SBI Materials Fuzzy-Inference Engines

8/12/2019 SBI Materials Fuzzy-Inference Engines

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Fuzzy Inference EnginesComposition and Individual-Rule BasedComposition, Non-Linear Mappings

Olaf Wolkenhauer

Control Systems Centre

UMIST

[email protected]

www.csc.umist.ac.uk/people/wolkenhauer.htm
mailto:[email protected]://www.csc.umist.ac.uk/people/wolkenhauer.htmhttp://www.csc.umist.ac.uk/people/wolkenhauer.htmmailto:[email protected]


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Contents

1 Approximate Reasoning 41.1 Modus Ponens . . . . . . . . . . . . . . . . . . . . . . 61.2 Compositional Rule of Inference . . . . . . . . . . . . . 71.3 Fuzzy Implication Operators . . . . . . . . . . . . . . 9

2 Composition-Based Inference 112.1 The Algorithm . . . . . . . . . . . . . . . . . . . . . . 12

3 Individual-Rule Based Inference 133.1 The Algorithm . . . . . . . . . . . . . . . . . . . . . . 143.2 Example: Individual-Rule Based Inference . . . . . . . 163.3 Minimum Inference Engine . . . . . . . . . . . . . . . 173.4 Product Inference Engine . . . . . . . . . . . . . . . . 183.5 Singleton Inputs . . . . . . . . . . . . . . . . . . . . . 193.6 Dienes-Rescher Inference Engine . . . . . . . . . . . . 213.7 Zadeh Inference Engine . . . . . . . . . . . . . . . . . 22

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3.8 Lukasiewicz Inference Engine . . . . . . . . . . . . . . 233.9 Singleton Input . . . . . . . . . . . . . . . . . . . . . . 24

4 Fuzzy Systems as Nonlinear Mappings 254.1 Defuzzication . . . . . . . . . . . . . . . . . . . . . . 264.2 Product Inference Engine with Singleton Input Data . 28

5 Comparison of Inference Engines 31

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Section 1: Approximate Reasoning 4

1. Approximate ReasoningLet proposition take the form

x is A

with fuzzy variable x taking values in X and A modelled by a fuzzyset dened on the universe of discourse X by membership function : X [0, 1].

A compound statement ,

x is A AND y is B

is a fuzzy set A B in X Y with

A B (x, y ) = T A (x), B (y)

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For the sake of simplicity we consider a single rule of type

IF x is A, THEN y is B

which can be regarded as a fuzzy relation

R : X Y [0, 1](x, y ) R(x, y )

where R(x, y ) is interpreted as the strength of relation between x andy. Viewed as a fuzzy set, with

R (x, y ) .= R(x, y )

denoting the degree of membership in the (fuzzy) subset R , R (x, y )is computed by means of a fuzzy implication .

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1.1. Modus Ponens

The (generalised) modus ponens provides a mechanism for inference :

Implication: IF x is A, THEN y is B .Premise: x is A .

Conclusion: y is B .

In terms of fuzzy relations the output fuzzy set B is obtained as therelational sup- t composition , B = A R .

The computation of the conclusion B (y) is realised on the basis

of what is called the compositional rule of inference .

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1.2. Compositional Rule of Inference

Given A (x), and R (x, y ), B (y) is found by generalising the crisprule (from functions to relations..)

IF x = a AND y = f (x ), THEN y = f (a )

The inference can be described in three steps :

1. Extension of A to X Y , i.e A ext (x, y ) = A (x ).2. Intersection of A ext with R , i.e

A ext R (x, y ) = T A ext (x, y ), R (x, y ) (x, y )

3. Projection of A ext R on Y , i.eB (y) = sup

x XA ext R (x, y )

= supx X

T A ext (x, y ), R (x, y ) (1)

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0

1

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1.3. Fuzzy Implication Operators

Dienes-Rescher implication:

R (x, y ) = max 1 A (x), B (y) . (2)

Zadeh implication:

R (x, y ) = max min A (x ), B (y) , 1 A (x) . (3)

Lukasiewicz implication:

R (x, y ) = min 1, 1 A (x) + B (y) (4)

G odel implication:

R (x, y ) = 1 if A (x) B (y),B (y) otherwise.

(5)

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Minimum implication:

R (x, y ) = min A (x ), B (y) (6)

Product implication:

R (x, y ) = A (x ) B (y) (7)

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Section 2: Composition-Based Inference 11

2. Composition-Based InferenceThe way rules are combined, depends on the interpretation for whata set of rules should mean. If rules are viewed as independent con-ditional statements , then a reasonable mechanism for aggregating nRindividual rules R i (fuzzy relations ) is the union :

R .=n R

i =1

R i

= S R 1 (x , y), . . . , R nR (x , y) . (8)

On the other hand, if rules are seen as strongly coupled conditional statements , their combination should employ an intersection opera-tor :

R .=n R

i =1

R i

= T R 1 (x , y ), . . . , R nR (x , y ) . (9)

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Section 2: Composition-Based Inference 12

2.1. The Algorithm

For the nR fuzzy if-then rules of the conjunctive linguistic model

R i : IF x1 is Ai 1 AND x2 is Ai 2 . . . AND xr is Air , THEN y is B i

Step 1: Determine the fuzzy set membership functions

A i 1 A ir (x1 , . . . , x r ) .= T A i 1 (x1 ), . . . , A ir (x r ) .(10)

Step 2: R i (x , y ), i = 1 , . . . , n R , is calculated according to any fuzzyimplication ( 2)-(7).

