## 4 Results and Discussion
### Quantitative Analysis  
When managing complex sales prediction issues, several analyses are established through a combination of techniques to complement the benefits of deep learning models and to enhance the accuracy. The deep learning models are always selected because of their high precision in solving the complex problems. Consequently, in the proposed Bi-GRU model, the forward GRU extracts prior information in the historical time-series data, and the backward GRU extracts future information in the historical time-series data to minimize the error performances (NRMSE, ND, RMSSE, MAE, MSE, and R2) for sales forecasting, which is shown in the below tables. Therefore, the efficacy of the Bi-GRU model was validated on the Rossmann and Walmart datasets. The numerical analysis of different models on the Rossmann and Walmart datasets is denoted in Tables 2 and 3. As specified in Tables 2 and 3, the efficacy of the Bi-GRU model was analyzed by comparing its results with other regression models, like linear regression, LSTM, Bi-Directional LSTM (Bi-LSTM), and GRU by means of NRMSE, ND, RMSSE, MAE, MSE, and R2. By viewing Table 2, we can see that the proposed Bi-GRU model obtains a high R2 value of 0.98 and a minimum NRMSE, ND, RMSSE, MAE, and MSE of 0.08, 0.07, 0.08, 0.05, and 0.04, respectively, on the Rossmann dataset. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Specifications** & **Walmart** & **Rossmann** \\ \hline Time granularity & Weekly & Daily \\ Window size of prediction & 6 & 12 \\ Window size of input & 30 & 30 \\ time covariates & 4 & 3 \\ features & 8 & 7 \\ time series & 2660 & 1115 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Data statistics of Rossmann and Walmart datasets. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{2}{c}{**Rossmann Dataset**} \\ \hline
**Forecasting Models** & **Measures** & **Without Feature Engineering** & **With Feature Engineering** \\ \hline \multirow{8}{*}{Linear regression} & NRMSE & 5.90 & 3.92 \\  & ND & 5.44 & 3.90 \\  & RMSSE & 4.97 & 3.68 \\  & MAE & 4.10 & 3.10 \\  & MSE & 4.04 & 2.96 \\  & R2 & 0.60 & 0.80 \\ \hline \multirow{8}{*}{LSTM} & NRMSE & 3.76 & 2.96 \\  & ND & 3.52 & 2.92 \\ \cline{1-1}  & RMSSE & 3.50 & 2.54 \\ \cline{1-1}  & MAE & 3.42 & 2.58 \\ \cline{1-1}  & MSE & 3.14 & 2.32 \\ \cline{1-1}  & R2 & 0.64 & 0.84 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Numerical analysis of different regression models on the Rossmann dataset. On the other hand, as seen in Table 3, the Bi-GRU model achieves a high R2 value of 0.99 and a minimum NRMSE, ND, RMSSE, MAE, and MSE of 0.01, 0.03, 0.08, 0.07, and 0.03 on the Walmart dataset, respectively. As compared to the other regression models--the linear regression, LSTM, Bi-LSTM, and GRU--the Bi-GRU model utilizes limited memory and is faster in regard to data processing. The Bi-GRU model is more precise while utilizing the datasets with longer sequences. In addition to this, the Bi-GRU model effectively addresses the problems of overfitting and vanishing gradient. \begin{table}
\begin{tabular}{c c c c} \hline  & \multicolumn{3}{c}{**Rossmann Dataset**} \\ \hline
**Forecasting Models** & **Measures** & **Without Feature Engineering** & **With Feature Engineering** \\ \hline \multirow{8}{*}{Bi-LSTM} & NRMSE & 2.90 & 2.67 \\  & ND & 2.72 & 2.42 \\  & RMSSE & 2.48 & 2.01 \\  & MAE & 2.44 & 1.96 \\  & MSE & 2.05 & 1.92 \\  & R2 & 0.76 & 0.91 \\ \hline \multirow{8}{*}{GRU} & NRMSE & 0.80 & 0.57 \\  & ND & 0.88 & 0.25 \\  & RMSSE & 0.86 & 0.33 \\  & MAE & 0.65 & 0.23 \\  & MSE & 0.58 & 0.22 \\  & R2 & 0.82 & 0.95 \\ \hline \multirow{8}{*}{Bi-GRU} & NRMSE & 0.16 & 0.08 \\  & ND & 0.15 & 0.07 \\ \cline{1-1}  & RMSSE & 0.22 & 0.08 \\ \cline{1-1}  & MAE & 0.16 & 0.05 \\ \cline{1-1}  & MSE & 0.12 & 0.04 \\ \cline{1-1}  & R2 & 0.88 & 0.98 \\ \hline \end{tabular}
\end{table} 
Table 2: _Cont._