 # Python Tutorial Neural Networks With Backpropagation For XOR Using One – Sem Seo 4 You

In the picture, we used the following definitions for the notations:

Here are the computations represented by the NN picture above:

\$\$ a_0^{(2)} = g(Theta_{00}^{(1)}x_0 + Theta_{01}^{(1)}x_1 + Theta_{02}^{(1)}x_2) = g(Theta_0^Tx) = g(z_0^{(2)}) \$\$ \$\$ a_1^{(2)} = g(Theta_{10}^{(1)}x_0 + Theta_{11}^{(1)}x_1 + Theta_{12}^{(1)}x_2) = g(Theta_1^Tx) = g(z_1^{(2)}) \$\$ \$\$ a_2^{(2)} = g(Theta_{20}^{(1)}x_0 + Theta_{21}^{(1)}x_1 + Theta_{22}^{(1)}x_2) = g(Theta_2^Tx) = g(z_2^{(2)}) \$\$ \$\$ h_Theta(x) = a_1^{(3)}=g(Theta_{10}^{(2)}a_0^{(2)} + Theta_{11}^{(2)}a_1^{(2)} + Theta_{12}^{(2)}a_2^{(2)}) \$\$

In the equations, the \$g\$ is sigmoid function that refers to the special case of the logistic function and defined by the formula:

\$\$ g(z) = frac{1}{1+e^{-z}} \$\$

Sigmoid functions

One of the reasons to use the sigmoid function (also called the logistic function) is it was the first one to be used. Its derivative has a very good property. In a lot of weight update algorithms, we need to know a derivative (sometimes even higher order derivatives). These can all be expressed as products of \$f\$ and \$1-f\$. In fact, it’s the only class of functions that satisfies \$f^{‘}(t)=f(t)(1-f(t))\$.

However, usually the weights are much more important than the particular function chosen. These sigmoid functions are very similar, and the output differences are small. Here’s a plot from Wikipedia-Sigmoid function. Note that all functions are normalized in such a way that their slope at the origin is 1.

Forward Propagation

If we use matrix notation, the equations of the previous section become:

\$\$ x = begin{bmatrix} x_0 \ x_1 \ x_2 \ end{bmatrix} z^{(2)} = begin{bmatrix} z_0^{(2)} \ z_1^{(2)} \ z_2^{(2)} \ end{bmatrix} \$\$ \$\$ z^{(2)} = Theta^{(1)}x = Theta^{(1)}a^{(1)} \$\$ \$\$ a^{(2)} = g(z^{(2)}) \$\$ \$\$ a_0^{(2)} = 1.0 \$\$ \$\$ z^{(3)} = Theta^{(2)}a^{(2)} \$\$ \$\$ h_Theta(x) = a^{(3)} = g(z^{(3)}) \$\$

The backpropagation learning algorithm can be divided into two phases: propagation and weight update.
– from wiki – Backpropagatio.

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