Nfis: Adaptive Neuro-Fuzzy Inference
ANFIS: Adaptive Neuro-Fuzzy Inference Systems
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
• • • •
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
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
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
• • • •
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
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