## 4 Methodology  
### Levenberg-Marquardt Neural Network  
The Levenberg-Marquardt neural network (LMNN) is a fast learning neural network. As shown in Figure 4, LMNN (pointed out as LM) outperforms close competitors such as the scaled conjugate gradient (SCG) back propagation algorithm and the one-step secant backpropagation (OSS) algorithm. Particularly, this figure evaluates the convergence speed on the sensor data, and tries to demonstrate the learning efficiency of the diverse strategies, justifying the considered selection. Additionally, the OSS algorithm required less storage capacity for computation, and it did not store the complete Hessian matrix. However, it needed more storage capacity for computations in terms of epochs when compared to SCG algorithm or LM. Hence, LMNN is capable of faster processing and computation than its close competitors, and thus, it was selected in this work.  
The Hessian matrix is calculated as follows:  
\[H\ \approx\ J^{T}\ J\ +\ \mu I \tag{11}\]  
where \(H\) is the Hessian matrix, \(J\) is the Jacobian matrix and \(\mu\) represents the combination coefficient. Equation (12) represents updated Levenberg-Marquardt algorithm.  
\[\omega_{k+1}\ =\ \omega_{k}\ (\int_{k}^{T}\ +\mu I\ )^{-1}\ J_{k}\ e_{k} \tag{12}\]  
The Algorithm \(1\) for LMNN is presented as follows.  
**Input:**\(x_{0}\in\mathbb{R}^{n},\mu>0,\gamma>0,\tilde{\xi}>0,\epsilon>0,\sigma_{1},\sigma_{2 }>0,r,\rho\in(0,1)\)  
**Step 0** Set \(k:=0\).  
**Step 1** Compute \(F_{k}+F(X_{k})\) and \(J_{k}+J(X_{k})\).  
If \(\parallel_{k}^{T}F_{k}\parallel\leq\epsilon\), stop. Otherwise \(\lambda_{k}\) by Equation (13)  
\[\lambda_{k}=\mu\parallel F_{k}\parallel^{kk}\text{ with }\delta_{k}=\left\{ \begin{array}{ll}\frac{1}{\parallel F_{k}\parallel}&\parallel F_{k} \parallel\geq 1,\\ 1&\text{otherwise}\end{array}\right. \tag{13}\]  
**Step 2** (a) Obtain \(d_{1k}\) by solving the following linear system  
\[(J_{k}^{T}J_{k}+\lambda_{k}I)d=-J_{k}^{T}F_{k}, \tag{14}\]  
(b) Solve the linear system  
\[(J_{k}^{T}J_{k}+\lambda_{k}I)d=-J_{k}^{T}F(y_{k}), \tag{15}\]  
to obtain \(d_{2k}\), where \(y_{k}=x_{k}+d_{1k}\).  
(c) Solve the linear system  
\[(J_{k}^{T}J_{k}+\lambda_{k}I)d=-J_{k}^{T}F(z_{k}), \tag{16}\]  
to obtain \(d_{3k}\), where \(z_{k}=y_{k}+d_{2k}\).  
(d) Set \(d_{k}=d_{1k}+d_{2k}+d_{3k}\)  
**Step 3** If  
\[\parallel F(x_{k}+d_{k})\parallel\leq\rho\parallel F_{k}\parallel, \tag{17}\]  
then take \(\alpha_{k}\) and go to Step 5. Otherwise go to step 4.  
**Step 4** Set  
\[d_{k}=\left\{\begin{array}{ll}d_{1k}+d_{2k}+d_{3k}&F_{k}^{T}J_{k}(d_{1k}+d_ {2k}+d_{3k})\leq-\gamma\\ d_{1k}&\text{otherwise}.\end{array}\right. \tag{18}\]  
Compute \(\alpha_{k}=max\{1,r^{1},r^{2},...\}\) with \(\alpha=r^{i}\) satisfying  
\[\parallel F(x_{k}+\alpha d_{k})\parallel^{2}\leq(1+\epsilon_{k})\parallel F_ {k}\parallel^{2}-\sigma_{1}\alpha^{2}\parallel d_{k}\parallel^{2}-\sigma_{2 }\alpha^{2}\parallel F_{k}\parallel^{2}, \tag{19}\]  
where the positive sequence \(\epsilon_{k}\).  
**Step 5** Set \(x_{k+1}=x_{k}+\alpha_{k}d_{k}\). Set \(k=k+1\) and goto step \(1\).  
**Algorithm 1** The Levenberg-Marquardt algorithm  
Figure 4: A comparison among LM, OSS and SCG.