Gasoline consumption

The data for this case study is weekly data on US finished motor gasoline products supplied (in thousands of barrels per day) from February 1991 to May 2005. It is part of the R package fpp2 and available from the EIA website. It is also bundled with the installation file of TSL. The dataset is used in the TBATS paper of De Livera, A.M., R.J. Hyndman, and R.D. Snyder (2011). Furthermore, the dataset is analysed by R.J. Hyndman on his blog. We quote from this blog post:

The TBATS model is preferable when the seasonality changes over time. The ARIMA approach is preferable if there are covariates that are useful predictors as these can be added as additional regressors.

This gasoline case study illustrates that you don't need to choose between the two methods when you work with Time Series Lab. TSL offers a modelling framework for complex seasonal patterns AND, at the same time, with the inclusion of covariates (explanatory variables). We show that it is possible for TSL to produce more accurate forecasts compared to the TBATS package. We deliberately compare with TBATS since this package shows accurate forecasts when complex seasonal patterns are present in the data.

In the figure below, the gasoline dataset is loaded into TSL and plotted. The upward trend and seasonality patterns are clearly visible in the data. The Data characteristics area shows T = 745 observations. At a later stage (Estimation step) we will split the time series into an Training sample and a test sample.

Local Linear Trend model

The seasonal pattern of this time series is important but for illustrative purposes, we start our analysis without a seasonal component and select the Local Linear Trend model which is a model with a trend component but no seasonal component. Select the Local Linear Trend model model on the Pre-built models page. Alternatively, you can go to the Build your own model page and select a time-varying level and a time-varying slope.
Our time series consist of a total of 745 observations (February 1991 to May 2005). For this case study we select the first 484 observations as Training sample and leaving 261 observations as Validation sample. Drag (and/or click) the sample bar on the Pre-built models page to set a Training sample of size 484, or alternatively, set the start and end of the Training sample to 1 and 484 on the Estimation page. Click the Process Dashboard button on the Pre-built models page or the Estimate button on the Estimation page. TSL estimates the model and if you go to the Text output page you see a green colored message informing us that:

All selected models and series were estimated successfully

Furthermore, at the bottom of the Text output page we find the Model fit. For the current model this is:

Variable: gasoline
Model: TSL003 Local Linear Trend
                                               TSL003
Log likelihood                              -3485.355   
Akaike Information Criterion (AIC)           6978.710   
Bias corrected AIC (AICc)                    6978.794   
Bayesian Information Criterion (BIC)         6995.439   
in-sample MSE                              1.1177e+05   
... RMSE                                      334.317   
... MAE                                       268.147   
... MAPE                                        3.480   
Sample size                                       484   
Effective sample size                             482   
* based on one-step-ahead forecast errors
                            

We report these numbers here to show the improvement of adding a seasonal component later. The graphical output of the current model is shown in the figure below.

With a smoothed level through the data, the seasonal pattern is even better visible. The triangular pattern in the level will appear later as well when we plot the forecasting performance of the model.
Let's assess the forecast performance of the model by going to the Model comparison page. This page can be viewed by clicking the Model comparison button in the button bar on the left of your screen. Note that this button is only visible when a Validation sample is specified. Click on the green Start loss calculation button in the top right of the window. Under User defined models, a new check-button appears which you should tick. The resulting TSL screen is shown below.

The pyramid shaped loss line can be explained by the fact that a forecast from the Local Linear Trend model is a straight line that is upward sloping for our data set. The forecasts do not take into account the seasonal pattern of the data so when the data is at the highest or lowest point in the seasonal cycle, the loss is the highest. Let's verify this. Go to the Forecasting page and select multi-step-ahead in the top left corner. Navigate to Plot options and Show forecast 150 periods ahead. The resulting window should look the one presented in the figure below.

Basic Structural Time Series model

It is time to introduce a seasonal component. We go to the Build your own model page and select a time-varying level, a time-varying slope, and a time-varying seasonal. The resulting model is called the Basic Structural Time Series model by Harvey (1990). Set the Seasonal period length to 365.25/7 ≈ 52.179 (weekly data taking leap years into account) and a Number of factors equal to 22.

The Build your own model page should look like the one in the figure below.

Estimate the model and go to the Text output page. We see that the model fit is improved by adding the seasonal component.

Variable: gasoline
Model: TSL004
                                               TSL004
Log likelihood                              -3218.400   
Akaike Information Criterion (AIC)           6532.799   
Bias corrected AIC (AICc)                    6543.613   
Bayesian Information Criterion (BIC)         6733.539   
in-sample MSE                              1.0083e+05   
... RMSE                                      317.539   
... MAE                                       252.502   
... MAPE                                        3.244   
Sample size                                       484   
Effective sample size                             430   
* based on one-step-ahead forecast errors
                            

A large improvement in forecasting performance, compared to the Local Linear Trend model, can be seen if we start (and plot) the loss calculation on the Model comparison page. This shows how important it can be to model the seasonal pattern of a time series. We will see an example of multiple seasonal patterns in a time series which makes the correct handling even more important. Plotting both RMSE losses leads to the following figure.

Next, go back to the Build your own model page and lower the number of factors of the seasonal component. Parsimonious models often perform better in forecasting. A better performing number of factors is 7 although other values might be even better for forecasting. A model comparison between 22 and 7 factors is made in the figure below. The loss corresponding to the model with 7 factors is lower than the one that is presented in Figure 2 of De Livera, A.M., R.J. Hyndman, and R.D. Snyder (2011) which is obtained with the TBATS package.

A figure with the extracted trend and seasonal pattern is obtained from the Graphics and Diagnostics page.