# The Cut-Off Frequency of High-Pass Filtering of Strong-Motion Records Based on Transfer Learning
## 4 Application of the Trained Network Models
### Analysis of Filtering Results  
To evaluate the accuracy of the high-pass cut-off frequencies obtained with different models, four error functions (Equations (3)-(6)) were applied to analyze the errors concerning the values obtained using the traditional method. The coefficient of determination \(R^{2}\) is widely used in regression problems to estimate the degree of linear correlation between the true value and the predicted value. The closer \(R^{2}\) is to 1, the better the performance of the model. The RMSE is used to measure the deviation between the predicted value and the true value. The MAE reflects the actual situation of the prediction error. The MAPE is the ratio of the prediction error to the true value. The closer these three values are to zero, the higher the performance of the model. The results of the calculations related to the four statistical parameters for each model are presented in Table 3.  
\[R^{2}=1-\frac{\sum\limits_{i=1}^{n}{{{({{{\hat{y}}_{i}}-{{y_{i}}})}^{2}}}}}{ \sum\limits_{i=1}^{n}{{{({{{\hat{y}}_{i}}-{{y_{i}}})}^{2}}}}} \tag{3}\]  
\[RMSE=\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}{{{({{{\hat{y}}_{i}}-{{y_{i}}})}^{2}}}}} \tag{4}\]  
\[MAE=\frac{1}{n}\sum\limits_{i=1}^{n}{{{|{{\hat{y}}_{i}}-{{y_{i}}}|}}} \tag{5}\]  
\[MAPE=\frac{100\%}{n}\sum\limits_{i=1}^{n}{{{|{{\hat{y}}_{i}}-{{y_{i}}}|}}} \tag{6}\]  
where \(y\) and \(\hat{y}\) are the high-pass cut-off frequencies obtained using the traditional method and the high-pass cut-off frequency obtained using the method of this paper, respectively, \(n\) represents the number of data samples, and \(\overline{y}\) denotes the mean value of \(y\).  
According to the analysis results in Table 3, using a CNN instead of artificial identification to judge whether a filtered displacement curve is qualified and thus obtain a high-pass cut-off frequency of strong-motion records can be satisfactory. The coefficient of determination \(R^{2}\) reached a maximum of 0.82 and a minimum of 0.57. The RMSE ranged from a minimum of 0.038 to a maximum of 0.059. The MAE ranged from a minimum of 0.026 to a maximum of 0.040. The MAPE ranged from a maximum of 25.34% to a minimum of 2.99%. In comparison, the VGG19-unfrozen model considering probabilities was used to achieve the highest accuracy.  
It can be concluded that the probability limit added to the prediction classification of the model enhanced its performance. \(R^{2}\) was improved by an average of 14.41%, of which VGG19-unfrozen had the most improvement at 26.15%. The RMSE, MAE, and  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & **Considering** & \multirow{2}{*}{\(R^{2}\)} & \multirow{2}{*}{**RMSE**} & \multirow{2}{*}{**MAE**} & \multirow{2}{*}{**MAPE (\%)**} \\  & **Probability?** & & & & \\ \hline \multirow{2}{*}{VGG19–frozen} & Yes & 0.64 & 0.054 & 0.035 & 21.03 \\  & No & 0.57 & 0.059 & 0.040 & 25.34 \\  & Yes & 0.82 & 0.038 & 0.026 & 2.99 \\  & No & 0.65 & 0.052 & 0.038 & 24.53 \\  & Yes & 0.74 & 0.046 & 0.031 & 18.54 \\  & No & 0.68 & 0.050 & 0.035 & 21.63 \\  & Yes & 0.68 & 0.051 & 0.034 & 20.43 \\  & No & 0.59 & 0.057 & 0.039 & 24.40 \\  & Yes & 0.69 & 0.050 & 0.034 & 20.49 \\  & No & 0.63 & 0.055 & 0.039 & 24.02 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Analysis of results.  
MAPE were reduced by 12.6%, 16.23%, and 30.01% on average. Among them, the three parameters of VGG19-unfrozen were reduced the most by 26.92%, 31.58%, and 87.81%. The performance of the VGG19-unfrozen model was most significantly improved by adding probabilistic restrictions.