# Forecasting Seasonal Sales with Many Drivers: Shrinkage or Dimensionality Reduction?
## 4 Empirical Study
### Dataset  
The empirical study is performed using a dataset of consumer goods from one of the largest stores of the leader in the supermarket segment in Portugal, operating more than 450 stores spread across 300 locations throughout the country. The store was chosen because of its complete coverage of products. The dataset contains product information at the SKU level, including unit sales, price and promotions for 173 weeks spanning between 3 January 2012 and 27 April 2015. We conducted the evaluation study based on 988 products from the six main categories including grocery, non-specialized perishables, specialized perishables, beverages, personal care and detergents & cleaning, covering a wide range of sales and promotional conditions. Consistent with previous studies [1; 47; 65] we focus on price promotional effects for each SKU, since other promotional data, such as display and weather data, are not available. Figure 1 plots the unit sales, price (in the second axis) and promotional periods (marked with dashed lines) of four typical SKUs. Two show a clear annual seasonal pattern that remains fairly constant during the sampled period. We observe that product sales spikes are associated with price reductions and calendar events such as Easter and Christmas. We also see that some products are heavily promoted, exhibiting high variations on their price while others have few promotions. Therefore, our study considers a wide variety of time series containing different features which are typical of this type of business. Any model needs to take these characteristics into account to effectively forecast the unit sales of each SKU. Figure 2 shows the impact of promotions on sales. It presents the distribution by category of the average weekly sales on promotional and non-promotional weeks for the selected SKUs, using violin plots, and shows the uplift of the sales on promotional weeks. Table 2 presents statistics of the sales volume with and without promotional activity for each product category during the 173-week period. The sales mean represents the average of the mean weekly unit sales on promotional and non-promotional  
weeks across all the SKUs of each category. The median is defined similarly. The promotion percentage indicates the percentage of promotional weeks within the 173-week time period for each category. The total number of SKUs in each category is also indicated. The dataset incorporates 43 covariates and calendar events, some with lead or lagged effects. * Price and a lag of order 1 (2 inputs). * Days promoted per week and a lag of order 1 (2 inputs); this variable indicates how many days in the week the SKU is under promotion. * Last week of the month and a lag of order 1 (24 inputs); this dummy variable captures the end of the month payday effect. * Binary indicators representing the following calendar events (15 inputs): New Year's Day, Carnival and the week before, Good Friday and Easter and the week before, Freedom's Day, Labor's Day, Corpus Christi week, Portugal's day, Assumption Day, Republic's day, All Saints' Day, Restoration of the Independence, Christmas and the week before. A logarithmic transformation of the sales time series is used, which helps model multiplicative effects of the aforementioned variables (proportional uplift of sales). At the end the logged target is transformed back to its original units. This transformation is not applied in the case of TBATS, since it already incorporates a Box-Cox transformation.