FIGURES FOR CHAPTER 7

NEURO FUZZY SYSTEMS

1

Fuzzy sets Fuzzy systemTraining

Fuzzy rules

Figure 7.1: Cooperative Neuro-fuzzy Type 1

Fuzzy rules Fuzzy systemTraining

Fuzzy sets

Figure 7.2: Cooperative Neuro-fuzzy Type 2

Parameterlearning

Errormeasure

Fuzzy system

Figure 7.3: Cooperative Neuro-fuzzy Type 3

2

Weightedrules

Errormeasure

Fuzzy system

Figure 4: Cooperative Neuro-fuzzy Type 4

X1

X2

X

X

X

X

X

XX X

XX

XX

R1 R2

R3

R4

X1

X2

R1

R2

R3

R4X

X

X

X

X

XX X

XX

XX

R1

R2

R3

R4

X1

X2

(1) (2) (3) Figure 7.5: Neural Network-Driven Fuzzy Reasoning

3

1x

2x

2x

1x

NeurofuzzySystem’s

InputNeurofuzzySystem’sOutput

CrispInput

CrispOutput

y

yFuzzifier module Inference module Defuzzifier module

Figure 7.6: Hybrid neuro-fuzzy Systems

4

y(k+1)

1x 2x

rule(1)

rule(2)

rule(3)

rule(4)

1x

1x

2x

2x

2x

1w

2w

3w

1x

4w

α α α= + +O x x1 1 11 1 1 2 2 0

α α α= + +O x x2 2 22 1 1 2 2 0

α α α= + +O x x3 3 33 1 1 2 2 0

α α α= + +O x x4 4 44 1 1 2 2 0

+ + +=

+ + +* w O w O w O w OO

w w w w1 1 2 2 3 3 4 4

1 2 3 4

= + + +*O w O w O w O w O1 1 2 2 3 3 4 4

µA11

µA2

1

µA1

2

µA2

2

µA31

µA32

µA4

1 µA4

2

∏

∏

∏

∏

N

N

N

N

O1

O2

O3

O4

x1

x2

*O∑

Fuzzificationlayer

T-normoperation

layer

Normalizationlayer

Consequentlayer

Summationlayer

ANFISoutput

ANFISinput

w2

w3

w4

x1 x2

x1 x2

x1 x2

x1 x2

w O1 1

w O2 2

w O3 3

w O4 4

1w 1w

2w

µA1

1

µA2

1

µA1

2

µA2

2

µA31

µA32

µA4

1

µA4

2

3w

4w

Figure 7.7: Five-layer ANFIS

5

∏

∏

∏

∏

O1

O2

O3

O4

x1

x2

*O∑

Fuzzificationlayer

T-normoperation

layer

Combined normalizationand consequent

layer

Summationlayer ANFIS

output

ANFISinput

∑

∑

µA1

1

µA2

1

µA1

2

µA2

2

µA3

1

µA3

2

µA4

1

µA4

2

1w

2w

3w

4w

Figure 7.8: Four-layer ANFIS

6

−2

0

2

−2

0

2

−5

0

5

x

Original Function (Peaks)

y −2

0

2

−2

0

2

−5

0

5

x

ANFIS Output Before Training

y

−2

0

2

−2

0

2

−5

0

5

x

ANFIS Output After Epoch 10

y −2

0

2

−2

0

2

−5

0

5

x

ANFIS Output After Training (100 Epochs)

y

Figure7.9 The Peaks function (original function and its ANFIS approximation)

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−3 −2 −1 0 1 2 3

0

0.2

0.4

0.6

0.8

1

x

Deg

ree

of m

embe

rshi

pMembership Functions Before Training

−3 −2 −1 0 1 2 3

0

0.2

0.4

0.6

0.8

1

y

Deg

ree

of m

embe

rshi

p

−3 −2 −1 0 1 2 3

0

0.2

0.4

0.6

0.8

1

x

Deg

ree

of m

embe

rshi

p

Membership Functions After Epoch 10

−3 −2 −1 0 1 2 3

0

0.2

0.4

0.6

0.8

1

y

Deg

ree

of m

embe

rshi

p

−3 −2 −1 0 1 2 3

0

0.2

0.4

0.6

0.8

1

x

Deg

ree

of m

embe

rshi

p

Membership Functions After Training (100 Epochs)

−3 −2 −1 0 1 2 3

0

0.2

0.4

0.6

0.8

1

y

Deg

ree

of m

embe

rshi

p

Figure 7.10 Evolution of the ANFIS membership functions

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−2

0

2

−2

0

2

−5

0

5

x

Error Before Training

y −2

0

2

−2

0

2

−5

0

5

x

Error After Epoch 10

y

−2

0

2

−2

0

2

−5

0

5

x

Error After Training (100 Epochs)

y

Figure 7.11 ANFIS output error quantification

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0 50 100 150 200 25010

−1

100

101

Error Comparision: ANFIS vs. MLP

Error in ANFISError in MLP

Figure 7.12 Performance of ANFIS versus MLP

Inpu

t 2

Input 1

rule1 rule2 rule3

rule4

rule5 rule6 rule7

(a)

rule1 rule2 rule3

rule4

rule5 rule6 rule7

(b)Input 1

Inpu

t 2

Figure 7.13: Crisp and fuzzy partitioning

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(a) (b)

Figure 7.14: Fixed (a) and adaptive (b) grid partitioning

Figure 7.15: Scatter partitioning

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