Question: Express the network function $\Phi$ in Figure 1.15 in terms of the primivitve functions $f_1, \ldots , f_4$ and of the weights $\alpha_1, \ldots , \alpha_5$.

Solution.

The capital F is used to denote the output of each function to reduce ambiguity:

\( F_1 = f_1 ( y ) \)
\( F_2 = f_2 ( x + \alpha_1 \cdot F_1 ) \)
\( F_3 = f_3 ( z + \alpha_2 \cdot F_1 + \alpha_3 \cdot F_2 ) \)
\( F_4 = f_4 ( \alpha_4 \cdot F_2 + \alpha_5 \cdot F_3 ) \)

\begin{equation} \Phi = F_4 \end{equation}

\begin{equation} f_4 ( \alpha_4 F_2 + \alpha_5 F_3 ) \end{equation}

\begin{equation} f_4 ( \alpha_4 F_2 + \alpha_5 f_3 ( z + \alpha_2 F_1 + \alpha_3 F_2 ) ) \end{equation}

\begin{equation} f_4 ( \alpha_4 f_2 ( x + \alpha_1 F_1 ) + \alpha_5 f_3 ( z + \alpha_2 F_1 + \alpha_3 f_2 ( x + \alpha_1 F_1 ) ) ) \end{equation}

\begin{equation} f_4 ( \alpha_4 f_2 ( x + \alpha_1 f_1 ( y ) ) + \alpha_5 f_3 ( z + \alpha_2 f_1 ( y ) + \alpha_3 f_2 ( x + \alpha_1 f_1 ( y ) ) ) ) \end{equation}