

![0_image_0.png](0_image_0.png)

Now we can combine hyperplanes f1/2, f1/3 and f2/3 to build up the new dividing hyperplanes g1, g2, and g3. The first hyperplane g1 combines the functions f1/2 and f1/3 so that g1 = f1/2 + f1/3. These functions are taken with weights of 1.0 because both functions f1/2 and f1/3 give the positive output values on the examples of class 1. Likewise, the second and third hyperplanes are as follows: g2 = f2/3 - f1/2 and g3 = - f1/3 - f2/3.

In practice each of hyperplanes g1, …, gC, can be implemented as a two-layer feedforward ANN with a given number of hidden neurons fully connected to the input nodes.

Then we can introduce the output neuron summing all outputs of the ANNs to make a final decision.

For example, the pairwise neural-network system depicted in Fig 3 consists of three neural networks performing the functions f1/2, f1/3, and f2/3. The three output neurons g1, g2, and g3 are connected to these networks with weights equal to (+1, +1), (–1, +1) and (–1, –
1), respectively.

![0_image_1.png](0_image_1.png)

In general, a pairwise neural-network system consists of C(C - 1)/2 neural networks, performing functions f1/2, …, fi/j, …, fC - 1/C, and C output neurons g1, …, gC, where i < j = 2, …, C. We can see that the weights of output neuron gi connected to the hidden neurons fi/k and fk/i should be equal to + 1 and –1, respectively.