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Exponential Smoothing Multiple Seasonal Decomposition (MSD) Control Chart

Exponential Smoothing is limited to a maximum seasonal frequency of 24. For higher frequencies use Exponential Smoothing – Multiple Seasonal Decomposition (MSD). The seasonal component is first removed through decomposition, a nonseasonal exponential smooth model fitted to the remainder (+trend), and then the seasonal component is added back in.

As the name implies, Multiple Seasonal Decomposition (MSD) also accommodates multiple seasonality, for example the half-hourly data with a seasonal frequency of 48 observations per day and 336 observations per week.

An Individuals control chart of the residuals is created for this forecast method. 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 exponential smoothing multiplicative, the moving 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.

  1. 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. The Multiple Seasonal Decomposition (MSD) option is not necessary for this data, but by way of introduction, we will use this to compare to the previous analysis.
  2. Click SigmaXL > Time Series Forecasting > Exponential Smoothing Control Chart > Multiple Seasonal Decomposition Control Chart. Ensure that the entire data table is selected. If not, Use Entire Data Table. Click Next.
  3. 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.


    EXPMSDCC3

  4. Click Model Options.

    EXPMSDCC4
  5. We will use the default Automatic Model Selection with AICc as the Model Selection Criterion. Click OK to return to the ARIMA Control Chart dialog. Click OK. The ARIMA (MSD) control charts are produced:

    EXPMSDCCStep5
    We can clearly see the out-of-control data points at 50, 51 and 100 on the Residuals Individuals chart. This is similar to what we observed previously with regular ARIMA Control Charts.

  6. Scroll down to view the ARIMA MSD Model header:

    EXPMSDCCStep6
    The model Additive Trend Method with Additive Errors (Holt’s Linear) (A, A, N) was automatically selected as the best fit for the deseasonalized Modified Ln Airline Passenger data based on the AICc criterion. There are no withhold periods.

  7. The Exponential Smoothing Model Summary is given as:

    EXPMSDCCStep7
    This is a summary of model information with Seasonal Frequency = 12 using Decomposition and Model Selection Criterion = “AICc”. The Box-Cox Transformation is “N/A”.
  8. We will not review the Parameter Estimates, Model Statistics and Forecast Accuracy as they are close to the ARIMA MSD values given earlier, although note that slight differences are due to the introduction of an outlier and a shift, as well now we are using all of the data, i.e., there are no withhold periods. Earlier we used a Box-Cox Transformation with Lambda=0 and here we are using Ln of the data.
  9. Open Half-Hourly Multiple Seasonal Electricity Demand - Taylor.xlsx (Sheet 1 tab). This is half-hourly electricity demand (MW) in England and Wales from Monday, June 5, 2000 to Sunday, August 27, 2000 (taylor, R forecast). This data has multiple seasonality with frequency = 48 (observations per day) and 336 (observations per week), with a total of 4032 observations. See the Run Chart, ACF/PACF Plots, Spectral Density Plot and Seasonal Trend Decomposition Plots for this data.
  10.  We will first construct a classical Individuals Control 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.
  11. Select Demand, click Numeric Data Variable (Y) >>. Click OK. An Individuals Control Chart is produced:

    ARIMAMSDCC5
    With the high frequency seasonality, this control chart is meaningless.

  12. Click SigmaXL > Time Series Forecasting > Exponential Control Chart > Multiple Seasonal Decomposition Control Chart. Ensure that the entire data table is selected. If not, check Use Entire Data Table. Click Next.
  13. Select Demand, click Numeric Time Series Data (Y) >>. Uncheck Display ACF/PACF/LB Plots and Display Residual Plots. Check Seasonal Frequency with Specify = 48 336. Leave Specify Model Periods and Box-Cox Transformation unchecked.

    EXPMSDCCStep13
  14. Click Model Options.

    EXPMSDCCStep14
  15. We will use the default Automatic Model Selection with AICc as the Model Selection Criterion. Click OK to return to the ARIMA MSD Control Chart dialog. Click OK. The ARIMA MSD control charts are produced:

    EXPMSDCCStep4
    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. You can enable scrolling or zoom in to view the last 30 points on the Control Charts. Here we will do the latter. Click SigmaXL Chart Tools > Show Last 30 Data Points.

    EXPMSDCCStep16A
    EXPMSDCCStep16B
    The out-of-control data point at 4026 is now clearly visible. Tip: The Y axis scaling for the Moving Limits chart makes the same point difficult to see.

    You may wish to change the maximum and minimum values, right click Format Axis, and change the Bounds Minimum Maximum values.

  17. To restore the chart, click SigmaXL Chart Tools > Show All Data Points.

    EXPMSDCCStep17
  18. Scroll down to view the Exponential Smoothing Model header:

    EXPMSDCCStep18
    The model Simple Exponential Smoothing with Multiplicative Errors (M, N, N) was automatically selected as the best fit for the deseasonalized Demand data based on the AICc criterion. There are no withhold periods.

    This is the same model used previously in Exponential Smoothing MSD Forecast, but that used a withhold sample of 96.

 


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