DSS- Fuzzy (literature)

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FUZZY RULE-BASED

SYSTEMS

Lecture 15

from: Fuzzy Rule-Based Modeling with

Applications to Geophysical, Biological and

Engineering Systems, by A. Bardossy and

L. Duckstein, CRC Press, 1995

A. Fuzzy Sets

• Dealing with uncertainty

–Probability theory and statistics –Bayesian statistics

–Fuzzy sets—nonfrequentistapproach

• Fuzzy set

–Set of objects without clearboundaries or not well defined—partial membership

1

• “Crisp” set

• Example:

–the set of “young persons” is afuzzy set

2

[ ]{ }( , ( ); , ( ) 0,1 A A x x x x µ µ = ∈ ∈A X

( ) 0 or 1 A x =

1 25

40 -( ) 25 40

25

0 > 40

A

x

x x x

x

µ

≤

= ≤ ≤

1

025 40

linear

x

3



• defines the “truth value” of

• Learning—process of removing orreducing fuzziness

• Cardinality

4

( ) A x

x ∈ A

{ }

1

1 1

( )

of ( , ),..., ( , )

I

i

i

I I

car

a a

µ

µ µ

=

=

=

∑A

A

• h level set

“too at least a degree”

so:

• compliment:

5

( ) | ( ) A A h x x h µ = ≥

1 2 1 2( ) ( ) if A h A h h h⊂ >

(0 1) A A h≡ < <cC A=

( ) 1 ( )C A x x µ = −

• intersection

• union:

6

D A B= ∩

( )( ) min ( ), ( ) D A B x x x µ µ µ =

B

x

AD

B

x

A E E A B= ∪

( )( ) max ( ), ( ) E A B x x x µ µ µ =

: algebraic productab

• Note: not necessarily

• t norm (intersection operator)

7

c A A∩ 0/

let D A B= ∩

( ) ( ) , ( ) D A B x t x x

a b

µ µ µ

=

(1 )(1 )

ab ab

a b ab a a b+ − + − −( )max 0, 1 ... many othersa b+ −



0 170 cm

185 -( ) 170 185 cm15

1 > 185

A

x

x x x

x

µ

≤

= < ≤

• t co-norm

• linguistic variables

– words, phrases, expressions

– e.g., “tall”

8

( , ) 1 (1 ,1 )

( , ) 1 (1 ,1 )

c x y t x y

t x y c x y

= − − −

= − − −

• Linguistic modifiers:

– VERY MOSTLY PRACTICALLY

– SORT OF NOT INDEED – SOMEWHAT ROUGHLY … etc.

• e.g.

– called “concentration”

– since actually has effectof decreasing membership

9

2VERY ( ) ( ) x x µ µ =

[ ]0,1 µ ∈

• “dilation” modifier

• “contrast” modifier

– to translate back to natural language,find closest Euclidean distance to

membership function of statement 10

MORE or LESS( ) ( ) x x µ µ =

( )

2

INDEED 2

2 ( ) 0 ( ) 0.5( )

1 2 1 ( ) 0.5 ( ) 1.0

x x x

x x

µ µ µ

µ µ

≤ <=

− − ≤ <

B. Fuzzy Numbers

• Special case of fuzzy sets

• Involves arithmetic operations

• A fuzzy subset A of real nos. is a

fuzzy number if:

– at least 1 z such that:[normality]

–for all

–[convexity]

∃ ( ) 1 A z =

( )

( ) min ( ), ( ) A A A

a c b

c a b µ µ µ

< <

≥

11



• Convexity insures h levels sets and

the support of the fuzzy no. are

intervals

• Membership value of real no. is“likeliness” of occurrence of that no.

– not “likelihood” 12

{ }supp( ) | ( ) 0 A A x x µ = >

1 1

convex nonconvex

• Level sets: different nos. with given

minimum “likeliness”

• Zadeh: is “possibility” of x .impossible totally possible

• A “crisp” number is a fuzzy no. with a

single point

13

( ) A x

• Triangular fuzzy nos. [symmetric or

asymmetric]

14

1

1

1 22 1

32 3

3 2

3

0

( )

0

A

x a

x a

a x aa a x

a xa x a

a a

a x

µ

≤

− < ≤−

= − < ≤

−

<

1

a1

xa2 a3

1 2 3( , , )T a a a

• Trapezoidal

fuzzy no.

