Exponential Smoothing Control Charts

Exponential Smoothing Control Charts

Use SigmaXL to create an exponential smoothing control chart in Excel from forecast residuals and moving limits.

An Individuals control chart is created using the residuals of the Exponential Smoothing forecast model.

The Moving Limits chart uses the one step prediction as the center line, so the control limits will move with the center line. If a Box-Cox transformation is used then an inverse transformation is applied to calculate the control limits. If the residuals are from a multiplicative exponential smoothing model, the control limits are approximate and out-of-control signals may not exactly match the Individuals Chart. If that occurs, the Individuals Chart should be used to determine what points are out-of-control.

The popular Add Data, Show Last 30 and Scroll features in SigmaXL Chart Tools are available for these control charts. For Add Data, the time series models are not refitted, but used to compute the residual values for the new data.

For further details and references, see the Appendix: Control Charts for Autocorrelated Data.

Note that a Moving Range Chart and Tests for Special Causes are not available here, but the user can store and select Residuals, then create with SigmaXL > Control Charts > Individuals & Moving Range.

1. Open Chemical Process Concentration - Series A.xlsx (Sheet 1 tab). This is the Series A data from Box and Jenkins, a set of 197 concentration values from a chemical process taken at two-hour intervals.


2. Earlier we saw that this process has significant autocorrelation. In order to see the impact on a control chart, we will construct an Individuals chart on the raw data. Click SigmaXL > Control Charts > Individuals. Ensure that the entire data table is selected. If not, check Use Entire Data Table. Click Next.


3. Select Concentration, click Numeric Data Variable (Y) >>. Click OK. An Individuals Control Chart is produced:

   
   
   There are 17 out-of-control data points, largely due to the autocorrelation. Searching for assignable causes using this chart as is would be futile.
   
   

4. Now click Sheet 1 tab and SigmaXL > Time Series Forecasting > Exponential Smoothing Control Chart > Control Chart. Ensure that the entire data table is selected. If not, check Use Entire Data Table. Click Next.


5. Select Concentration, click Numeric Time Series Data (Y) >>. Uncheck Display ACF/PACF/LB Plots. Leave Display Residual Plots, Specify Model Periods, Seasonal Frequency and Box-Cox Transformation unchecked.
   
   
   Since we will be running the same (A, N, N) model as used earlier, we will not need the ACF/PACF/LB and Residuals Plots.


6. Click Model Options. Select Specify Model.
   
   
   
7. We will use the default Error: Additive and Trend: None, which is a simple exponential smoothing model, or Exponentially Weighted Moving Average (EWMA). Click OK to return to the Exponential Smoothing Control Chart dialog. Click OK. The exponential smoothing control charts are produced:
   
   
   
   Now we only have two out-of-control data points on the Individuals chart to investigate. The Moving Limits chart uses the one step prediction as the center line, so the control limits move with the center line.
   
   
8. You can scroll through the chart data points. Click SigmaXL Chart Tools > Enable Scrolling.
   
   
   You may be prompted with a warning message that custom formatting on the chart will be cleared. You can avoid seeing this warning by checking Save this choice as default and do not show this form again.
   
   
9. Click OK. The scroll dialog appears allowing you to specify the Start Subgroup and Window Width. Enter Start Subgroup = 40 and Window Width = 30 to view the two out-of-control data points.
   
   
   
10. Click OK. This allows us to zoom in on the out-of-control points at 43 and 64.
    
    
    
    Observation 43 is lower than expected from the exponential smoothing forecast model. Observation 64 is higher than expected.
    
    
11. Click Cancel to exit the scroll dialog.


12. Now we will add a new data point to the Series A Concentration Data. The residuals will be computed using the same model as above without re-estimation of the model parameters or recalculation of the control limits. This is also known as the Phase II application of a Control Chart, where an out-of-control signal should lead to an investigation into the assignable cause and corrective action or process adjustment applied. Click Sheet1, enter the value 19 as shown in cell B199 (and optionally Observation number 198 in cell A199).
    
    
    
13. Click Exp. Smoothing Control Charts tab (if more than one control chart sheet exists in the workbook, please select the chart where the data will be added).


