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Exponential Smoothing – Multiple Seasonal Decomposition (MSD) Forecast

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. For forecasting, a naïve seasonal forecast is used on the seasonal component. Note that the prediction intervals are derived from the exponential smoothing model and do not include uncertainty in the seasonal component.

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. When using MSD, it is recommended to limit the forecast period to 2*dominant seasonal frequency.

  1. Open Monthly Airline Passengers - Series G.xlsx (Sheet 1 tab). This is the Series G data from Box and Jenkins, monthly total international airline passengers for January 1949 to December 1960. See the Run Chart, ACF/PACF Plots, Spectral Density Plot and Seasonal Trend Decomposition Plots for this data. 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 Forecast > Multiple Seasonal Decomposition Forecast. Ensure that the entire data table is selected. If not, check Use Entire Data Table. Click Next.
  3. Select Monthly Airline Passengers, click Numeric Time Series Data (Y) >>. Select Date, click Optional Time Axis Labels >>. Check Display ACF/PACF/LB Plots and Display Residual Plots. Check Specify Model Periods. Set Withhold Periods = 24. Select Withhold Forecast Type: Multi-Step-Ahead with Prediction Interval at Start of Withhold. Select Seasonal Frequency Specify and enter 12. Check Box-Cox Transformation and select Rounded Lambda. We will use the default Prediction Interval = 95.0 %.


    ExpMSDFORECASTStep3
    Seasonal Frequency can have multiple entries however, we recommend no more than 3 values.

  4. Click Model Options.

    ExpMSDFORECASTStep4
  5. We will use the default Automatic Model Selection with AICc as the Model Selection Criterion. Click OK to return to the ARIMA MSD Forecast dialog. Click OK. The exponential smoothing forecast report is given:

    ExpMSDFORECASTStep5
  6. Scroll down to view the ARIMA Model header:

    ExpMSDFORECASTStep6
    After Multiple Seasonal Decomposition, the model is Additive Trend Method with Additive Errors (Holt’s Linear) (A, A, N).

    (A, A, N) was automatically selected as the best fit for the deseasonalized Airline Passenger data based on the AICc criterion.

    The header also includes the number of specified withhold periods.

  7. The Exponential Smoothing Model Summary is given as:

    ExpMSDFORECASTStep7
    This is a summary of the model information with Seasonal Frequency = 12 using Decomposition and Model Selection Criterion = “AICc”. The Box-Cox Transformation is “Rounded Lambda” with Lambda = 0 (Ln transformation).

  8. The Parameter Estimates for the deseasonalized Airline Passenger data are:

    ExpMSDFORECASTStep8
    • Error includes the smoothing parameter alpha and initial level value (l). The error is additive, but on the Ln transformed data.

    • Trend adds a smoothing parameter (beta) and initial trend value (b).

    • Seasonal smoothing parameter (gamma) and initial seasonal values are not computed because the data has been deseasonalized.

  9. The Exponential Smoothing Model Statistics are:

    ExpMSDFORECASTStep9
    • The number of observations, n = 144 – 24 (withhold) = 120

    • Degrees of freedom (DF) = 120 (n) – 3 (2 terms in the model, 1 nonseasonal difference) = 117

    • Note that the model statistics are based on the Ln transformed data, not the original data.

    • The model statistics are better than the previous analysis: lower StDev and Variance, higher Log-Likelihood and lower AICc, AIC and BIC. This is to be expected because the data has been deseasonalized so the seasonal error component is not included.

  10. The Forecast Accuracy metrics are:

    ExpMSDFORECASTStep10
    Comparing to our earlier analysis, both the In-Sample (Estimation) One-Step-Ahead Forecast errors and Out-of Sample (Withhold) Multi-Step-Ahead Forecast errors are slightly smaller. (This was not expected, typically a Seasonal Exponential Smoothing or ARIMA model would give a more accurate forecast, based on comparison of methods using forecast competition data). However, given that we are forecasting out for two years, both models look very good..

    Forecast Accuracy metrics are calculated using the actual raw data versus inverse transformed forecast as displayed in the Forecast Chart and Table, so allow comparison across all model types and transformations.

  11. Click on the Exp Smooth MSD ACF PACF LB sheet to view the ACF/PACF/LB Plots:

    ExpMSDFORECASTStep11A

    ExpMSDFORECASTStep11B
    The ACF/PACF Residuals Plots are similar to the previous analysis and indicate that almost all of the autocorrelation has been accounted for in the model, however the Ljung-Box plot confirms that this is a better fit, with most P-Values being blue (> .05).

  12. Click on the Exp Smooth MSD Residuals sheet to view the Residual Plots:

    ExpMSDFORECAST12
    The residuals are approximately normally distributed, with a roughly straight line on the normal probability plot. There are no obvious extreme outliers or patterns in the charts.

    Note that Residuals for MSD are the final observed -predicted values, so there are no scaling differences if the model uses additive or multiplicative error. Since a Box-Cox transformation was used, the residuals are in Ln transformed units.

 


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