15

1

11 2

2 1

2 3

43 4

4 3

4

0

1( )

0

A

x a

x aa x a

a a

a x a xa x

a x aa a

a x

µ

≤

− < ≤−

< ≤= −

< ≤−

<

1

a1 xa2 a3 a4



• Fuzzy mean

–we often need to “defuzzify” a fuzzy

set—i.e., replace with a crisp value –could take element with highest

membership value—but may not be

unique or representative of fuzzy set

16

1

3 x

6 9

e.g.

(3,3,9)T

• Fuzzy mean:

for

17

( )

( )

( )

A

A

t t dt

M A

t dt

µ

µ

+∞

−∞

+∞

−∞

=

∫

∫1 2 3( , , )T A a a a=

1 2 3( )

3

a a a M A

+ +=

Note: not all fuzzy sets have fuzzy

means: e.g., 1 2( , , )T a a ∞

• Fuzzy mean for piecewise linear

fuzzy membership function

18

( )

( )

1

1 11 1

0

11

0

( )

2 2( ) ( )6 6

( ) ( )

2

L

L

M A

x x x x x x x x

x x x x

µ µ

µ µ

−

+ ++ +

=

++

=

=

+ + − +

+−

∑

∑

0 1for L x x x< < ⋅⋅ ⋅ < breakpoints

• Advantage of fuzzy mean:

–continuous—i.e., differences in

membership functions produce

smooth changes in fuzzy mean• Fuzzy median:

–disadvantage: not continuous—also,not necessarily UNIQUE

19

( )

( )

( ) ( )

m A

A A

m A

t dt t dt µ µ +∞

−∞

=∫ ∫



–advantage: good for fuzzy sets

defined on DISCRETE ordered

sets X • Expressing similarity between fuzzy

sets:

–similar to Euclidean distance—butonly applicable to TRIANGULAR

membership functions20

( )

2

2 2 21 1 2 2 3 3

( , )

( ) ( ) ( )

d A B

a b a b a b

=

− + − + −

C. Fuzzy Rules

• Problem of imprecise information and

measurements• Crisp systems tolerate no exceptions

and no errors

• Wide differences in opinion

–a statement AND its opposite may

both be true to a certain degree

21

• Structure of fuzzy rule:

or simplified:

[different rules with different

consequences can be applied to samepremises]

22

1 1 2 2IF: is is is

THEN:

i i k ik

i

a A a A a A

B

⋅ ⋅ ⋅

{ AND OR XOR

1 2 THEN:i i ik i A A A B⋅⋅⋅

consequencepremises

• Degree of fulfillment [DOF]

–Product inference

23

1

1 2

1 2

1 2

1 2

1 2

1 11 2 1 2

1 2 1 2

1 2

1 2 1 2

1 2

(not ) 1 ( )

( AND ) ( ) ( )

( OR ) ( ) ( )

( ) ( )

( XOR ) ( ) ( )

2 ( ) ( )

A A A

A A

A A

A A

A A

A a

A A a a

A A a a

a a

A A a a

a a

µ µ

µ µ

µ µ

µ µ

µ µ

= −

= ⋅

= +

− ⋅

= +

− ⋅



–min-max inference

[both reproduce Boolean truth table

for crisp or 0-1 case]24

( )( )( )

( )

1 2

1 2

1 2

1 2

1 2 1 2

1 2 1 2

1 2 1 2

1 2

( AND ) min ( ), ( )( OR ) max ( ), ( )

( XOR ) max min 1 ( ), ( ) ,

min ( ),1 ( ) )

(

A A

A A

A A

A A

A A a a A A a a

A A a a

a a

µ µ µ µ

µ µ

µ µ

=

=

= −

−

• Example:

IF: (1,2,3)T AND

((1,2,6)T

OR (4,5,7)T)

AND (2.5,4,4.5)T

THEN: (0,1,2)T

If: a1=1.5 0.5

a2=4 0.5

a3=4.8 0.8

a4=3 0.333product gives DOF=0.150

min-max gives DOF=0.33325

0.8

–min-max: accounts for only limiting

or extreme arguments

–product: accounts for fulfillment of

ALL arguments

26

• Degree of fulfillment for “AND” coupling

• Degree of fulfillment for “OR” coupling

2 clauses

multiple clauses—recursive “OR”

