Sales forecasting of Walmart stores

Link to the Kaggle competition: https://www.kaggle.com/c/walmart-recruiting-store-sales-forecasting/overview

Historical sales data for 45 Walmart stores located in different regions were provided. Each store contains many departments. The goal is to predict the future weekly sales for each department in each store based on the historical data.

The training file provides the weekly sales data for various stores and departments from 2010- 02 to 2011-02. For each fold, the goal is to predict sales data for 8 weeks in the future (i.e. the forecast horizon is 2 months) based on sales data available up to then. This is done through calling the mypredict() function. The actual sales data corresponding with the current forecast horizon is then added into the training set for the next fold.

The mypredict() function

When the mypredict() function is called, it will first extend the training set with two additional months of sales data. The first fold is an exception since it would only use data from 2010-02 to 2011-02.

The function will then perform individual forecast for each department in the test set. At each iteration, both the test matrix and training matrix used for the forecasting are derived. The test matrix has a structure where the rows are dates corresponding to the forecast horizon of the current fold, and columns representing the stores associated with the department to be forecasted. The training matrix has a similar structure except the rows correspond to the dates from the training set available to the current fold. The restructuring is required in order to use the forecasting functions available from the forecast package. Next, the linear_model() function is called by passing in the training matrix and an empty test matrix. It will then output the test matrix with the predicted sale prices.

In fold 5, the sale prices are shifted in order to account for the slight shift of the Christmas shopping season from 2010 to 2011. After the predicted sale prices for every store and department have been obtained, it will transform the data back to the original format of the test matrix before returning it.

Linear regression model

In the linear_model() function, the tslm command from the forecast package is applied over each column of the training matrix. This is the same as fitting a linear regression model on the time series of a given store, and then the sales of this store for the next two months are forecasted using the fitted model. This is done for every store of a given department.

A linear regression model is selected because it can learn time series patterns through feature engineering. Additionally, it has some advantages over naïve or seasonal naïve models because it is able to learn both the trend and seasonality of the time series data. Although there is not much trend in the Walmart time series data since there are only up to two years of training data when forecasting for sales price in 2012, it might still be useful to include. Trend is modelled by including a degrees of freedom associated with time in the linear model. For instance, a linear regression model that considers only the trend is given by $y_t=\beta_0 + \beta_1 t + \epsilon_t$ (where $t$ denotes the time and $y_t$ the observed value at $t$).

In contrast, the Walmart time series data exhibits a strong seasonality. Seasonality can be modelled by using seasonal dummy variables in the linear regression model. The categorical variable modelling the seasonality has 52 levels corresponding to the 52 weeks in a year. Hence, there are 52 dummy variables in the model. Note that when forecasting the sales price for the weeks of 2011, the training data contains just one season (i.e. 2010) so the linear regression model is equivalent to the seasonal naïve model. This is because the linear regression model will just memorize the value from the last season for that week and predict the same value as that. On the other hand, forecasting the sales price for the weeks of 2012 will use data from the previous two seasons (i.e. 2010 and 2011).

Shifting

Fold 5 has a high WMAE because it contains two holiday weeks and therefore receives higher weights in WMAE. The high WMAE for fold 5 is due to a slight shift of the Christmas shopping season from 2010 to 2011.

The forecast model assumes the time series has a frequency of 52 weeks per year. For the most part this worked well since the data is short. Super Bowl is always on a Sunday and Thanksgiving is always on a Thursday, so those events have a fixed relationship to the week boundaries. Labor Day is not in the test data, so it is irrelevant here. However, Christmas occurs on a fixed date so its day of the week changes.

In 2010, Christmas occurs on a Saturday (where each week ends on Friday). That causes all of its associated sales to fall into the week before. However, in 2011, it occurs on a Sunday, so there is one pre-Christmas shopping day in week 52.

In order to account for this, a post-forecast adjustment was made for fold 5 (weeks 46 to 52). If the average sales for weeks 49, 50 and 51 of a given department were at least 10% higher than for weeks 48 and 52, then circularly shift a fraction of the sales from weeks 48 through 52 into the next week (and from 52 back to 48). Since the tslm model is fitted based on training data from 2010 when predicting the holidays of 2011, the sale for a given week is circularly shifted by 1/7 to the next week.

Results

The weighted mean absolute error (WMAE) of the time series linear model (tslm) was evaluated for each of the ten folds. The results are summarized below,

Fold 1	Fold 2	Fold 3	Fold 4	Fold 5	Fold 6	Fold 7	Fold 8	Fold 9	Fold 10
2042	1440	1435	1597	2029	1674	1719	1421	1431	1447