# Walking Back the Data Quantity Assumption to Improve Time Series Prediction in Deep Learning
## 4 Dataset Details and Performance Metrics
### Error Metrics  
The error metrics enable us to effectively assess the accuracy of the models. In this study, we chose the metrics that are most frequently employed in neural network-based time series forecasting [59].  
Root mean squared error (RMSE):  
\[RMSE=\sqrt{\frac{1}{N}\sum_{t=1}^{N}(x(t)-\hat{x}(t))} \tag{9}\]  
Mean squared error (MSE):  
\[MSE=\frac{1}{N}\sum_{t=1}^{N}(x(t)-\hat{x}(t))^{2} \tag{10}\]  
Mean absolute percentage error (MAPE):  
\[MAPE=\frac{1}{N}\sum_{t=1}^{N}\frac{|x(t)-\hat{x}(t)|}{|x(t)|}\times 100 \tag{11}\]  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline
**Time** & **1 st Iteration** & **2 nd Iteration** & **3 rd Iteration** & **4 th Iteration** & **5 th Iteration** & **6 th Iteration** & \multicolumn{2}{c}{**Total**} & \multicolumn{2}{c}{**Observations**} \\ \hline  & **Train** & **Test** & **Train** & **Test** & **Train** & **Test** & **Train** & **Test** & **Train** & **Test** & & \\ \hline
**GOLD** & 740 & 186 & 888 & 223 & 1036 & 260 & 1184 & 297 & 1332 & 334 & 1480 & 371 & 1851 \\ \hline
**WTI** & 4115 & 1029 & 4938 & 1235 & 5761 & 1441 & 6584 & 1647 & 7408 & 1852 & 8230 & 2058 & 10,288 \\ \hline
**Apple** & 2088 & 522 & 2505 & 627 & 2923 & 731 & 3340 & 836 & 3758 & 940 & 4175 & 1044 & 5219 \\ \hline
**OMIE** & 16,612 & 4153 & 19,380 & 4846 & 22,149 & 5538 & 24,918 & 6230 & 27,686 & 6922 & 27,686 & 6922 & 34,608 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Dataset division.  
R-squared (R\({}^{2}\)):  
\[R^{2}=1-\frac{\sum_{t=1}^{\mathrm{N}}(x(t)-\hat{x}(t))^{2}}{\sum_{t=1}^{\mathrm{ N}}(x(t)-\overline{x}(t))^{2}} \tag{12}\]  
In these equations, \(x(t)\) represents the \(t\)-th element of the time series, while \(\hat{x}(t)\) denotes the predicted value, \(\overline{x}(t)\) indicates the mean value of the time series, and N is the total number of elements in the series.  
The error in the predictions is measured using the MAPE and RMSE metrics, which allow us to know the accuracy of the forecasting model, the ideal being the lowest possible value since it represents the least error in the predictions.  
The accuracy of the predictions is assessed using the MAPE and RMSE metrics, which help determine the effectiveness of the forecasting model, with lower values indicating less error in the predictions. Similarly, MSE and R\({}^{2}\) provide insights into model performance by analyzing the variation of the dependent variable relative to the predictions of the independent variable. A value closer to 1 reflects superior performance according to the R\({}^{2}\) metric.  
R\({}^{2}\) is commonly employed to evaluate the fitting performance of nonlinear models and can be complemented by other metrics such as MAPE, MSE, and RMSE. This allows for an intuitive and straightforward comparison with previous studies, facilitating the interpretation of results for a broader audience [60]. Although R\({}^{2}\) is traditionally associated with linear models, its application to nonlinear models can yield valuable insights regarding the proportion of variability explained by the model. Recognized and valued as an important metric, R\({}^{2}\) is the comprehensibility of results. When used alongside MAPE, MSE, and RMSE, it provides a robust and comprehensive evaluation of model performance while addressing potential limitations by offering multiple perspectives on model accuracy and error [61; 62]. In some cases, R\({}^{2}\) may even yield a negative value when the model poorly fits the problem at hand [63] and in cases of a poor fit to the problem posed [64].  
The decision to utilize various metrics stems from the recommendation to apply multiple performance indicators to ensure the validity of machine learning models, as this approach can help mitigate the shortcomings of individual metrics [65]. These metrics are widely used in the literature for various types of time series, including those characterized by significant trends [66; 67; 68].