27

,

1 1

( )i k

K K

i A k k

k k

D a ν = =

= =∏ ∏

,1 ,2 ,1 ,21 2 1 2( ) ( ) ( ) ( )

i i i ii A A A A D a a a a µ µ µ = + − ⋅

1 1 1( ,..., ) ( ( ,..., ), )i K i i K K D D Dν ν ν ν ν −=



• Degree of fulfillment for “MOST OF”

• Degree of fulfillment for “AT LEAST AFEW”

p=1: perfect compensation

p=∞: no compensation

p=2: good compromise 28

( ),

1

1

1

1 1 ( )i k

K p p

i A k k

D a K µ =

= − − ∑

( ),

1

1

1( )

i k

K p p

i A k

k

D a K

µ =

=

∑

p-norm

• Example: suppose we have the

following 5 rule arguments for given

set of facts :

29

1

2

3

4

5

1 1

2 2

3 3

4 4

5 5

( ) 0.9

( ) 0.9

( ) 0.5

( ) 0.1

( ) 0

A

A

A

A

A

a

a

a

a

a

ν µ

ν µ

ν µ

ν µ

ν µ

= =

= =

= =

= =

= =

1 2 3 4 5, , , ,a a a a a

For “AND” coupling

For “OR” coupling

30

0.9 0.9 0.5 0.1 0.0 0i D = ⋅ ⋅ ⋅ ⋅ =

( )( )( )

( )

0.9 0.9 0.9 0.9

0.5 0.9 0.9 0.9 0.9 0.5

0.1 0.9 0.9 0.9 0.9

0.5 0.9 0.9 0.9 0.9 0.5 0.1

0.9955

(

)

i D = + − ⋅

+ − + − ⋅ ⋅

+ − + − ⋅

+ − + − ⋅ ⋅ ⋅

=

For “MOST OF” coupling ( p=2)

For “AT LEAST A FEW” coupling

31

( ) ( ) ( ) ( ) ( )1

2 2 2 2 2 2

1

1 0.9 1 0.9 1 0.5 1 0.1 1 0

5

i D

0.355

= −

− + − − − −

=

( ) ( ) ( ) ( ) ( )1

2 2 2 2 2 20.9 0.9 0.5 0.1 0

5

i D

0.613

+ + + + =

=



D. Combination of Fuzzy Rule

Responses

• With fuzzy rules (unlike “crisp” rules),several rules can be applied from the

same premise vector

• Need to be able to specify an “overall

response”

• DOF of rule i:

1( ,..., ) K a a

1( ,..., ) (response)i i K i D a a Bν = →

32

• Problem: define fuzzy set B based on

the individual

i.e.,[combination operator]

• Assume all rules have consequences

which are fuzzy subsets of same set

• 3 methods

– minimum

– maximum

– additive 33

( , )i i B ν

( ) ( )( )1 1, ,..., , I I B C B Bν ν =

• Minimum combination

• Cresting minimum combination

disadvantage: requires much care—can

easily get

• Maximum combination:

34

0( ) min ( )

i

i

B i B x xν

ν µ >

=

( ){ }0

( ) min min , ( )i

i

B i B x xν

µ ν µ >

=

( ) 0 B x =

1,...,( ) max ( )

i B i Bi I

x xν µ =

= ⋅

• Cresting Maximum combination

tolerates disagreements—but does notemphasize agreement--vague

35

( ){ }1,...,

( ) max min , ( )i

B i Bi I

x x µ ν µ =

=



• Additive combination:

1) Weighted sum:

A rule is better if its consequence ismore specific—i.e., not vague

36

1I

i=1

( )

( )

max ( )

i

i

I

i B

i B

i Bu

x

x

u

ν µ

µ

ν µ

=

⋅

=

⋅

∑

∑

2) Normed weighted sum:

37

1( ) [continuous]

1car( ) ( ) [discrete]

i B

i

i ii j

x dx

B j

µ β

µ β

∞

−∞

=

= =

∫

∑

1

=1

( )

