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Adaptive Neuro Fuzzy Inferencing System

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					               Adaptive Neuro Fuzzy Inferencing System (ANFIS)

An Adaptive Neuro Fuzzy Inferencing System (ANFIS) is an adaptive network that learns the rules
and membership functions from the given data. It consists of a network of nodes and directional
links. The adaptive part of the name is in place as some or all of the nodes have parameters that
affect the output.




ANFIS uses a two pass learning algorithm which consists of a forward pass and a backward pass
through the network. Below is a look at the forward pass at each layer of the network in turn. The
backward pass sends the error back through the network in a similar way to back propagation.
Backpropagation is an example of a learning rule. For simplicity it is assumed that the ANFIS has
two inputs and one output:




Layer 1 - Premise Parameters


Every node (i) in this layer is an adaptive node with a node function.
The O,i(x) is essentially the membership grade of the fuzzy sets A and B and it specifies the degree to
which x or y satisfy the quantifier.


Layer 2 - T-norm Operator : Each node output represents the firing strength of a rule.




The T-norm operator that performs fuzzy AND for j = 1,2,...,n number of inputs




Layer 3 - Outputs of layer 3 are normalized firing strengths




The i th node calculates the ratio of the i th rule's firing strength to the sum of all the rules' firing
strength.


Layer 4 - Consequent Parameters


Every node (i) in this layer is an adaptive node with a node function.




p(i) q(i) r(i) is the parameter set for the output of Layer 3.


Layer 5 - Crisp Output


The single node ∑ computes the overall output as the summation of all incoming signals.




                                                      2
Example Layer 1


Question: For x = 3 and y = 4 find the crisp output of the Sugeno fuzzy system.


For simplicity it is assumed that the ANFIS has two inputs and one output:




The membership functions could be anything but the following bell shaped function will be used,
given by:




where ai ,bi ,ci are parameters to be learnt (the premise parameters). A1, A2 and B1, B2 are fuzzy sets.




All the x's are changed to the value 3 and all the y's are changed to the value 4. Then the calculations
can take place to give the output of Layer 1



                                                   3
                                           Example Layer 2


     The outputs of layer 1 are taken and put into place for layer 2 to continue the calculations.




Example Layer 3


The outputs of layer 2 are taken and put into place for layer 3 to continue the calculations.




                                                   4
                                       Example Layer 4


 The outputs of layer 3 are taken and put into place for layer 4 to continue the calculations.




                                       Example Layer 5


Finally the outputs of layer 4 are taken and put into place for layer 5 to find the answer to the
                                       original question.




       Therefore if x = 3 and y = 4 the crisp output of the Sugeno fuzzy system is 5.23


                              Try yourself with different values.




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