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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