## 27 January 2009

### ANFIS

Adaptive network-based fuzzy inference system used a feed forward network to search for fuzzy decision rules that perform well on a given task. Using a given input-output data set ANFIS creates a fuzzy inference system whose membership function parameters are adjusted using a backpropagation algorithm alone or combination between a backpropagation algorithm with a least squares method. This allows the fuzzy systems to learn from the data being modeled. ANFIS provide a method for the fuzzy modeling procedure to learn information from the data set, followed by creating the membership function parameters that best performing the given task. Consider a first order Takagi-Sugeno fuzzy model with a two input, one output system having two membership functions for each input.
The functioning of ANFIS is a five layered feed forward neural structure and the functionality of the nodes in these layers can be summarized as follows:
Layer 1: Every node i in this layer is an adaptive node with a node output defined by:

Picture: Equation(1)

where x(or y) is the input to the node; Ai (or Bi-2) is a fuzzy set associated with this node, characterized by the shape of the membership function in this node and can be any appropriate functions that are continuous and piecewise differentiable such as Gaussian , generalized bell shaped, trapezoidal shaped and triangular shaped functions. Assuming a bell shaped function as the membership function, Ai can be computed as,

Picture: Equation (2)

ai and ci are the parameter set. Parameters in this layer are referred to as premise (antecedent) parameters.
Layer 2: Every node in this layer is a fixed node labeled Π , which multiplies the incoming signals and outputs the product. For instance,

Picture: Equation(3)

Each node output represents the firing strength of a rule.
Layer 3: Every node in this layer is a circle node labeled N. The ith node calculates the ratio of the ith rule's firing strength to the sum of all rule's firing strengths. Output of this layer will be called normalized firing strengths.

Picture: Equation(4)

Layer 4: Node i in this layer compute the contribution of the ith rule towards the model output, with the following node functions:

Picture: Equation(5)

Layer 5: The single node in this layer is a fixed node labeled that computes the overall output as the summation of all incoming signals.
Overall output =

Picture : Equation (6)

The parameters are adjusted using a learning rule :

delta(p_i) = - (eta) d(e_T)/d(p_i) , i = 1, 2, 3, ....... ; /eta/ < 1,
e_T = sum square error

and data learning.