Step 3: R (x , y ) is determined according to ( 8) or (9).

Step 4: Finally, for an input A , the output B is

B (y) = supx X

T A (x ), R (x , y ) . (11)

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S i 3 I di id l R l B d I f 13


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Section 3: Individual-Rule Based Inference 13

3. Individual-Rule Based InferenceGiven input fuzzy set A in X , the fuzzy set B i in Y is given by thegeneralised modus ponens ( 1), i.e

B i (y) = supx X

T A (x ), R i (x , y) i = 1 , . . . , n R (12)

The output of the fuzzy inference engine from the unionB (y) = S B 1 (y), . . . , B r (y) (13)

or intersection

B (y) = T B1(y), . . . , B

r(y) (14)

of the individual output fuzzy sets B1 , . . . , B r .

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S ti 3 I di id l R l B d I f 14


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3.1. The Algorithm

Step 1: Determine the fuzzy set membership functions

A i 1 A ir (x1 , . . . , x r ) .= T A i 1 (x1 ), . . . , A ir (x r ) .(10)

Step 2: Equation ( 10) is viewed as the fuzzy set A in the fuzzy im-

plications ( 2)-(7) and R i (x , y), i = 1 , . . . , n R , is calculatedaccording to any of the implications.

Step 3: For a given input fuzzy set A in X , determine the outputfuzzy set B i in Y for each rule R i according to the generalisedmodus ponens ( 1), i.e

B i (y) = supx X

T A (x ), R i (x , y ) (12)

for i = 1 , . . . , n R .

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Section 3: Individual Rule Based Inference 15


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Step 4: The output of the fuzzy inference engine is obtained fromeither the union

B (y) = S B 1 (y), . . . , B r (y) (13)or intersection

B (y) = T B 1 (y), . . . , B r (y) (14)

of the individual output fuzzy sets B1 , . . . , B r .

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3.2. Example: Individual-Rule Based Inference

Minimum inference.

Singleton input.

Union intersection.

R i : IF x1 isZ

1 0 +1

AND x2 isP

1 0 +1

x 1 x2

minT (

)

w (6)

w (6) THENY

B 6

y (6)0

for all R i ,...Y

B 6

y

B

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3.3. Minimum Inference Engine

Individual-rule based inference.

Union combination (13).

Fuzzy implication ( 6).

Max. for all the t-conorm operators.

Using (6) and the min for for all t-norm operators.

We obtain from ( 12) and ( 13) :

B (y) = maxi =1 ,... ,n R supx X min A (x ), A i 1 (x 1), . . . , A ir (x r ), B i (y)(15)

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3.4. Product Inference Engine

Individual-rule based inference.

Union combination (13).

Fuzzy implication ( 7).

Max. for all the t-conorm operators.

Using (7) and the algebraic product for all t-norm operators

We obtain from ( 12) and ( 13) :

B (y) = maxi =1 ,... ,n R supx X A (x )

r

k =1A ik (x k ) B i (y) (16)

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3.5. Singleton Inputs

Let the fuzzy set A is a singleton , that is, if we consider crisp inputdata,

A (x ) = 1 if x = x0 otherwise,

(17)

where x is some point in X . Substituting ( 17) in (15) (Minimum Inference Engine ) and

(16) (Product Inference Engines ),

we nd that the maximumsup

x X

is achieved at

x =

x .

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Hence, the Minimum Inference Engine (15) reduces to,

B (y) = maxi=1

,... ,n Rmin A i 1 (x1), . . . , A ir (x r ), B i (y) (18)

and the Product Inference Engine (16) reduces to

B (y) = maxi =1 ,... ,n R

r

k =1

A ik (x k ) B i (y) (19)

A disadvantage of the minimum and product inference enginesis that

for some x X , A i k (x k ) is very small,

then B (y) obtained from ( 15) and (16) will be very small.

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3.6. Dienes-Rescher Inference Engine

Using individual-rule based inference.

Intersection combination (14).

Implication ( 2).

Using the min t-norm in ( 14) and ( 10).

We obtain from ( 12) :

B (y) = mini =1 ,... ,n R

supx X

min A (x ),

max 1 mink =1 ,... ,r A ik (x k ) , B i (y)

(20)

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3.7. Zadeh Inference Engine



Implication ( 3).

Using t-norm min in ( 14) and ( 10).


