Nfis: Adaptive Neuro-Fuzzy Inference

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ANFIS: Adaptive Neuro-Fuzzy Inference Systems
Adriano Cruz Mestrado NCE, IM, UFRJ

Logica Nebulosa – p. 1/3

• • • • •

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

Logica Nebulosa – p. 2/3

• • • •

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
• • • •

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



W1 X A2 Y


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
• •

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 =



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


W1f2 x y




y B2


Logica Nebulosa – p. 13/3

Learning Algorithm

Logica Nebulosa – p. 14/3

Hybrid Learning Algorithm - I

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

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