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Seasonal Trend Decomposition Plots

  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. Click SigmaXL > Time Series Forecasting > Seasonal Trend Decomposition Plots. Ensure that the entire data table is selected. If not, Use Entire Data Table. Click Next.

  3. Click Concentration, click Numeric Time Series Data (Y) >>. Seasonal Frequency and Box-Cox Transformation should be unchecked as shown.


  4. Click OK. A Trend Decomposition Plot for Concentration is produced.

    We can clearly see the “wandering mean” in this process. As discussed earlier, this can be modeled with exponential smoothing or with ARIMA after differencing.

  5. A Decomposition Summary report is also included to the right of the plot, indicating seasonal frequency and Box-Cox parameters (if applicable):

    Seasonal Frequency = 1 denotes a nonseasonal process.

  6. The Smoothed Trend and Remainder Plots are also produced as shown:


The Seasonal Trend Decomposition Plots are useful to visually distinguish trend and seasonal components in the time series data. If the Seasonal Frequency is unchecked, a Trend Decomposition Plot is produced as the first chart, showing the raw data and the trend. The trend component uses data smoothing, rather than a linear trend so that it may display either a linear trend or cyclical patterns. If a single seasonal frequency is specified, a Seasonal Trend Decomposition Plot is produced, showing the data, smoothed trend and seasonal component. If a multiple seasonal frequency is specified, a Multiple Seasonal Trend Decomposition Plot is produced, showing the data, smoothed trend and multiple seasonal component.

The second chart shows just the smoothed trend; the third chart (if applicable) shows just the seasonal or multiple seasonal component. The final chart is the remainder component.

This is an additive decomposition model, so the sum of the trend value + seasonal value(s) + remainder value gives the original data value. A multiplicative equivalent may be obtained by specifying the Box-Cox Transformation with Lambda = 0, which is a Ln transformation, but the charts will display the transformed data to maintain an additive model. Rounded or Optimal Lambda may also be used, but will only consider the range of values 0 to 1 (this conservative approach is used in time series forecasting, unlike regular Box-Cox in SigmaXL which uses a range of -5 to 5). See Appendix: Box-Cox Transformation.

The decomposition algorithms used here are the same as used in Exponential Smoothing – Multiple Seasonal Decomposition (MSD), and ARIMA – 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. This is mainly used for high seasonal frequency and/or multiple seasonal frequency time series. For further details and formulas, see Appendix: Seasonal Trend Decomposition.

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