# Long-Term Structural State Trend Forecasting Based on an FFT-Informer Model
## 4 Conclusions  
In order to construct machine-learning-based long-term forecasting methods to obtain the long-term dependence between historical data and prediction while maintaining high forecasting accuracy, a long-term structural state trend forecasting method based on LSTF with the improved Informer model integrated with FFT is proposed, named the FFT-Informer model. In this method, FFT extracts amplitude and phase, and the improved Informer model forecasts trends, fully considering trend development and the impact of \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{**Metric**} & \multicolumn{3}{c}{**SMC**} & \multicolumn{3}{c}{**ASCE**} \\ \cline{3-8}  & & **5** & **10** & **15** & **5** & **10** & **15** \\ \hline \multirow{3}{*}{FFT–Informer} & MAE & **0.084** & **0.119** & **0.132** & **0.091** & **0.098** & **0.113** \\  & MSE & **0.017** & **0.028** & **0.042** & **0.021** & 0.036 & **0.048** \\ \multirow{3}{*}{Informer} & MAE & 0.111 & 0.135 & 0.147 & 0.117 & 0.121 & 0.134 \\  & MSE & 0.024 & 0.029 & 0.046 & 0.027 & **0.031** & 0.049 \\ \multirow{3}{*}{ARIMA} & MAE & 0.216 & 0.181 & 0.298 & 0.196 & 0.165 & 0.234 \\  & MSE & 0.053 & 0.062 & 0.069 & 0.055 & 0.062 & 0.072 \\ \multirow{3}{*}{ESN} & MAE & 0.174 & 0.197 & 0.213 & 0.186 & 0.192 & 0.226 \\  & MSE & 0.046 & 0.052 & 0.072 & 0.051 & 0.059 & 0.079 \\ \multirow{3}{*}{LSTM} & MAE & 0.142 & 0.191 & 0.231 & 0.163 & 0.212 & 0.246 \\  & MSE & 0.037 & 0.041 & 0.062 & 0.045 & 0.057 & 0.069 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of long-term forecasting methods. (Bold value is the minimum value). \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{**Metric**} & \multicolumn{3}{c}{**SMC**} & \multicolumn{3}{c}{**ASCE**} \\ \cline{3-6}  & & **5** & **10** & **5** & **10** \\ \hline \multirow{3}{*}{FFT–Informer} & MAE & **0.084** & **0.119** & **0.091** & **0.098** \\  & MSE & **0.017** & **0.028** & **0.029** & **0.031** \\ \multirow{3}{*}{Least-squares approximation *} & MAE & 0.095 & 0.131 & 0.034 & 0.041 \\  & MSE & 0.101 & 0.126 & 0.03 & 0.039 \\ \multirow{3}{*}{Direct signal extraction *} & MAE & 0.123 & 0.167 & 0.146 & 0.175 \\  & MSE & 0.049 & 0.062 & 0.067 & 0.053 \\ \multirow{3}{*}{Fourier *} & MAE & 0.114 & 0.127 & 0.114 & 0.132 \\  & MSE & 0.048 & 0.052 & 0.054 & 0.062 \\ \multirow{3}{*}{Wavelet transform *} & MAE & 0.121 & 0.123 & 0.110 & 0.118 \\ \multirow{3}{*}{Informer} & MSE & 0.044 & 0.052 & 0.031 & 0.056 \\  & MAE & 0.111 & 0.135 & 0.117 & 0.121 \\ \multirow{3}{*}{Informer} & MSE & 0.024 & 0.029 & 0.027 & 0.031 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of the prediction results of the combination of each feature extraction method for the long-term forecasting algorithm and Informer. (* represents the combination with Informer model. 5 and 10 represents five and ten feature windows. Bold value is the minimum value). abnormal deviation. The method has a good performance to capture long-term dependence between input and predicted values. Experimental results on two classical data sets show that the FFT-Informer model achieves high and stable accuracy and outperforms the comparative models in forecasting accuracy, which indicates that this model can effectively forecast the long-term state trend change of structures and is proposed to be applied to structural state trend forecasting and early damage warning. J.M. ; writing--review and editing, J.D. ; visualization. All authors have read and agreed to the published version of the manuscript. The work described in this paper is supported by Humanities and Social Science Planning Fund (grant no. 21 YJAZH013 and 20 YJAZH132) from the Ministry of Education, China. Not applicable. The authors declare that they have no known competing financial interest or personal relationships that could have appeared to influence the work reported in this paper.