Q1) The ramp and  sigmoid functions are shown in the following figure.

One of the usages of neural networks is function approximation We want to approximate the following function using a neural network

a) For this purpose, first show that the above function can be approximated as a summation of four ramp functions.

a) Express each of them in the form a_i(Ramp(b_iu+q_i)) i=1..4, and draw them .

b) Then construct a single input, single output neural network using four neurons with  output function f(a)=Ramp(a) in the first layer, and one neuron with f(a)=a in the output layer. Indicate clearly the connections and the values of connection weights and q for each neuron in the network.

c) Draw the function generated a neural network, having the same structure, connection weights and q values, but Sig(a) is used instead of Ramp(a) in the first layer.

Solution

     x =1.25 ramp(0.5u+1.25) - 1.25 ramp(0.5u-1.5) + 0.5 ramp(2u-1.5) -0.5 ramp (2u-2)

b)