Q1)2 input 4x4 Kohonen is given below network. Next to it, are the current connection weights from the  input layer to the neurons at the Kohonen layer .

The weight update rule is given by the formula

where w_j(t) is the connection weight vector of the j^th neuron at Kohonen layer,  h(t) is the current learning rate and N(j,t) is the neighbourhood function, whose value is determined according to the position of the neuron j with respect to the winner. The current value of the neighbourhood function around the winner is given above, where N(j,t)=1 is corresponding to the winner and the others are the 8 neurons around it. On the graph given below each Kohonen neuron is positioned on the w₁,w₂  space (equivalently u₁,u₂ space) and they are connected to its four neighbouring neurons.

a)      If u=[14,14]^T  is applied at the input (which is marked as x on the graph given below) determine which one is the winner:

b)      Assuming that current learning rate is h(t) is 0.5, find out the new connection weights, and then mark them below together with the lines connecting to the four neighbours

c)      Explain briefly what you observed  with this weight update.

d)      If the inputs are uniformly distributed within the rectangle 0£w₁£40, 0£w₂£40 and the learning rate is decreased very slowly and the neighborhood function is sharpened accordingly, what would be the final positions of the neurons on the weight space. Show also the regions to be clustered by each neuron.

e)      If always the input u=[14,14] instead of the inputs explained in (d) was applied at the input, what would happen?