( )

max ( )

i

i

I

i i B

i B I

i i Bu

i

x

x

u

ν β µ

µ

ν β µ

=

⋅ ⋅

=

⋅ ⋅

∑

∑

3) Cresting weighted sum:

38

( )

( )1

=1

min , ( )

( )

max min , ( )

i

i

I

i B

i B I

i Bu

i

x

x

u

ν µ

µ

ν µ

==

∑

∑

4) Cresting normed weighted sum:

rules with “crisper” answers carry

more weight

39

( )

( )1

=1

min , ( )

( )

max min , ( )

i

i

I

i i B

i B I

i i Bu

i

x

x

u

β ν µ

µ

β ν µ

=

⋅

=

⋅

∑

∑



• Example:

minimum combination:

40

1 1

2 2

Rule 1: 0.4 (0,2,4)

Rule 2: 0.5 (3,4,5)

T

T

B

B

ν

ν

= =

= =

4min 0.4 ,0.5 ( 3)

2

4 23 0.4 0.5 ( 3) @

2 7

x x

x x x

− ⋅ ⋅ −

− ⋅ = ⋅ − =

41

( )

230.5( 3) 3

74 23

0.4 42 7

B

x x

x x

x µ

− < ≤

= − < <

cresting minimum

42

( )

4 4min 0.4,

2 2

min 0.5, ( 3) 34 10

3 @2 3

x x

x x x

x x

− − =

− = −−

= − =

( )

10( 3) 3

3

4 10 42 3

B

x x

x x x

µ

− < ≤

= − < <

1

1 x2 3 4 5 6

0.5 crestmin

mincomb

1

1 x2 3 4 5 6

0.5crest

max

maxcomb

43



weighted sum combination

44

0.4 0< 224

0.4 2 3( ) 2

4 0.4 +0.5( -3) 3 4

20.5(5 ) 4 5

x x

x x

f x

x x x

x x

≤

− < ≤

= −

< ≤

− < ≤Sum of membership functions · ν i

imax ( ) 0.5

so... ( ) 2 ( )

i

i

B

B

u

x f x

ν µ

µ

⋅ =

=

∑

normed weighted sum:

11

22

1area under : 2

1area under : 1

B

B

β

β

=

=

45

0.2 0< 22

40.2 2 3

( ) 24

0.2 +0.5( -3) 3 42

0.5(5 ) 4 5

x x

x x

f x x

x x

x x

≤

− < ≤

= −< ≤

− < ≤

( ) 2 ( ) B x f x=

46

1

1 x2 3 4 5 6

0.5norm. wtd.

sum

wtd.sum

1

1 x2 3 4 5 6

0.5

crest.wtd.sum

crest.norm.

wtd. sum

47



• Advantage of additive methods for

combination

– same support

• Disadvantage of additive methods

– for repeating rules, gives different

µ

48

E. Defuzzification

• Often necessary to replace fuzzy

consequence with single crispconsequence

• Example: prediction for forecasting

or control decision:

• Methods

1) maximum2) mean

3) median

Bb

( ) f b D B=

49

• Defuzzification by maximum

– not necessarily unique

• Defuzzification by mean

50

( ) max ( ) B B x

b x µ =

( ) ( ) f b D B M B= =

1

1

1( )

( )1

I

i iii

I

iii

B

M B

ν β

ν β

=

=

=

∑

∑

– for weighted

sum comb.

easy!

• Defuzzification by normed weightedsum:

• For weighted sum combination andmean defuzzification, rules can be:

51

1

1

( )

( )

I

i i

i

I i

i

B

M B

ν

ν

=

=

⋅

=

∑

∑

,1 ,IF AND ANDTHEN: ( ( ), )

i i K

i i

A M B β

⋅⋅ ⋅



• That is, we only have to represent

fuzzy consequence by its mean

and area under• For normed weighted sum

combination and mean

defuzzification, only need

• Defuzzification by median:

52

i B

( )i B : i µ β

( )i B

( ) ( ) f b D B m B= =

• Best methods are:

– product inference

– additive combinations – fuzzy mean defuzzification

53

F. Rule Systems

• Rule system

–this does not imply that allarguments used for every rule:

