solution of multi-layer ann with sigmoid

Question

Solution of multi-layer ann with sigmoid units:

Assume here that we input the values 10, 30, 20 with the three input units and from top to bottom. So after then the weighted sum coming into H₁ will be as: 

S_H1 = (0.2 * 10) + (-0.1 * 30) + (0.4 * 20) = 2 -3 + 8 = 7. 

After then the σ function is applied to S_H1 to give as: 

σ(S_H1) = 1/(1+e-7) = 1/(1+0.000912) = 0.999 

Here don't forget to negate S. Also the weighted sum coming into H₂ will be as: 

S_H2 = (0.7 * 10) + (-1.2 * 30) + (1.2 * 20) = 7 - 36 + 24 = -5 

Now next σ applied to S_H2 gives as: 

σ(S_H2) = 1/(1+e5) = 1/(1+148.4) = 0.0067 

Furthermore from this we can see there that H₁ has fired, but H₂ has not. So now we can calculate the weights sum going in to output in unit O1 will be as: 

SO₁ = (1.1 * 0.999) + (0.1*0.0067) = 1.0996 

And here next the weighted sum going in to output in unit O₂ will be as: 

SO₂ = (3.1 * 0.999) + (1.17*0.0067) = 3.1047 

However the output sigmoid unit in O₁ will now calculate the output values from the network for O₁ as:

 σ(SO₁) = 1/(1+e^-1.0996) = 1/(1+0.333) = 0.750 

and the output from the network for O2: 

σ(SO₂) = 1/(1+e^-3.1047) = 1/(1+0.045) = 0.957 

However therefore if this network represented the learned rules for a categorisation problem then the input triple (10,30,20) would be categorised into the category associated with O2 it means it has the larger output.