ANFIS(Adaptive Network Fuzzy

Inference system)G.Anuradha

Introduction

• Conventional mathematical tools are quantitative in nature

• They are not well suited for uncertain problems• FIS on the other hand can model qualitative

aspects without employing precise quantitative analyses.

• Though FIS has more practical applications it lack behind– Standard methods for transformation into rule base– Effective methods for tuning MFs for better

performance index

So……

ANFIS serve as a basis for constructing a set of fuzzy if-then rules with appropriate membership functions to generate the stipulated input-output pairs

Fuzzy if-then rules and Fuzzy Inference systems

• Fuzzy if-then rules are of the form IF A THEN B where A and B are labels of fuzzy sets.

• Example – “if pressure is high then volume is small”

Linguisticvariables

Linguistic values

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 = p1x + q1y + r1

Rule2:

If x is A2 and y is B2, then f2 = p2x + q2y + r2

Fuzzy Inference system

Blocks of FIS

Steps of fuzzy reasoning

Types of fuzzy reasoning

• Type 1: The overall output is the weighted average of each rule’s firing strength and output membership functions.

• Type 2: The overall output is derived by applying the “max” operation to the qualified fuzzy outputs. The final crisp output can be obtained using some defuzzification methods

• Type 3: Takegi and Sugeno fuzzy if-then rules are used. The output of each rule is a linear combination of input variables plus a constant term and the final output is the weighted average of each rule’s output

Adaptive Networks – Architecture and Learning

Has parameters

Has no parameters

Adaptive Networks – Architecture and Learning

• Superset of all feedforward NN with supervised learning capability

• Has nodes and directional links connecting different nodes

• Part or all the nodes are adaptive(each output of these nodes depends on parameters pertaining to this node) and learning rule specifies how these parameters should be changed to minimize a error measure

Learning rule

• The basic learning rule is gradient descent and chain rule

• Because of the problem of slowness and being trapped in local minima a hybrid learning rule is proposed

• This learning rule comes in two modes– Batch learning– Pattern learning

Architecture and basic learning

• An adaptive network is a multi-layer feedforward network in which each node performs a particular function on the incoming signals

• The nature and the choice of the node function depends on the overall input-output function

• No weights are associated with links and the links just indicate the flow

Architecture and basic learning Contd…

• To achieve desired i/p-o/p mapping the parameters are updated according to training data and gradient-based learning procedure

Gradient based learning procedure• Given adaptive network has L layers

• k-th layer has #k nodes

• (k,i)- ith node in the kth layer

Node function- ith node in the k-layer

Node output depends on its incoming signals and its parameter set and a,b,c etc. are parameters pertaining to this node

Learning paradigms for Adaptive networks

• Batch learning:-Update action takes place only after the whole training data set has been presented(After an epoch)

• On-line learning:-parameters are updated immediately after each input-output pair has been presented.

Hybrid Learning Rule-Batch-Off line learning rule

• Combines gradient method and least square estimator to identify parameters

Where I is a set of input variables and S is the set of parameters

If there exists a function H such that the composite function HoF is linear in someof the elements of S, then these elements can be identified by the least square Method.

• Using least square estimator we have

For systems with changing characteristics, X can be iteratively calculated with the formulae given below. Usually used for online version

Si is the covariance matrix. The initial conditions to the equation are X0=0 and where is a positive large number and I is the identity matrix

ANFIS(Adaptive Network based fuzzy

inference system)

• It is functionally equivalent to FIS

• It has minimum constraints so very popular

• It should be feedforward and piecewise differentiable