# Robust Sales forecasting Using Deep Learning with Static and Dynamic Covariates
## 4 Results and Discussion  
Table 1 displays the results obtained from an empirical study employing the DeepAR model across various feature combinations, organized by the number of feature groups included. The seasonal naive model is used as a benchmark. This simple time series forecasting model assumes that the future value of a series is equal to the most recent observed value from the same season. In this context, assuming a seasonal period of 7 days with daily data, as is the case here, the forecast for all future Monday values, for instance, mirrors the last observed Monday value, and so forth. Table 1 highlights the most effective combination of features in boldface within the RMSSE and MASE columns. Let us highlight the key observations in the results. Firstly, regardless of the error measure used, the DeepAR model with or without features consistently exhibits a significantly better performance compared to the univariate benchmark. The improvements in forecast accuracy are substantial, amounting to approximately 25.8% based on RMSSE. This underscores the advanced capabilities of the DeepAR model in time series forecasting. Secondly, it is important to note that there are differences in the results between RMSSE and MASE. This difference is understandable since these error measures concentrate on distinct aspects of the distribution of the target variable. Thirdly, the incorporation of time-, event-, and ID-related features consistently enhances the accuracy of the baseline DeepAR model, which lacks any covariates. Thisimprovement aligns with expectations, given the significance of special events, sales dates, and item attributes in retail forecasting. Fourthly, in general, the inclusion of price-related features does not lead to performance improvement in DeepAR models across various feature combinations. This result is unsurprising, especially in the context of a retailer like Walmart, where price distributions tend to remain relatively stable over the years. Finally, the usefulness of ID-related features significantly surpasses those of time- and event-related features. However, the best performance is achieved when all three of these groups of features are used together, suggesting that the individual relevance of each type of feature is emphasized when the information is given jointly. This model's performance (DeepAR + Events + Time + IDs) improved by 1.8% for RMSSE and 6.5% for MASE when compared to the baseline model without features. Additionally, in Figure 6, we offer a comparison of how features related to prices and IDs affect the performance of DeepAR models across different feature combinations. Hollow dots represent errors from models without prices/IDs, while filled dots represent errors from models with prices/IDs. These figures unequivocally demonstrate that, regardless of the error measure used, the inclusion of price-related features never enhances the accuracy of any DeepAR model, regardless of the feature combination. Conversely, the addition of ID-related features consistently improves the forecasting performance of a DeepAR model, regardless of the feature combination used. The influence of time- and event-related features closely resembles that of the IDs, albeit with a less pronounced improvement (figures not displayed). In the M5 dataset, we only had five distinct ID-related features: item ID, category ID, department ID, store ID, and state ID. However, in a real retail setting, there would be numerous additional attributes and features associated with each product. These may include details such as the product type, brand, size, and attributes that are relevant within specific subcategories (for example, \"white vs. red\" matters for wine but less so for beer, even though both fall under the category of \"alcoholic beverages\"). Leveraging these additional attributes is likely to result in even more significant improvements compared to solely relying on the five item IDs provided by the M5 dataset.