## 6 Example Analysis
### Comparison and Analysis of Prediction Results
#### 6.4.3 Comparison of Evaluation Metrics  
The error evaluation metrics for each prediction model were computed based on the test results from the test set. Table 2 provides a summary of the error evaluation metrics for all the models.  
Compared with the single Transformer model, the Transformer-BiLSTM model improves the Transformer decoder part to BiLSTM on this basis, which solves the long-term dependency problem in the input sequence data, and the bidirectional data processing makes the message acquisition more comprehensive. The MAE, MAPE, MSE, and RMSE are reduced by 20.85%, 15.10%, 30.24%, and 16.48%, and the R2 increases by 31.89%. It can be seen that the R2 of all four models is greater than 0.4, and all of them have good fitting effect.  
Compared with the Transformer-BiLSTM model, the VMD-Transformer-BiLSTM model adds the VMD on the basis of this model, removes the redundant noise signals in  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & **MAE** & **MAPE** & **MSE** & **RMSE** & \(\mathbf{R^{2}}\) \\ \hline Transformer & 3.160 & 1.503 & 15.471 & 3.933 & 0.486 \\ Transformer-BiLSTM & 2.501 & 1.276 & 10.793 & 3.285 & 0.641 \\ VMD-Transformer-BiLSTM & 1.536 & 0.427 & 3.960 & 1.990 & 0.830 \\ VMD-NRBO-Transformer-BiLSTM & 0.885 & 0.199 & 1.032 & 1.016 & 0.956 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Evaluation metric results of each model.  
Figure 12: Prediction errors of each model on the test set. (**a**) Prediction errors of the Transformer model on the test set; (**b**) prediction errors of the Transformer–BiLSTM model on the test set; (**c**) prediction errors of the VMD-Transformer-BiLSTM model on the test set; and (**d**) prediction errors of the VMD-NRBO-Transformer-BiLSTM model on the test set.  
the original data, and retains the key information, which optimizes the prediction effect. The MAE, MAPE, MSE, and RMSE are reduced by 38.58%, 66.54%, 63.31%, and 39.42%, and the R\({}^{2}\) increased by 29.49%.  
The various error evaluation indexes of the VMD-NRBO-Transformer-BiLSTM model are optimal among all models, compared with the VMD-Transformer-BiLSTM model, and the NRBO algorithm is added, which is a reasonable optimization for the initial hyperparameters of the network model, and it improves the efficiency of the model training and the prediction accuracy. The MAE, MAPE, MSE, and RMSE were reduced by 42.38%, 53.40%, 73.94%, and 48.94%, respectively, and the R\({}^{2}\) increased by 15.18%.  
To visually illustrate the differences in the error evaluation metrics across the models and emphasize the advantages of the VMD-NRBO-Transformer-BiLSTM model, the evaluation metric results are presented in compass diagrams. Prior to plotting, the values are normalized to a common scale, and each value is represented as a proportion of the total sum for the corresponding metric. The compass diagrams depicting the error evaluation metrics are shown in Figure 13.  
Combining the results of the above studies, the VMD-NRBO-Transformer-BiLSTM prediction model proposed in this paper provides a good solution to the problems of input data feature acquisition, capturing long-term dependencies and gradient explosions of time-series data, as well as network model hyperparameter selection. In previous studies, the volatility and nonlinearity of PV output data make prediction difficult, and the denoising of nonlinear complex signals using the VMD technique removes signal redundancy and retains key information. The combination of BiLSTM and Transformer makes Transformer more adept at handling the prediction of time-series data and improves the information capturing ability. Previously, model hyperparameter selection relied on human experience, and the introduction of the NRBO algorithm makes the hyperparameter selection of the network model more suitable for the predicted data features.  
Figure 13: _Cont._