–e.g., assign for all

arguments k not used in rule i

ℜ

,1 ,2 ,IF THEN

(for 1,..., )

i i i K i A A A B

i I

⋅ ⋅ ⋅

=

,( ) 1

i k A x =

, fuzzy subsets offuzzy subsets of

i k k

i

A X B Y

54

• Numerical Rule System :

• To calculate response of rule system:

–inference method

–combination method

–defuzzification method

• complete if for every premise vector

, responseis nonempty fuzzy set

55

,IF: and fuzzy numbersi k i A B

ℜ

ℜ

( )1,..., K a a ∈ A ( )1,..., K a a



• Rule system with maximum or

additive combination methods is

complete on Q iff:

[minimum does not work]

56

1

rule such that

( ,..., ) 0i k

i

D a a

∀ ∃

>

a

ℜ • Example:

not complete under minimum

combination

for support [2,3]

• It is complete for maximum or additive

57

1,1 1

2,1 1

(1, 2,3) (1,2,3)

(2,3,4) (3,4,5)

T T

T T

A B

A B

= =

= =

( )1 2

1 2min ( ), ( ) 0 B B x xν µ ν µ ⋅ ⋅ =

• For maximum and additive

combinations, all we need for

completeness is:

58

1

supp( )

e.g., above 1 3; 2 4

1 4

I

i

A i

x x

x

=

⊂

≤ ≤ ≤ ≤

⇒ ≤ ≤∪

∪

• Rule system is nondegenerate if:

• If nondegenerate and complete—then mean defuzzification response is

continuous function of A.

59

ℜ

i,k , Aand and ,

are continuous

i k ii A B∀ ∀

ℜ



• Example:

complete and nondegenerate on

interval [1, 5]

Use normed weighted sum

combination with mean defuzzification

60

if (1, 2,3) then (0,1,2)

if (3, 4,5) then (2,3,4)if (0,3,6) then (0,2, 4)

T T

T T

T T

61

for 1 2

values are ( 1),0, resp.3

for 2 3 values are (3 ), 0, resp.

3for 3 4

6 values are 0, ( 3), resp.

3for 4 5

6 values are 0, (4 ), resp.

3

i

i

i

i

x x

D x

x x D x

x x

D x

x x

D x

≤ ≤

−

≤ ≤−

≤ ≤−

−

≤ ≤−

−

1( 1) 3(0) 25 33

for 1 2 :4 3

( 1) 03

1(3 ) 3(0) 293

for 2 3 :9 2

(3 ) 03

x x

x x

x x x

x x

x x

x x

− + +

− ≤ ≤ =

− − + +

− + + −

≤ ≤ =−

− + +

62

61(0) 3( 3) 2

7 153for 3 4 :

6 2 30 3

36

1(0) 3(5 ) 257 113

for 4 5 :6 21 4

0 53

x x

x x

x x x

x x

x x

x x x

− + − +

− ≤ ≤ =

− − + − +

−

+ − + −

≤ ≤ =− −

+ − +

63



0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5

x

y

“defuzzified”response

• 3 crisp rules would give 3 different values

as consequences

• Objects in discrete categories (e.g., HIGH,

MEDIUM, LOW) can be given continuous

representation64

( )

1 1

1

1

If: = , ,

and ( , ..., ) is a continuousfunction on , then for any 0,

any inference combinations and any

defuzzification method, a rule system

such that: ,...,

K K

K

K

a a a a

f a a

f a a

A

A ε

− + − + ×⋅⋅⋅×

>

∃

ℜ − ℜ( )

( )1

1

,...,

,...,

K

K

a a

a a

ε <

∀

• Proposition:

65

• Corollary:

Every nondegenerate rule system,

inference and combination method

with mean defuzzification can bereplaced by a rule system using:

– AND operator

– product inference

– weighted combination

– fuzzy mean defuzzification

66

• Use of closed form functions:

– have specific shape

• e.g., linear

– parameters assessed with specifictechnique

• e.g., least-squares

– try to approximate observed data

– parameters may not have physical

meaning

67



• Advantage of rule systems:

– rules can define function in

specific neighborhoods – errors in coefficients of polynomial

can be disastrous

– errors in a single rule only

influence function on the support

of that rule

68

G. Membership Functions in

Rule Systems

• What is a good rule?