14. Click SigmaXL Chart Tools > Add Data to this Control Chart.
    
    
    
15. The Residuals Individuals Control Chart and Moving Limits Charts are now updated with the new data, showing this as an out-of-control data point:
    
    
    Now we have a chart that can be used to identify assignable causes. The number of out-of-control signals have been dramatically reduced.
    
    
16. We recommend renaming the workbook to Chemical Process Concentration Series A_AddData.xlsx, so that later use of the Concentration data does not include the added data point.

Monthly Airline Passengers - Modified for Control Charts

17. Open Monthly Airline Passengers - Modified for Control Charts.xlsx (Sheet 1 tab). This is based on the Series G data from Box and Jenkins, monthly total international airline passengers for January 1949 to December 1960. A Ln transformation is applied (avoiding the need for a Box-Cox transformation), a negative outlier is added at 50 (-.25) and a level shift applied (+.25), starting at 100. Coded variables were added to help distinguish an outlier versus a shift, but they will be analyzed later using ARIMA Forecast with Predictors. Exponential Smoothing does not support predictors.


18. Earlier we saw that this process has significant autocorrelation with a strong trend and seasonality. In order to see the impact on a control chart, we will construct an Individuals chart on the raw data. Click SigmaXL > Control Charts > Individuals. Ensure that the entire data table is selected. If not, check Use Entire Data Table. Click Next.


19. Select Ln (Airline Passengers-Modified), click Numeric Data Variable (Y) >>. Click OK. An Individuals Control Chart is produced:



With strong trend, seasonality and positive autocorrelation, this control chart is meaningless.


20. Now click Sheet 1 tab and SigmaXL > Time Series Forecasting > Exponential Smoothing Control Chart > Control Chart. Ensure that the entire data table is selected. If not, check Use Entire Data Table. Click Next.


21. Select Ln(Airline Passengers-Modified), click Numeric Time Series Data (Y) >>. Uncheck Display ACF/PACF/LB Plots and Display Residual Plots. Check Seasonal Frequency with Specify = 12. Leave Specify Model Periods and Box-Cox Transformation unchecked.




22. Click Model Options.



23. We will use the default Automatic Model Selection with AICc as the Model Selection Criterion. Click OK to return to the Exponential Smoothing Control Chart dialog. Click OK. The exponential smoothing control charts are produced:



Now we can clearly see the out-of-control data points at 50, 51 and 100 on the Individuals chart. In order to view the points on the Moving Limits chart we will use scrolling.


24. Click SigmaXL Chart Tools > Enable Scrolling



You may be prompted with a warning message that custom formatting on the chart will be cleared. You can avoid seeing this warning by checking Save this choice as default and do not show this form again.


25. Click OK. The scroll dialog appears allowing you to specify the Start Subgroup and Window Width. Enter Start Subgroup = 40 and Window Width = 20 to view the first two out-of-control data points.




26. Click OK. This allows us to zoom in on the out-of-control points at 50 and 51.



Observation 50 is lower than expected from the exponential smoothing forecast model. Observation 51 is higher than expected. Later investigation will reveal that this is a single negative outlier.

Tip: Scrolling keeps the original Y axis minimum and maximum setting. You may wish to change this to auto by clicking on the Y axis, right click Format Axis, click Bounds Minimum Reset and Bounds Maximum Reset. This changes the axis settings to Auto so when you scroll or Update the Y axis will automatically adjust as well.


27. Now enter Start Subgroup = 90 and Window Width = 20 to view the third out-of-control data point.




28. Click Update.



Observation 100 is higher than expected from the exponential smoothing forecast model. Later investigation will reveal that this is a shift in the mean.


29. Click Cancel to exit the scroll dialog.


30. Scroll down to view the Exponential Smoothing Model header:



The model Additive Trend, Additive Seasonal Method with Additive Errors (Holt-Winters) (A, A, A) was automatically selected as the best fit for the Modified Ln Airline Passenger data based on the AICc criterion.


31. The Parameter Estimates and Exponential Smoothing Model Statistics are slightly different than our earlier analysis because we have introduced an outlier and a shift, as well here we are using all of the data, i.e., there are no withhold periods. Note that earlier we used a Box-Cox Transformation with Lambda=0 and here we are using Ln of the data.


32. The Forecast Accuracy metrics are given as:



Note that these forecast errors are very different than our earlier analysis where the forecast errors were calculated on the raw data versus final predicted values, but here we are using Ln of the Airline Passenger data.