## 4 Experimental Results and Analysis
### Introduction of Evaluation Index  
In this study, the evaluation of the proposed MDFFE model's effectiveness in forecasting crude oil and hot-rolled coil futures prices is based on several key metrics: mean absolute error (MAE), root mean square error (RMSE), mean absolute percentage error (MAPE), coefficient of determination (R-Squared), and sum of squared errors (SSE). These metrics provide a multidimensional perspective on the accuracy and fit of the model's predictions compared to actual values, which is crucial for a thorough and detailed performance analysis. MAE serves to quantify the average absolute difference between the predicted and actual prices. It offers a straightforward measure of the typical deviation in the predictions from the true values. The formula for MAE is as follows:  
\[MAE=\frac{1}{n}{\sum_{i=1}^{n}}|y_{i}-\hat{y_{i}}| \tag{20}\]  
Among them, \(y_{i}\) represents the real observation value of the \(i\) sample, \(\hat{y_{i}}\) represents the model prediction value of the \(i\) sample, and \(n\) is the number of samples. The smaller the value of MAE is, the smaller the average deviation between the predicted value and the real value of the model is, and the higher the prediction accuracy of the model is. RMSE serves to quantify the sample standard deviation of the differences between the forecasted and actual values for crude oil and hot-rolled coil futures prices. The formula is as follows:  
\[RMSE=\sqrt{\frac{1}{n}{\sum_{i=1}^{n}}(y_{i}-\hat{y_{i}})^{2}}. \tag{21}\]  
RMSE is closely tied to mean squared error (MSE), as it is derived by taking the square root of MSE. This square root operation brings the error back to the same scale as the original data, making it more interpretable in practical contexts. For instance, if futures price data are denominated in a specific currency, the RMSE value indicates the average prediction error per sample in that currency. RMSE takes into account the error across all samples and provides a clear picture of the average deviation between predicted and actual values, complementing MAE to offer a more comprehensive assessment of model accuracy. MAPE, on the other hand, measures the average percentage error of predictions relative to the actual values, and its formula is as follows:  
\[MAPE=\frac{1}{n}{\sum_{i=1}^{n}}\frac{|y_{i}-\hat{y_{i}}|}{y_{i}}\times 100 \%. \tag{22}\]  
The key benefit of MAPE lies in its ability to neutralize dimensional effects and express prediction errors as percentages. This feature allows for a consistent and straightforward assessment criterion, which is particularly useful when dealing with crude oil and hot-rolled coil futures prices of varying scales or when comparing multiple forecasting models. No matter the scale of the futures prices, MAPE enables a clear comparison of the relative errors from model predictions. The coefficient of determination, known as R-Squared, is utilized to gauge how well the model fits the data. Its formula is as follows:  
\[R-Squared=1-\frac{\sum_{i=1}^{n}(y_{i}-\hat{y_{i}})^{2}}{\sum_{i=1}^{n}(y_{i}- \overline{y})^{2}} \tag{23}\]  
where \(\overline{y}\) represents the average of the actual observations. The value of \(\mathrm{R}-\mathrm{Squared}\), when closer to 1, indicates a superior fit of the model to the data, suggesting that the model effectively captures the majority of the data's variability. Conversely, if B is near 0, the model's fit is inadequate, potentially due to issues such as an illogical model structure or imprecise parameter estimates, which can lead to unreliable predictions for crude oil and hot-rolled coil futures prices. SSE quantifies the total squared discrepancy between the predicted and actual values. The formula is as follows:  
\[SSE=\sum_{i=1}^{n}y_{i}-\hat{y_{i}}^{2}. \tag{24}\]  
SSE is centered on the summation of the squared discrepancies between all sample predictions and their actual values. A higher SSE value signifies a greater accumulation of overall prediction errors, indicating that the model's forecasts significantly diverge from the true values.