• How crisp should arguments and

responses be?

• What shape for membership

functions?

• If we are using normed weighted sumcombination and mean

defuzzification—DOESN’T MATTER!69

• Arguments with narrow (i.e., crisp)

supports

– need many rules

• Very wide supports – nonspecific responses

• Triangular or trapezoidal

membership function are the most

popular

70

H. Fuzzy Rule Construction

• Assessment of rules: knowledge

and/or available data encoded into

rules• Ways:

1) rules known by experts*

2) rules from experts—but updated from

data

3) not known—but variables specified4) only observations available*

71



• Example: algae growth model

x : available nutrients

y : algae biomass[use scale between 0 and 1]

States described by

high (H )

low (L)

Experts: HH LH LL HL HH

[ represents time transition]72

nutrientsalgae biomass

Expert Rules: Starting @ HH – if

algae high, nutrients reduced to L @

t +1; then since nutrients insufficient,

algae becomes L in next period; as

algae breaks down, nutrients

replenished and become H again;

then algae becomes H again in thenext time period.

73

74

ˆ ˆ

ˆ

ˆ

T

T

Let and be triangular

fuzzy numbers:

(0.4,1.0,1.0)

(0.0,0.0,0.7)

H L

H =

L =

1

0.4 0.5 0.7 1.0

H L

ˆ

ˆ

Calculate means:

0.4+1.0+1.0 2.4= = 0.8

3 3

0.0+0.0+0.7 0.7= = 0.233

3 3

Construct state vector trajectory :

Let initial state be :

( (0), (0)) = (0.5,0.6)

M(H)=

M(L)=

x y

75



{ }2

ν

ε ν µ µ

⋅

L

H

L

H

2(0.5) =

for 7

1 nutrients x(0.5) =

61

(0.6) =for 7

2 algae y(0.6) =

6

Use for DOF: AND rule fulfillment grade

for is: i j

µ

µ

µ

µ

(i, j) :

(i, j) L,H (i, j)= (0.5) (0.6)

product infere nce

76

2/492/21

1/421/18(0.5)H

µ

(0.5)L µ

(0.6)H µ (0.6)L µ

DOF:

1/42HL HH

2/49LL HL

2/21LH LL

1/18HH LH

DOFRule

Transition table:

77

there are

really 8 rules:

HHL AND H[x] [y]

⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅

⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅

1 0.7 2 0.7 2 2.4 1 2.40.35 0.35+ 0.3 + 0.318 3 21 3 49 3 42 3(1)=

1 2 2 10.35 + 0.35+ 0.3+ 0.3

18 21 49 42

=

1 2.4 2 0.7 2 0.7 1 2.40.3 0.35+ 0.35+ 0.3

18 3 21 3 49 3 42 3(1) =

1 2 2 10.3 + 0.35+ 0.35+ 0.318 21 49 42

x

y

0.4319

= 0.4222

Fuzzy mean combination of rules:

replaced with mean value, and:( )i B µ x

= =1 1

0.3 0.35H L β β

78

1

0.60.5

nutrients x (t )algae y (t )

Same procedure used to obtain x (t +1),

y (t +1) as function of x (t ), y (t ), etc.

oscillatory behavior

79

t

same as solution ofcoupled diff. eqs.



I. Deriving Rule Systems from

Data Sets

• Training set:

• Fuzzy rules can be used to:

– simplify complex models

– can generate synthetic training

sets

80

( ){ }1( ),..., ( ), ( ) ; 1,..., K a s a s b s s S T = =

( ), , , ,

,

, , for each

1where mean ( )

i

i k i k i k i k T

i k k i s R

A

a s

N

α α α

α

− +

∈

= ∑

• Counting algorithm:

1. Define support:

2. Assume

81

( ), , ,, for eachi k i k i k Aα α − +

{

( )1

, ,

th

where ( ),..., ( ), ( )

such that ( ) , ,

1,...,

denotes the set of all those premisevectors that fulfill at least the

rule; forms a subset of the trainingse

i K

k i k i k

i

R a s a s b s

s

k K

R

i

in part

T

α α α − +

= ∈

∈

=

t ; is the number of elementsin .

i

i

N R

T

82

3. Define support:

83

( ), ,i i iT

β β β − +

min ( )

1

( )

max ( )

i

i

i

i s R

ii s R

i s R

b s

b s N

b s

β

β

β

−

∈

∈+

∈

=

=

=∑

( ) ( )( )1 1 1

Rule System is:

IF , , AND , ,

THEN: , ,

i i i iK iK iK T T

i i iT

α α α α α α

β β β

− + − +

− +

⋅ ⋅ ⋅ ⋅ ⋅ ⋅



• Questions:

– How to define supports?

