materi anfis

7/22/2019 Materi Anfis

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Summary

Introduction

ANFIS Architecture Hybrid Learning Algorithm

ANFIS as a Universal Approximatior

Simulation Examples

Lo ica Nebulosa . 2/


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Introduction

ANFIS: Artificial Neuro-Fuzzy Inference Systems

ANFIS are a class of adaptive networks that arefuncionally equivalent to fuzzy inference systems.

ANFIS represent Sugeno e Tsukamoto fuzzy

models. ANFIS uses a hybrid learning algorithm



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Sugeno Model

Assume that the fuzzy inference system has two

inputs xandy and one output z. A first-order Sugeno fuzzy model has rules as the

following:

Rule1:If xis A1 andy is B1, then f1=p1x+q1y+r1

Rule2:

If xis A2 andy is B2, then f2=p2x+q2y+r2



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Sugeno Model - I

x

X

A1

A2

y

X

B1

B2

W1

W2

f1=p1x+q1y+r1 f2=p2x+q2y+r2 f=w1.f1+w2.f2

w1+w2

Y

Y



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ANFIS Architecture

A1

A2

x

y

B1

B2

Prod

Prod

Norm

Norm

Layer1 Layer2 Layer3

W1

W2

x y

x y

Sum

W1f1

W1f2

f

Layer4 Layer5



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Layer 1 - I

Ol,i is the output of the ith node of the layer l.

Every node iin this layer is an adaptive node witha node functionO1,i=Ai(x)fori= 1, 2,or

O1,i=Bi2(x)fori= 3, 4 x(ory) is the input node iandAi (or Bi2) is a

linguistic label associated with this node

Therefore O1,i is the membership grade of a fuzzyset(A1, A2, B1, B2).



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Layer 1 - II

Typical membership function:

A(x) = 1

1 +|xciai |2bi

ai, bi, ci is the parameter set. Parameters are referred to as premise parameters.



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Layer 2

Every node in this layer is a fixed node labeled

Prod. The output is the product of all the incoming

signals.

O2,i=wi=Ai(x)Bi(y), i= 1, 2 Each node represents the fire strength of the rule

Any other T-norm operator that perform the AN Doperator can be used



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Layer 3

Every node in this layer is a fixed node labeled

Norm. Theith node calculates the ratio of the ith rulets

firing strenght to the sum of all rulets firing

strengths. O3,i=wi =

wiw1+w2

, i= 1, 2

Outputs are called normalized firing strengths.



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Layer 4

Every node iin this layer is an adaptive node with

a node function:O4,1=wifi=wi(px+qiy+ri)

wi is the normalized firing strenght from layer 3. {pi, qi, ri}is the parameter set of this node.

These are referred to as consequent parameters.



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Layer 5

The single node in this layer is a fixed node

labeledsum, which computes the overall output asthe summation of all incoming signals:

overall output=O5,1=

iwifi=

iwifiiwi



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Alternative Structures

There are other structures

A1

A2

x

y

B1

B2

Prod

Prod

Layer1 Layer2 Layer3

W1

W2

x y

x y

Sum

W1f1

W1f2

f

Layer4 Layer5

W1f1+W2f2

/

Sum



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Learning Algorithm



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Hybrid Learning Algorithm - I

The ANFIS can be trained by a hybrid learning

algorithm presented by Jang in the chapter 8 ofthe book.

In the forward pass the algorithm uses

least-squares method to identify the consequentparameters on the layer 4.

In the backward pass the errors are propagated

backward and the premise parameters areupdated by gradient descent.



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Hybrid Learning Algorithm - II

Forward Pass Backward PassPremise Parameters Fixed Gradient Descent

Consequent Parameters Least-squares estimator Fixed

Signals Node outputs Error signals

Two passes in the hybrid learning algorithm for ANFIS.



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Universal Aproximator



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ANFIS is a Universal Aproximator

When the number of rules is not restricted, a

zero-order Sugeno model has unlimitedapproximation power for matching any nonlinearfunction arbitrarily well on a compact set.

This can be proved using the Stone-Weierstrasstheorem.

LetDbe a compact space of Ndimensions, andlet Fbe a set of continuous real-valued functionsonDsatisfying the following criteria:



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Stone-Weierstrauss theorem - I

LetDbe a compact space of Ndimensions, and

let Fbe a set of continuous real-valued functionsonDsatisfying the following criteria:

Indentity function: The constant f(x) = 1is in F.

Separability: For any two points x1=x2 inD, there is anf inFsuch that f(x1)=f(x2).

Algebraic closure: Iffandg are any two functions in F,then f g andaf+bg are in Ffor any two realnumbers aandb.



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Stone-Weierstrauss theorem - II

Then Fis dense on C(D), the set of continuous

real-valued functions onD. For any >0and any function g inC(D), there is a

function f inFsuch that|g(x)f(x)|< for allx

D.

The ANFIS satisfies all these requirements.



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Anfis and Matlab



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Matlab

It is possible to use a graphics user interface

Command anfisedit.

It is possible to use the command line interface orm-file programs.

There are functions to generate, train, test and use

these systems.



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ANFIS gui



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Applying

Initializing

Training Testing

Using



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Initializing - GENFIS1 - 1

FIS = GENFIS1(DATA)generates asingle-output Sugeno-type fuzzy inference system(FIS) using a grid partition on the data (noclustering).

FISis used to provide initial conditions forposterior ANFIS training.

DATAis a matrix with N+1 columns where the firstN columns contain data for each FISinput, andthe last column contains the output data.



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By default GENFIS1uses two gbellmftypemembership functions for each input.

Each rule generated has one output membershipfunction, which is of type linear by default.

It is possible to define these parameters usingFIS = GENFIS1(DATA, NUMMFS, INPUTMF,

OUTPUTMF)

fis = genfis1(data, [3 7],char(pimf, trimf));



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data = [rand(10,1) 10*rand(10,1)-5 rand(10,1)];

fis = genfis1(data, [3 7], char(pimf,trimf));

[x,mf] = plotmf(fis,input,1);

subplot(2,1,1), plot(x,mf);

xlabel(input 1 (pimf));

[x,mf] = plotmf(fis,input,2);subplot(2,1,2), plot(x,mf);

xlabel(input 2 (trimf));



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0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

input 1 (pimf)

2 1 0 1 2 3 40

0.2

0.4

0.6

0.8

1

input 2 (trimf)



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Initializing - GENFIS2

GENFIS2generates a Sugeno-type FIS usingsubtractive clustering.

GENFIS2extracts a set of rules that models thedata behavior.

The rule extraction method first determines thenumber of rules and antecedent membershipfunctions and then uses linear least squaresestimation to determine each rules consequentequations.



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Training

ANFISuses a hybrid learning algorithm to identifythe membership function parameters ofsingle-output, Sugeno type fuzzy inferencesystems (FIS).

There are many ways of using this function.

Some examples: [FIS,ERROR] = ANFIS(TRNDATA)

[FIS,ERROR] =ANFIS(TRNDATA,INITFIS)



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Using

EVALFISevaluates a fuzzy inference system.

Y = EVALFIS(U,FIS)simulates the FuzzyInference System FIS for the input data U andreturns the output data Y.



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Example

run exemplo06_03.m



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The End


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