1. ANFIS (Adaptive Network Fuzzy Inference system)
G.Anuradha

2. 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

3. 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

4. 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

5. 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

6. Fuzzy Inference system

7. Blocks of FIS

8. Steps of fuzzy reasoning

9. Types of fuzzy reasoning

10. 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

11. Adaptive Networks – Architecture and Learning
Has parameters
Has no parameters

12. 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

13. 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

14. 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

15. 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

16. 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

19. 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.

20. 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 some
of the elements of S, then these elements can be identified by the least square
Method.

21. 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

22. 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