## 4 Case Study: Damage Identification of a Jacket Platform
### Damage Identification Using Different Training Samples  
The structure of the BP neural network is constructed in the form of 18 \(\times\) 17 \(\times\) 8 according to Equation (11), and the training samples are used to train the network. The Levenberg-Marquardt algorithm is selected as the training function. The maximum number of training is 500 times and the training accuracy is set as \(1\times 10^{-4}\). The minimum gradient is \(1\times 10^{-20}\). The learning rate is 0.01 and the momentum factor is 0.9.  
To study the influence of the training sample size on the accuracy of prediction, this paper selects random training samples from the 20,000 groups of sample data. The training sample size is increased from 1000 to 20,000 at an interval of 1000, and 20 groups of training samples are used to train the networks. In this work, four groups of test samples listed in Table 3 are generated randomly using the trained networks. The accuracy of the predicted value is analyzed using the mean squared errors (MSE) which calculates the summed errors between the predicted and original values of the test sample. If \(N\) data are divided into \(r\) groups, and the sample variance of group \(i\) is \(s_{i}^{2}\), then the formula of MSE can be written as follows:  
\[\text{MSE}=\frac{\sum_{i=1}^{r}(n_{i}-1)s_{i}^{2}}{N-r} \tag{12}\]  
where MSE is the sum of the squared errors, and (\(N-r\)) is the degrees of freedom.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Sample No.** & \multicolumn{4}{c}{**Input Data (RSF and AO) and Output Data (AE)**} \\ \hline \multirow{2}{*}{\(1\)} & \multicolumn{4}{c}{RSF = [0.02660, 0.02784, 0.00787, 0.0314, 0.03357, 0.00413]} \\  & \(\Delta\Theta=[-2.1006\), \(-2.7346\), \(-0.1784\), \(-0.9464\), \(-1.4298\), \(-2.9443\), \(-3.3809\), \(-3.4088\), \(-1.0199\), \(-1.7558\), \(-2.0844\)]} \\  & \(\Delta\Theta=[0.7266\), 0.9215, 0.659, 0.8071, 0.8876, 0.8537, 0.5484, 0.7897] \\ \hline \multirow{2}{*}{\(10\),000} & \multirow{2}{*}{\(\Delta\Theta=[0.014\), 0.0355, \(-0.0049\), \(1.2194\), \(1.0664\), \(0.8378\), \(-1.0295\), \(-1.0162\), \(-0.8293\), \(0.0041\), \(0.0189\), \(0.0062\)} \\  & & \(\Delta\Theta=[0.8992\), 0.5597, 0.718, 0.6769, 0.9547, 0.88496, 0.9812, 0.7855] \\ \hline \multirow{2}{*}{\(20\),000} & \multirow{2}{*}{\(\Delta\Theta=[0.062\), 0.0168, \(-0.0159\), \(1.258\), \(1.107\), \(0.8558\), \(-1.2725\), \(-1.0879\), \(-0.9145\), \(-0.0132\), \(0.0158\), \(-0.0181\)} \\  & & \(\Delta\Theta=[0.5501\), 0.5125, 0.8262, 0.8889, 0.7276, 0.9428, 0.5593, 0.7297] \\ \hline \hline \end{tabular}
\end{table}
Table 2: Selected training sample data.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Test Sample**} & \multicolumn{4}{c}{**Member Unit and Corresponding Elastic Modulus**} \\ \cline{2-10}  & **E1,2** & **E2,3** & **E7,8** & **E8,9** & **E13,14** & **E14,15** & **E19,20** & **E20,21** \\ \hline D1 & 0.9236 & 0.7286 & 0.694 & 0.8462 & 0.7688 & 0.809 & 0.9239 & 0.5017 \\ D2 & 0.7345 & 0.6601 & 0.8964 & 0.7911 & 0.8476 & 0.535 & 0.6736 & 0.5562 \\ D3 & 0.7649 & 0.9947 & 0.8158 & 0.5754 & 0.886 & 0.7773 & 0.7388 & 0.9021 \\ D4 & 0.9182 & 0.5591 & 0.8795 & 0.8568 & 0.7954 & 0.5317 & 0.6839 & 0.511 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Test samples.  
Figure 5 shows the variation of the MSE with the sample size of the training data. As shown, MSE changes between 0.0005 and 0.0015. Generally, the prediction error tends to decrease with the increase of the sample size, and the prediction accuracy gets quite stable when the sample size reaches 17,000. From this sensitivity study, it is revealed that a sample number of 7000 yields minimum error, and the corresponding MSE value is 0.000542.