To see what is meant by the text above: switch from Smoothing to Filtering. You now see that the Level checkbox is unchecked because the level that is currently plotted corresponds to the Smoothed level and not the Filtered level. The reason the Smoothed level in the plot does not automatically switch to a Filtered level on changing is that we sometimes want to compare Smoothed, Filtered, and Predicted components in one plot. If you click on level, the resulting graph should look like the following figure.

Missing data

We continue this Case study with a version of the Nile data with missing values to illustrate one of the many advantages of using TSL, namely the capability of easily handling missing values. Missing values in time series can occur due a variety of reasons and for some time series algorithms it is problematic.

Go back to the Database page of TSL and select the Nile_missing time series by clicking on the name. We see that the Data characteristics are updated by selecting the new time series. It shows us 40 missing values, among other characteristics. The TSL window should now look like the following figure.

We will estimate and compare two models with each other. Click on the Pre-built models button in the button bar at the left of your screen and switch on, the models Exponential Smoothing and Local Level. Make sure these are the only selected models, see also the Model selection summary in the blue pane in the bottom of the screen. Select an 100%/0% ratio for Training and Validation sample and click the Process Dashboard button which is the green arrow located at the bottom right of your screen.

Comparing results

Go to the Graphics and diagnostics page and click the Clear all button (eraser icon, bottom right) to start with a clean graph window. From the Individual tab select Y data to plot the Nile_missing time series. From the drop-down menu select the Exp Smoothing model and plot the Total signal from the Composite tab. Next, from the drop-down menu select the Local Level model and make sure the Type in the top left corner says Predicting, followed by plotting the Total signal from the Composite tab. The resulting graph should look the figure below. We see that the Local Level model reacts stronger to changes in the time series after missing values periods.

We can also see the difference in model fit expressed in numbers. Go to the Text output page where at the end of the estimation, model fit of the selected models is summarized. Looking at in-sample MSE we see that the loss of the Local Level model is lower.

Variable: Nile_missing
Model(s): 
TSL005 Exp Smoothing
TSL006 Local Level

                                               TSL005         TSL006
Log likelihood                                      -        -380.01   
Akaike Information Criterion (AIC)                  -         766.02   
Bias corrected AIC (AICc)                           -         766.44   
Bayesian Information Criterion (BIC)                -         772.30   
in-sample MSE                                23735.33       23069.80   
... RMSE                                       154.06         151.89   
... MAE                                        119.86         118.60   
... MAPE                                        14.06          13.70   
Sample size                                       100            100   
Effective sample size                              99             99   
* based on one-step-ahead forecast errors
                            

If you want to compare models and conclude something like "model A is better than model B", it is important to note that only looking at in-sample (Training sample) model fit can be misleading. It is often a good idea to take forecast performance into account as well. If model A performs better on both model fit and forecast performance, it is a good indication of model A being preferred over model B. We see examples of comparing forecast performance in other Case studies. The forecasts of both our models can be visually inspected on the Forecasting page. The figure below plots the forecasts of both model in one graph. Since no new data is coming in, the forecasts are just straight lines but the level (height) of the lines differ per model. Note that the local level model is not just a theoretical model, it has practical value as well. For example for inflation modelling, the local level model is a strong contender. We will see more complex forecasting patterns in other case studies.

Outliers and Structural breaks

Intervention analysis, also called anomaly detection, is an important part of time series analysis. We distinguish two types of anomalies, Outliers and Structural breaks. For example, early warning systems rely on outlier and break detection. Could a catastrophic event have been seen in advance? Take for example sensor readings from an important piece of heavy machinery. The breaking down of this machine would cost a company a lot of money. If anomalies were detected in the sensor reading, preventive maintenance might have saved the company from a break-down of the machine. Intervention variables are dummy (or indicator) variables which are used to take account of outlying observations and structural breaks. These data irregularities are usually thought of as arising from a specific event, for example a strike in the case of an outlier or a change in policy in the case of a structural break. An outlier can be thought of as an unusually large value of the irregular disturbance at a particular time. It can be captured by an impulse intervention variable which takes the value one at the time of the outlier and zero elsewhere. A structural break in which the level of the series shifts up or down is modelled by a step intervention variable which is zero before the event and one after. Alternatively it can be modelled in exactly the same way by adding an outlying intervention to the level equation. In other words the break is identified with an unusually large value of the level disturbance. TSL is able to propose a set of potential outliers and structural breaks for time series. It is an effective multi-step procedure based on the auxiliary residuals, see also Harvey and Koopman (1992) for details. First the selected model is estimated and the diagnostics are investigated. Then a first (larger) set of potential outliers and trend breaks are selected from the auxiliary residuals. After re-estimation of the model, only those interventions survive that are sufficiently significant. After the automatic selection, the results are reported. All considered outliers and breaks are kept in the intervention dialog and they can be deleted from the model or added to the model.
The Nile time series has some interesting features with regard to Intervention analysis. To see this, go back to the Database page and select the Nile time series again without missing values. Next, go to the Build your own model page and select a time-varying level and time-varying slope. These two model components correspond to a model with the name Local Linear Trend model. On top of that, select Intervention variables with the automatic setting. Next, go to the Estimation page, make sure the sample starts at $t = 1$ and ends at $t = 100$ and click the green Estimate button. Once TSL is done estimating, you should see the graph as presented in the figure below. We see from the figure that TSL finds a structural break and an outlier. We can also inspect these in more detail by looking at the Text output page where we see

            Beta                        Value       Std.Err         t-stat          Prob
            beta_outlier_1913-01-01    -389.4        123.92         -3.143        0.0022
            beta_break_1899-01-01      -265.5         43.67         -6.079    2.4458e-08
                            

TSL finds the location of the structural break at 1899 which is very plausible since the year 1899 corresponds to the building of a dam at Aswan. Interestingly, the addition of the outlier and structural break remove certain dynamics from the data which we can see from the straight lines in the graph which are the result of the (close to) zero variances from the Level and Slope component.