Radial Basis Function Network(RBFN)

Radial basis function network can be used for approximating function and recognizing patterns. It uses Gaussian Potential functions.

Architecture

The architecture radial basis function network consists of three layers, the input,hidden and the output layer as shown in below figure. The architecture of RBFN is a multi layer feed forward network. There exist ‘n’ number of input neuron and ‘m’ number of output neurons with the hidden layer existing between the input and output layer. The interconnection between the input layer and hidden layer forms hypothetical connection and between the hidden and output layer forms weighted connections. The training algorithm is used for updation of weights in all the interconnections.

Training algorithm for RBFN

The training algorithm for the radial basis function network is given below.  Radial basis function uses Gaussian activation function. The response of such function is non-negative for all value of x. The function is defined as, f(x)=exp(-x²)

Step 1: Initialize the weights (set to small random values)

Step 2: While stopping is false do step 3-10

Step 3: For each input do steps 4 – 9

Step 4: Each input unit (x_i ,i=1,2.....n) receives input signal to all unit in the layer above(hidden unit)

Step 5: Calculate the radial basis function

Step 6: Choose the centres for the radial basis functions. The centres are chosen from the  set of input vectors. A sufficient number of centers have to be selected in order to ensure adequate sampling of the input vector space.

Step 7: The output of i_m unit v_i(x_i) in the hidden layer.      

$$ v_i(x_i)=exp(-\sum_{j=1}^r[{{x_{ji}-\bar{x}_{ji}]^2}\over{\sigma^2_i}}) $$
Where $\bar x_{ji}$ = Centre of the RBF unit for input variables
$\sigma _i$ = Width of the RBF unit
$x_{ji}$ = $j_{th}$ variable of unit pattern

Step 8: Initialize the weights in the output layer of the network to some small random value.

Step 9: Calculate the output of the neural network
$$ y_{net}=e\sum_{i=1}^Hw_{im}v_i(x_i)+w_0$$
Where, H = Number of hidden layer nodes(RBF function)
$y_{net}$ = Output value of the $m_{th}$ mode in output layer for the $n_{th}$ incoming pattern.
$w_{im}$ = Weight between  $i_{th}$ RBF unit and  $m_{th}$ output node
$w_0$ = Biasing term at $n_{th}$ output node

Step 10: calculate error and test stopping condition.

The stopping condition may be the weight change, number of epochs, etc.