## Appendix A
\begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Symbolic Definition of the BPANN**} \\ \hline Target vector of the BPANN & \(T_{k}=(y_{1},y_{2},\ldots,y_{q})\) \\ where \(q\) represents the number of target vectors of the BPANN & \\ \hline Input vector of the middle layer unit & \(S_{k}=(s_{1},s_{2},\ldots,s_{p})\) \\ Output vector of the middle layer unit & \(B_{k}=(b_{1},b_{2},\ldots,b_{p})\) \\ where \(p\) represents the number of vectors of the middle layer unit & \\ \hline Input vector of the output layer unit & \(L_{k}=(l_{1},l_{2},\ldots,l_{q})\) \\ Output vector of the output layer unit & \(C_{k}=(c_{1},c_{2},\ldots,c_{q})\) \\ where the number of vectors of the output layer unit is equal to that of the target vectors of the ANN \\ \hline Connection weight of the input layer to the middle layer & \(w_{ij}\), \(i=1,2,\ldots,n,j=1,2,\ldots,p\) \\ Connection weight of the middle layer to the output layer & \(v_{j\mu}\), \(j=1,2,\ldots,p\), \(t=1,2,\ldots,q\) \\ Output threshold of each unit in the middle layer & \(\theta_{j}\), \(j=1,2,\ldots,p\) \\ Output threshold of each unit in the output layer & \(\gamma_{t}\), \(t=1,2,\ldots,q\) \\ Parameter \(k\) & \(k=1,2,\ldots,m\) \\ where \(m\) represents the number of parameters \(k\) & \\ \hline \multicolumn{2}{c}{**Learning Steps of the BPANN**} \\ \hline Step & Description \\ \hline
1 & Initialize and subsequently assign a random value within the interval \((-1,1)\) to each \(w_{ij}\), \(v_{j\mu}\), \(\theta_{j}\) and \(\gamma_{t}\). \\ \hline
2 & Randomly select and apply a set of input and target vectors (i.e., \(P_{k}\) and \(T_{k}\)) to the network. \\ \hline
3 & Calculate the input and output value \(s_{j}\) and \(b_{j}\) of each unit in the middle layer by the \\  & transfer function according to the formula \(\left\{\begin{array}{cc}s_{j}=\sum\limits_{i=1}^{n}w_{ij}a_{i}-\theta_{j}& \\ b_{j}=f(s_{j})&\\ \end{array}\right.\)\(j=1,2,\ldots,p\). \\ \hline
4 & Calculate the input and output value \(l_{t}\) and \(c_{t}\) of each unit in the output layer by the \\  & transfer function according to the formula \(\left\{\begin{array}{cc}l_{t}=\sum\limits_{j=1}^{p}v_{j\mu}a_{t}-r_{t}&\\ c_{t}=f(l_{t})&\\ \end{array}\right.\)\(t=1,2,\ldots,q\). \\ \hline
5 & Calculate the generalized error of each unit of the output layer \(d_{t}^{k}\) according to the formula \(d_{t}^{k}=(y_{t}^{k}-c_{t})\times c_{t}(1-c_{t})\)\(t=1,2,\ldots,q\). \\ \hline
6 & Calculate the generalized error of each unit of the middle layer \(c_{j}^{k}\) according to the formula \(c_{j}^{k}=\left[\sum\limits_{l=1}^{q}d_{t}\times\gamma_{j\mu}\right]\times b_{ j}(1-b_{j})\)\(t=1,2,\ldots,q\). \\ \hline
7 & Modify the connection weights \(v_{\mu}\) and the thresholds \(\gamma_{t}\) according to formulas \\  & \(\left\{\begin{array}{cc}v_{j\mu}(N+1)=v_{j\mu}(N)+\alpha\times d_{t}^{k} \times b_{j}&\\ \gamma_{t}(N+1)=\gamma_{t}(N)+\alpha\times d_{t}^{k}&\\ \end{array}\right.\)\(t=1,2,\ldots,q,\)\(j=1,2,\ldots,p,\)\(0\prec\alpha\prec 1\). \\ \hline
8 & Modify the connection weights \(w_{ij}\) and the thresholds \(\theta_{j}\) according to formulas \\  & \(\left\{\begin{array}{cc}w_{ij}(N+1)=w_{ij}(N)+\beta\times c_{j}^{k}\times d_{j }^{k}&\\ \theta_{j}(N+1)=\theta_{j}(N)+\beta\times c_{j}^{k}&\\ \end{array}\right.\)\(i=1,2,\ldots,n,\)\(j=1,2,\ldots,p,\)\(0\prec\beta\prec 1\). \\ \hline
9 & Randomly select and apply the next learning sample vectors to the network and return to Step 3 until the completion of \(m\) training samples. \\ \hline
10 & Re-select a set of input and target sample vectors from \(m\) study samples and return to Step 3 until the network global error \(E\) is less than a pre-set minimum value, i.e., the network \\  & converges. Go to Step 11. If the number of training steps is greater than the pre-set value, the network cannot converge. \\ \hline
11 & End. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: \(Cont\).