of fuzzy expert systems

$$ General Method

$$ Build a set of fuzzy rules

$$ Fuzzify input $/$ output conditions

$$ Calculate membership grades of input

$$ Fire applicable rules

$$ Defuzzify to get crisp answer

EXPERT SYSTEMS

$$ Rules

$$ derived from observations

$$ fuzzy if $-$ then rules are unqualified

$$ Reasoning method

$$ inference rules, modus ponens, etc.

$$ making sequence of deductions from

input facts

$$ Shell $=$ Inference Engine $+$ User Interface

E XAMPLE: FUZZY If $-$ then RULES

$1.$ If Age is Young and Education is Low, then Income

is Low

$2.$ If Age is Young and Education is High, then Income

is Medium

$3.$ If Age is Middle and Education is Low, then income

is Low

$4.$ If Age is Middle and Education is High, then income

is High

$5.$ If Age is Old and Education is Low, then income

is Medium

$6.$ If Age is Old and Education is High, then income

is High

RULE TABLE

a p e Low High

Young L M

Middle L H

Old M H

QUERY: Example

Q: What is the mean income of a $32-$year$-$old

person with a $2-$year college degree?

Solution:

step $1-$ Calculate membership grades

$32$ year old: $0\mathrm{.}6$ in Young and $0\mathrm{.}2$ in Middle

$2-$yr degree: $0\mathrm{.}5$ in Low and $0\mathrm{.}5$ in High

FUZZY SETS FOR AGE$0$

$0\mathrm{.}25$

$0\mathrm{.}5$

$0\mathrm{.}75$

$1$

$202530354045505560$

Young

Mid

Old

FUZZY AGE CALCULATIONS

$1$ if x $<=30$

Young$($x$)=\{(35$ x$)/5$ if $30<$ x $<35$

$0$ if x $>=35$

$0$ if x $<=30$ or x $>=50$

Middle$($x$)=\{($x $-30)/10$ if $30<$ x $<40$

$(50$ x$)/10$ if $40<=$ x $<50$

Example: a $32-$years$-$old person belongs to

Young with membership grade $0\mathrm{.}6$

Middle with membership grade $0\mathrm{.}2$

FUZZY SETS FOR EDUCATION$0$

$0\mathrm{.}25$

$0\mathrm{.}5$

$0\mathrm{.}75$

$1$

$1234$

Lower

Higher

VeryGood

$1$: high school $2$: $2-$yr college $3$: bachelor's $4$: graduate degree

LLLLL

FUZZY SETS FOR INCOME$0$

$0\mathrm{.}25$

$0\mathrm{.}5$

$0\mathrm{.}75$

$1$

$1020304050607080$

Low

Medium

High

Medium $=\{\mathrm{.}4/20+\mathrm{.}6/30+1/40+\mathrm{.}3/50+\mathrm{.}1/60+0/$rest $\}$

QUERY: Example $1$

step $2-$ Fire all applicable rules

a p e Low$(0\mathrm{.}5)$ High$(0\mathrm{.}5)$

$(0\mathrm{.}6)$Young

L M

$(0\mathrm{.}2)$Middle

L H

Old M H

step $2-$ Fire all applicable rules

a p e Low$(0\mathrm{.}5)$ High$(0\mathrm{.}5)$

$(0\mathrm{.}6)$Young

L M

$(0\mathrm{.}2)$Middle

L H

Old M H

QUERY: Example $1$

step $3$ Find firing strengths of rules

L rule $1($Young & Low$)$

firing strength is $0\mathrm{.}5=$ min $\{0\mathrm{.}6,0\mathrm{.}5\}$

M rule $2($Young & High$)$

$0\mathrm{.}5=$ min $\{0\mathrm{.}6,0\mathrm{.}5\}$L rule $3($Middle & Low$)$

$0\mathrm{.}2=$ min $\{0\mathrm{.}2,0\mathrm{.}5\}$

H rule $4($Middle & High$)$

$0\mathrm{.}2=$ min $\{0\mathrm{.}2,0\mathrm{.}5\}$

QUERY: Example $1$

step $4-$ Defuzzify to get answer:L $(10$ K$)*0\mathrm{.}5=5$ KM $(40$ K$)*0\mathrm{.}5=20$ KL $(10$ K$)*0\mathrm{.}2=2$ K

$+$

H $(80$ K$)*0\mathrm{.}2=16$ K

total: $43$K

Different algorithms may give different answers

QUERY: Example $2$

Q: What is the mean income of a Very Young

person with a Very Good education?

$$ We desire an answer more useful $/$ specific than

$$Very High$$ or $$Probably High$$

$$ Approximate reasoning algorithms can be designed

similarly

$$ Answer in the form of a fuzzy set, e$.$g$.$

$\{\mathrm{.}1/30+\mathrm{.}2/40+\mathrm{.}5/50+\mathrm{.}6/60+\mathrm{.}8/70+\mathrm{.}8/80+\mathrm{.}5/90\}$

FUZZY SET FOR

VERY YOUNG$0$

$0\mathrm{.}25$

$0\mathrm{.}5$

$0\mathrm{.}75$

$1$

$202530354045505560$

Young

Mid

Old

VeryYoung

FUZZY EXPERT SYSTEMS: SUMMARY

$$ Rules are easier to formulate

$$ fuzzy rules are qualitative

$$ obtained empirically and from experts

$$ crisp rules are OK

$$ Performance can be fine tuned

$$ adjust the fuzzy sets $($size$,$ shape, number, etc.$)$

$$ different inference algorithms

$$ validation by experts

ASSIGNMENT

Propose a way to find an answer for the query of Example $2$