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:
SH1 = (0.2 * 10) + (-0.1 * 30) + (0.4 * 20) = 2 -3 + 8 = 7.
After then the σ function is applied to SH1 to give as:
σ(SH1) = 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:
SH2 = (0.7 * 10) + (-1.2 * 30) + (1.2 * 20) = 7 - 36 + 24 = -5
Now next σ applied to SH2 gives as:
σ(SH2) = 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.