# Long-Term Structural State Trend Forecasting Based on an FFT-Informer Model
## 3 Experiments and Discussion
### Long-Term Structural State Trend Forecasting  
The Informer model obtains the correlation of data in and between windows through the overall training of windows.The prediction length can be set to different values, but it needs to be less than the training window length. The one-forward prediction is carried out through the window. First, feature vectors of the data set are obtained through the feature extraction by FFT in the previous step, and they are converted into data tensor. The next step is to add location information in the attention window, which can capture a correlation between \begin{table}
\begin{tabular}{c c c c c} \hline
**Feature Extraction** & **Least-Squares** & **Direct Signal** & **Fourier** & **Wavelet** \\
**by FFT** & **Approximation** & **Extraction** & **Polynomial** & **Transform** \\ \hline
3 & 5 & 1 or 1 & 20 & 12 \\ \hline \end{tabular}
\end{table} 
Table 2: Number of features collected in unit window for each feature extraction method. Figure 7: Fitting curve of feature vectors by FFT. the data in the training windows. Finally, the model parameters with an Adam optimizer are optimized. Adam optimizer [24] is a gradient descent algorithm with an adaptive learning rate, which can not only adapt to a sparse gradient, but also alleviate the problem of gradient oscillation. The training error decreases rapidly and becomes stable as training proceeds and parameter adjustment in epochs as shown in Figure 8.  
By the early data processing and parameters optimization, the parameters of the model are determined before the prediction. Like training, the data required by the Informer model forecasting is still in tensor form, and the size of the forecasting windows is the same as that of the training windows. The forecasting feature vectors can be obtained by adjusting the length of the model in a one-forward operation. In this experiment, the vibration state data collected by SMC sensors are predicted directly, and the prediction length is set to five eigenvectors, as shown in Figure 9. The feature forecasting is carried out section by section in the form of windows, and the change of actual value is well captured. In order to verify its flexibility and consider the impact of noise, the data of the same sensor separated by half a year are selected to learn the model, which predicts different structural states under different structural condition, as shown in Figure 10, the change of structural state can still be captured well. The forecasting windows fit the changes of data well and achieve good forecasting results. Figure 8: Training error change during the training phase. Figure 9: Structural state trend forecasting based on the FFT–Informer on the SMC data set. (Time represents the number of the predicted value). In order to fully verify the effectiveness of the model, we also forecast the structural state trend using ambient vibration data in the ASCE data set. Different from the data in the SMC, the data in the ASCE fluctuate more obviously, so the windows size of feature extraction is reduced to 50 to better fit trend features. The forecasting length is still five window lengths, as shown in the Figure 11. In order to effectively prove our model applicable to general cases, the predicted position is also adjusted as in the previous experiment to verify that the prediction is not sensitive to the change of structural state, as shown in Figure 12. Experimental results show that the model can be well-applied for a general case and have good forecasting performance. The Informer model is a nonautoregressive predictor through a forward operation. The length of prediction can also be adjusted directly to predict long time series. As shown in Figure 13, in order to show long-term forecasting performance, the forecasting length is adjusted to 10 and 15, in which each represents the feature vectors of 100 values. When changing the data set, the size of the feature extraction window and the forecasting length, the FFT-Informer model still performs well in long time series forecasting.