ANFIS Adaptive Network Fuzzy Inference system G Anuradha

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” Linguistic variables 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: Rule 1: If x is A 1 and y is B 1, then f 1 = p 1 x + q 1 y + r 1 Rule 2: If x is A 2 and y is B 2, then f 2 = p 2 x + q 2 y + r 2

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 inputoutput 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 Ho. F is linear in some of 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 X 0=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