## 4 Comparison of Load Spectrum
### Comparison of Methods  
In order to better evaluate the load extrapolation effect based on the LSTM model. We use five sets of unused extrusion cycle data to compare the load spectra under the two methods. For the rain flow extrapolation method, we extract the load amplitude and mean value from the original data. Through the joint distribution function of load amplitude and mean value, the times of original load data in each load area can be calculated. The frequency calculation formula is:  
\[N_{ij}=N_{w}\int_{S_{\alpha_{i}}}^{S_{\alpha_{i+1}}}\int_{S_{\alpha_{i}}}^{S_{ \alpha_{i+1}}}f(x,t)dxdt \tag{12}\]  
where the total cumulative frequency of load obtained from the data set is \(N_{w}\), \(S_{\alpha_{i}}\) and \(S_{\alpha_{i+1}}\) is the upper and lower limit of each group's load amplitude, \(S_{m_{j}}\) and \(S_{m_{j+1}}\) is the upper and lower limit of the average load of each group, \(N_{ij}\) is the frequency of load amplitude falling in interval \(i\) and load mean falling in interval \(j\). To obtain the load spectrum of the extruder in a specific extrusion cycle on this basis, the total cumulative frequency needs to be divided by the corresponding number of extrusion cycles. The expression for averaging the above load frequency to \(M\) extrusion cycles is:  
\[N^{*}{}_{ij}=\frac{N_{ij}}{M} \tag{13}\]  
The data of 25 extrusion cycles are drawn by the rainflow counting method to obtain the statistical histogram of the mean and amplitude, and then the statistical distribution function is fitted. The fitting results of load amplitude and mean are shown in Figure 9. The parameter estimation results of the load amplitude and the mean value show that the correlation coefficient of the amplitude following the Weibull distribution is 0.92, and the correlation coefficient of the mean following the normal distribution is 0.97, which further confirms the distribution fitting results of the load amplitude and the mean. The short-term load spectrum compiled by the frequency calculated by the joint distribution function averaged to five extrusion cycles is shown in Table 3. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Load Type** & \multicolumn{6}{c}{**Scale Factor of Load Class Division**} \\ \hline Amplitude & 1 & 0.95 & 0.85 & 0.725 & 0.575 & 0.425 & 0.275 & 0.125 \\ Mean & 1 & 0.875 & 0.75 & 0.625 & 0.5 & 0.375 & 0.25 & 0.125 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Scale factor for the grade division of load cycle amplitude and mean. Figure 9: Distribution fitting of load amplitude and mean. For the time domain extrapolation based on LSTM model, the trained model of 25 groups of original extrusion cycle data can be directly predicted. This method does not need artificial calculation of multiple links, and the load data about time series can be obtained. The predicted data and real data of five groups of extrusion cycles are counted by rain flow counting method. The eight-level load spectra are shown in Tables 4 and 5, respectively. The frequency distribution of the eight-level load spectrum drawn based on the three types of data are shown in Figure 10a-c. It is found by comparing the short-term load spectrum compiled based on the rain flow extrapolation method and LSTM with the actual load spectrum. For the frequency distribution of load amplitude and mean value at all levels, the results predicted by the LSTM model are more consistent with the actual situation than the rain flow extrapolation method. Because the load spectrum compiled by the rain flow extrapolation method is only a statistical analysis of the original data and no new load information is regenerated on the basis of the original data. Faced with the random load generated by the 5 MN metal extruder, the effect of predicting the trend of the load change is not significant. In addition, artificially fitting the joint distribution