Adriano Cruz Mestrado NCE, IM, UFRJ

Logica Nebulosa – p. 1/3

Summary

• • • • •

Introduction ANFIS Architecture Hybrid Learning Algorithm ANFIS as a Universal Approximatior Simulation Examples

Logica Nebulosa – p. 2/3

Introduction

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ANFIS: Artiﬁcial Neuro-Fuzzy Inference Systems ANFIS are a class of adaptive networks that are funcionally equivalent to fuzzy inference systems. ANFIS represent Sugeno e Tsukamoto fuzzy models. ANFIS uses a hybrid learning algorithm

Logica Nebulosa – p. 3/3

Sugeno Model

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Assume that the fuzzy inference system has two inputs x and y and one output z . A ﬁrst-order Sugeno fuzzy model has rules as the following: Rule1: If x is A1 and y is B1 , then f1 = p1 x + q1 y + r1 Rule2: If x is A2 and y is B2 , then f2 = p2 x + q2 y + r2

Logica Nebulosa – p. 4/3

Sugeno Model - I

A1

B1

W1 X A2 Y

B2

W2 X x f1=p1x+q1y+r1 f2=p2x+q2y+r2 f= Y y w1.f1+w2.f2 w1+w2

Logica Nebulosa – p. 5/3

ANFIS Architecture

Layer1 Layer2 Layer3 Layer4 x W1 Prod A2 W2 Prod B1 Norm x y W1f2 Norm f Sum y W1f1 Layer5

A1 x

y B2

Logica Nebulosa – p. 6/3

Layer 1 - I

• Ol,i •

is the output of the ith node of the layer l.

Every node i in this layer is an adaptive node with a node function O1,i = µAi (x) for i = 1, 2, or O1,i = µBi−2 (x) for i = 3, 4 (or y ) is the input node i and Ai (or Bi−2 ) is a linguistic label associated with this node Therefore O1,i is the membership grade of a fuzzy set (A1 , A2 , B1 , B2 ).

• x •

Logica Nebulosa – p. 7/3

Layer 1 - II

•

Typical membership function:

µA (x) = 1 1 + | x−ci |2bi ai

• ai , bi , ci •

is the parameter set.

Parameters are referred to as premise parameters.

Logica Nebulosa – p. 8/3

Layer 2

• •

Every node in this layer is a ﬁxed node labeled Prod. The output is the product of all the incoming signals. Each node represents the ﬁre strength of the rule Any other T-norm operator that perform the AN D operator can be used

• O2,i = wi = µAi (x) · µBi (y), i = 1, 2 • •

Logica Nebulosa – p. 9/3

Layer 3

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Every node in this layer is a ﬁxed node labeled Norm. The ith node calculates the ratio of the ith rulet’s ﬁring strenght to the sum of all rulet’s ﬁring strengths. Outputs are called normalized ﬁring strengths.

• O3,i = w i = wi , i = 1, 2 w1 +w2 •

Logica Nebulosa – p. 10/3

Layer 4

•

Every node i in this layer is an adaptive node with a node function: O4,1 = wi fi = w i (px + qi y + ri )

• wi

is the normalized ﬁring strenght from layer 3. is the parameter set of this node.

• {pi , qi , ri } •

These are referred to as consequent parameters.

Logica Nebulosa – p. 11/3

Layer 5

•

The single node in this layer is a ﬁxed node labeled sum, which computes the overall output as the summation of all incoming signals: i w i fi

• overall output = O5,1 =

=

i

wi fi i wi

Logica Nebulosa – p. 12/3

Alternative Structures

•

There are other structures

Layer1 Layer2 Layer3 x W1 y W1f1 W1f1+W2f2 A2 W2 Layer4 Layer5

A1 x

Prod

Sum

Prod

W1f2 x y

/

f

B1

y B2

Sum

Logica Nebulosa – p. 13/3

Learning Algorithm

Logica Nebulosa – p. 14/3

Hybrid Learning Algorithm - I

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The ANFIS can be trained by a hybrid learning algorithm presented by Jang in the chapter 8 of the book. In the forward pass the algorithm uses least-squares method to identify the consequent parameters on the layer 4. In the backward pass the errors are propagated backward and the premise parameters are updated by gradient descent.

•

•

Logica Nebulosa – p. 15/3

Hybrid Learning Algorithm - II

Forward Pass Premise Parameters Consequent Parameters Signals Fixed Least-squares estimator Node outputs

Backward Pass Gradient Descent Fixed Error signals

Two passes in the hybrid learning algorithm for ANFIS.

Logica Nebulosa – p. 16/3

Universal...