– Use split sampling methods?• Weighted counting algorithm

– considers minimal DOF

– calculate DOF’s

– For each rule i :

84

( )i sν

( )min ( )i

i s

b sν ε

β −>

=

( )

( )

( )

( ) ( )

( )

max ( )

i

i

i

i s

i

i s

i s

s b s

s

b s

ν ε

ν ε

ν ε

υ

β

υ

β

>

>

+

>

⋅

=

=

∑

∑

85

The higher the ε value, the fewer

elements used to define the responses,and the crisper are the assessed rules

• Least Squares Method:

86

( )

( )

2

1

11

1

min ( ),..., ( ) ( )

( ) ( )

( ),..., ( )

( )

K s

I

i i

i K I

ii

R a s a s b s

s M B

R a s a s

s

ν

ν

=

=

−

⋅

=

∑

∑

∑

Rule response fornormed weighted

sum combination:

If we are using normed weighted sum

combination and mean

defuzzification, then the unknowns

are:Shapes of premise membership

functions are assumed in order to get

the

1( ),..., ( ) I B M B

iν

87

Note: the other methods can be used

for any combination and defuzzification



Example:

• For comparison of the algorithms

• Training set with 25 sets ofobservation data (a1(s),b(s)),

s=1,…,25

• Develop rule system for the interval

[0,8]

• Highly variable, nonlinear behavior

88

TrainingSet

89

• Rule system from counting algorithm

• 7 rules; supports of equal length

• 2 or 3 rules applicable to each a(s)

90

• Rule system from weighted counting

algorithm

• For ε = 0.5; smaller supports; crisper

91



• Least-squares algorithm

• For ε = 0.5; smaller supports; crisper

( )( )1( ) ( ) cos 12

L x R x xπ = = +

1

L R

L-R fuzzy nos. –

smooth

approximations

to triangular

fuzzy numbers

92

• Rule system from least-squares

algorithm

93

x training setcounting algorithm

weighted countingalgorithm

94

x training setleast-squares

6th

order polynomial

Least-squares

[correlation = 0.96]

6th order

polynomial

[correlation = 0.89]

95



J. Application: Reservoir

Operation*

• Premises:

– Reservoir pool elevations

– Inflows [net]

– Forecasted demands (power)

– Time of year

96

•Shresha, B., L. Duckstein, and E. Stakhiv. (1996).“Fuzzy Rule-Based Modeling of Reservoir

Operation,” ASCE Journal of Water Resources

Planning and Management , 122(4), 262-269.

• Consequence:

– Actual release

• Typical Rule:IF pool elevation is Ai ,1 AND

Net inflows is Ai ,2 AND

Forecast demand is Ai ,3 AND

Time of year is Ai ,4

THEN release is Bi

97

• Uses split sampling

• Calibration:

– Training set

– Weighted counting algorithm• Storage elevation premises

calculated from mass balance:

St+1=St + I t – R t - Lt

98

consequence

• Following constraints imposed:

St,min < St < St,max

R t,min < R t < R t,max

• Applied to Tenkiller Lake inOklahoma

• Project purposes:

– flood control

– water supply

– hydropower

99

– recreation

– habitat



• Capacity: 371,000 AF

• Daily data from 1980 to 1992

• Training set uses 1989• 9 TFN’s cover elevation 620 ft. to

677.2 ft.

• 8 TFN’s: inflows from 0 to 180,000

day-second-ft

• Supports cover entire range—25%overlap

100

• Training for each month separately

• Power demands: low, medium-low,

medium, medium-high, high• Product inference used [danger of

incompleteness

• Additive combination

101

102 103

DSS- Fuzzy (literature)

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