In this example, we want to approximate the following scatter plot with a single layer neural network. Blue points are the training set given by an input \( x_i \) and an expected output \( y′_i \). The red line is the output of the network \( y=f(x) \) after training.
The following perceptron will be used for the single layer network:
\( x \) is the input. The activation function is given by \( f(x)=x \).
As explained on the previous page, the weights will be updated according to this formula:
$$ w_i'= w_i + \eta(y'-y)x_i $$
Let's detail for each weight \( w_1 \) and \( w_2 \):
$$ w_1'= w_1 + \eta(y'-y)x $$ $$ w_2'= w_2 + \eta(y'-y) $$