B (y) = mini =1 ,... ,n R

supx X

min A (x ), max

min A i 1 (x 1), . . . , A ir (x r ), B i (y) ,

1 mink =1 ,... ,r

A ik (x k ) (21)

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3.8. Lukasiewicz Inference Engine



Implication ( 4).

Using the min t-norm in ( 14) and ( 10).


B (y) = mini =1 ,... ,n R

supx X

min A (x ),

min 1,1

mink =1 ,... ,r

A ik (

xk ) +

B i (

y)

= mini =1 ,... ,n R

supx X

min A (x ),

1 mink =1 ,... ,r

A ik (x k ) + B i (y) (22)

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3.9. Singleton Input

If the fuzzy set A is a singleton, substituting ( 17) into the equationsof the inference engines ( 20)-(22), the sup x X is obtained at x = x ,leading to the following singleton input inference engines :

From the Dienes-Rescher Inference Engine ( 20) :

B (y) = mini =1 ,... ,n R

max 1 mink =1 ,... ,r

A ik (xk) , B i (y)

From the Zadeh Inference Engine ( 21) :

B (y) = mini =1 ,... ,n R

max min A i 1 (x 1), . . . ,

A ir (x r ), B i (y) , 1 mink =1 ,... ,r A ik (x i )

From the Lukasiewicz Inference Engine ( 22) :

B (y) = mini =1 ,... ,n R

1, 1 mink =1 ,... ,r

A ik (x i ) + B i (y)

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Section 4: Fuzzy Systems as Nonlinear Mappings 25


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4. Fuzzy Systems as Nonlinear Mappings

Linguistic model:R i : IF x1 is Ai 1 AND x2 is Ai 2 . . . AND xr is Air , THEN y is B i

Let the input data be crisp, i.e substituting ( 17) into the productinference engine (16), we have

B (y) = maxi =1 ,...,n R

r

k =1

A ik (x k ) B i (y) . (23)

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4.1. Defuzzication

A defuzzier is a mapping from the fuzzy set B in Y to a pointy in Y .

To obtain a single-valued numerical output from the inferenceengines, one has to somehow capture the information given inB (y) by a single number.

The centre of gravity defuzzier determines y as the centre of the area under the membership function B (y) :

y .=

Y B (y) y dy

Y B (y) dy

(24)

The main problem with this defuzzier is the calculation of theintegral for irregular shapes of B (y).

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Since the fuzzy set B is the union or intersection of nR fuzzysets, the weighted average of the centres of the nR fuzzy setsprovides a reasonable approximation of ( 24).

Let y( i )0 be the centre of the ith fuzzy set and w(i ) be its height,

the center average defuzzier calculates y as

y .=

n R

i =1y( i )0 w

( i )

n R

i =1w( i )

. (25)

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4.2. Product Inference Engine with Singleton Input Data

Use the centre average defuzzier ( 25).

The centre of the fuzzy set A ik (x k ) B i (y) determines thecentre of B i , denoted y(

i )0 in (25).

The height of the ith fuzzy set in (23) isr

k =1

A ik (x k ) B i (y( i )0 ) =r

k =1

A ik (x k )

and equals w( i ) in (25).

This reduces the fuzzy system to

y =

n R

i =1y( i )0

r

k =1A ik (x k )

n R

i =1

r

k =1A ik (x k )

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... or in general, we nd that the fuzzy system is a nonlinearmapping

f : X Y x f (x )

where x X R r maps to f (x ) Y R , a weighted average of theconsequent fuzzy sets :

f (x ) =

n R

i =1y( i )0

r

k =1A ik (x k )

n R

i =1

r

k =1A ik (x k )

. (26)

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Similar to ( 26), we obtain for a fuzzy system, with

minimum inference engine ( 15),

singleton input ( 17) and

centre average defuzzier ( 25),

f (x ) =

n R

i =1y( i )0

r

mink =1 A ik (x k )n R

i =1

rmink =1

A ik (x k ). (27)

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Section 5: Comparison of Inference Engines 32


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1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Figure 1: Gaussian and trapecoidal input fuzzy sets.

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1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Figure 2: Gaussian and trapecoidal outputs sets B i .

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x1

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Figure 4: Minimimum inference with Gaussian and trapecoidal sets.

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1 0.5 0 0.5 1 1

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x1

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x1

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Figure 5: Contourplots for minimum inference (left) vs product inference (right).

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1 0.5 0 0.5 1 1

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x1

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Figure 6: Minimum inference with input fuzzy partition that does not have fully overlapping fuzzy sets.

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1 0.5 0 0.5 1 1

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x1

x 2

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Figure 7: Product inference with non-overlapping input fuzzy partition.

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References[1] Kruse, R., Gebhardt, J. and Klawonn, F. : Foundations of Fuzzy

Systems. Wiley, 1994.[2] Wang, L.-X. : A Course in Fuzzy Systems and Control.

Prentice Hall, 1997.

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SBI Materials Fuzzy-Inference Engines