# Long-Term Forecasting Using MAMTF: A Matrix Attention Model Based on the Time and Frequency Domains
## 5 Conclusions  
We adopted the idea of seasonal trend decomposition for long-term multivariate time series forecasting to decompose the complex and variable time series data into seasonal terms and trend terms. We proposed a matrix attention mechanism, used it as a base module to build a frequency domain block and a time domain block for extracting the seasonal and trend term information, and constructed a time-frequency dual-domain model. After converting seasonal information with poor regularity into the frequency domain, it is beneficial to extract useful information to assist in prediction. Therefore, the seasonal information was feature-extracted using the frequency domain block, while the trend information was feature-extracted using the time domain block. We used three public datasets and one private dataset for the experimental demonstration, and in the comparison experiments, the M-MAMFT model achieved the best prediction results in comparison with other models, which proved the effectiveness of our model architecture. In the visual analysis, we graphically analyzed the difference between the prediction results of the M-MAMFT model and the prediction results of the other models, proving the ability of the M-MAMFT model to better capture the information from the data. In the ablation experiments, the superior performance of our model architecture was verified, as well as the superior performance of the matrix attention mechanism over other self-attention families. At the same time, they further proved that in addition to the models built from the family of self-attention mechanisms, which can achieve a high accuracy in long-term time series prediction, some other methods can also achieve the same or even better prediction results.  
Conceptualization: [PERSON]; methodology: [PERSON]; conducting experiments: [PERSON] and [PERSON]; writing--original draft preparation: [PERSON]; writing--review and editing: [PERSON] and [PERSON]; supervision: [PERSON]. All authors have read and agreed to the published version of the manuscript.  
This research was supported by the Academic Research Projects of Beijing Union University (No. ZK80202103).  
Not applicable.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Models**} & \multirow{2}{*}{**Metric**} & \multicolumn{4}{c}{**ETTh2**} & \multicolumn{4}{c}{**Weather**} & \multicolumn{4}{c}{**BCAT**} \\  & & **96** & **192** & **336** & **720** & **96** & **192** & **336** & **720** & **96** & **192** & **336** & **720** \\ \hline \multirow{2}{*}{M-MAMFT} & MSE & 0.176 & 0.208 & 0.227 & 0.266 & 0.151 & 0.202 & 0.247 & 0.310 & 0.290 & 0.313 & 0.330 & 0.378 \\  & MAE & 0.293 & 0.323 & 0.337 & 0.379 & 0.209 & 0.259 & 0.296 & 0.341 & 0.362 & 0.379 & 0.389 & 0.417 \\ \hline \multirow{2}{*}{Auto-Correlation} & MSE & 0.193 & 0.246 & 0.268 & 0.319 & 0.153 & 0.203 & 0.251 & 0.313 & 0.298 & 0.335 & 0.375 & 0.400 \\  & MAE & 0.314 & 0.352 & 0.378 & 0.424 & 0.209 & 0.261 & 0.300 & 0.342 & 0.364 & 0.383 & 0.426 & 0.420 \\ \hline \multirow{2}{*}{Full-Attention} & MSE & 0.228 & 0.284 & 0.298 & 0.363 & 0.175 & 0.222 & 0.281 & 0.338 & 0.302 & 0.341 & 0.336 & 0.418 \\  & MAE & 0.348 & 0.403 & 0.411 & 0.442 & 0.246 & 0.287 & 0.338 & 0.385 & 0.366 & 0.390 & 0.393 & 0.418 \\ \hline \multirow{2}{*}{FEA-f} & MSE & 0.330 & 0.359 & 0.383 & 0.462 & 0.160 & 0.217 & 0.260 & 0.339 & 0.299 & 0.332 & 0.339 & 0.406 \\  & MAE & 0.438 & 0.464 & 0.449 & 0.510 & 0.245 & 0.301 & 0.330 & 0.408 & 0.371 & 0.386 & 0.404 & 0.429 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of ablation studies with different attentional mechanisms under the same model structure.  
Data Availability Statement: The data presented in this study are available on request from the corresponding author due to privacy.