Exponential Smoothing Forecast
Simple Exponential Smoothing
Simple Exponential Smoothing forecasts are calculated using weighted averages, where the weights decrease exponentially as observations come from further in the past with the smallest weights associated with the oldest observations:
Error, Trend, Seasonal (ETS) Models
Error, Trend, Seasonal (ETS) models expand on simple exponential smoothing to accommodate trend and seasonal components as well as additive or multiplicative errors. Simple Exponential Smoothing is an Error Model. Error, Trend model is Holt’s Linear, also known as double exponential smoothing. Error, Trend, Seasonal model is HoltWinters, also known as triple exponential smoothing. Rob Hyndman has developed a complete taxonomy that describes all of the combinations of exponential smooth models in a consistent manner (see fpp2):
Summary of ETS Models in SigmaXL:
Exponential Smoothing Parameter Estimation, Model Statistics and Information Criteria for Model Comparison
Model parameters are solved using nonlinear maximization of the LogLikelihood function. LogLikelihood is related to Ln(SumofSquares Error). Information Criteria AICc, AIC and BIC are calculated using 2*LogLikelihood and incorporate a penalty for the number of terms in the model, so smaller is better. These are used in automatic model selection. AICc is the default Information Criterion, based on forecast error performance with competition data.
Missing Values
Exponential Smoothing handles missing values with seasonally adjusted linear interpolation. While there is robustness against some missing values, if the number of missing values is large then model estimation and forecast accuracy will be degraded. Upon selection of a time series with missing values, you will see a popup “Warning: Missing values detected. Seasonally adjusted linear interpolation will be used.”
where scale is the MAE of the insample naïve or seasonal naïve forecast (set all forecasts to be the value of the last observation/period). Note that scale for nonseasonal is identical to MRbar of an Individuals Moving Range chart. A scaled error is less than one if it arises from a better forecast than the average naïve/seasonal naïve forecast. Conversely, it is greater than one if the forecast is worse than the average naïve forecast. MASE is recommended over the popular MAPE, because MAPE becomes infinite if any y_t=0.
InSample is also referred to as the “Train” data. The same metrics may be applied to OutofSample (Withhold), also referred to as the “Test” data and may be OneStepAhead or MultiStepAhead. This will be demonstrated later. OutofSample (Withhold) data is not used in the model parameter estimation, so is a much better indicator of true forecast accuracy. Insample accuracy metrics can be biased due to overfitting. Scale is always computed using the insample data.
 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 twohour intervals. See the Run Chart, ACF/PACF Plots, Spectral Density Plot and Seasonal Trend Decomposition Plots for this data.
 Click SigmaXL > Time Series Forecasting > Exponential Smoothing Forecast > Forecast. Ensure that the entire data table is selected. If not, check Use Entire Data Table. Click Next.
 Select Concentration, click
Numeric Data
Variable (Y) >>. Check Display
ACF/PACF/LB Plots and Display Residual Plots.
Leave Specify Model Periods, Seasonal Frequency
and BoxCox Transformation unchecked. We will
use the default No. of Forecast Periods = 24
and Prediction Interval = 95.0 %.
• Optional Time Axis Labels will be displayed on the forecast chart time axis. If used, dates for the forecast periods should also be included, otherwise the time axis will be blank for the forecast periods.
• No. of Forecast Periods are the number of time series values to be predicted (forecast horizon). The most accurate forecast will be for the first predicted value (onestepahead).
• Prediction Interval % is the confidence level for the individual predictions. For example, a 95% prediction interval contains a range of values which should include the actual future value with 95% probability. The interval will get larger the further out you predict.
• Model Options opens another dialog which allows you to set automatic options or to specify a model.
• Display ACF/PACF/LB option will produce ACF and PACF plots for the raw data as well as for the model residuals. The LB plot is a plot of LjungBox test PValues for various lags and is used to determine if a group of autocorrelations are significant, (i.e., the autocorrelations do not come from a white noise series).
• Display Residual Plots will produce a table of model residuals and the usual model residual plots: histogram, normal probability plot, residuals versus data order, and residuals vs forecast value. Note that if a BoxCox transformation is applied, the residuals are transformed so will not be equal to forecast  actual.
• Specify Model Periods are used to specify a start period, end period or withhold sample. The withheld data is not used in model estimation, so this is very useful for model validation and comparison. This will be used in a later example.
• Seasonal Frequency and BoxCox Transformation will be used in a later example.
 Click Model Options.
• Automatic Model Selection will be used later. It is the default selection.
• Model Selection Criterion is the information criterion metric to be used in automatic model selection. AICc is the default selection.
• Clicking OK accepts the settings and returns you to the previous dialog. Clicking Cancel will cancel any changes and return you to the previous dialog.
 Select Specify Model.
• Specify Model allows you to manually specify the ErrorTrendSeasonal model. Seasonal options are greyed out because the Seasonal Frequency option in the main dialog is unchecked. A summary description of the model is also given. This will be included in the forecast report.
 We will use the default Error:
Additive and Trend: None, which is a
simple exponential smoothing model, the same as was demonstrated
previously. Click OK to return to the
Exponential Smoothing Forecast dialog. Click OK.
The exponential smoothing forecast chart is given:
This is very similar to the exponential smooth plot demonstrated above, showing the raw Concentration data (black) and onestepahead forecast values (red), but with the addition of a 24period forecast and the 95% prediction interval.
 You can zoom in to view the last 30
points on the Forecast Chart. Click SigmaXL Chart Tools
> Show Last 30 Data Points.
 To restore the chart, click
SigmaXL Chart Tools > Show All Data Points.
 You can also scroll through the data.
Click SigmaXL Chart Tools > Enable Scrolling
You are 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.
 Click OK. The scroll
dialog appears allowing you to specify the Start Period
and Window Width:
At any point, you can click Restore/Show All Data Points or Freeze Chart. Freezing the chart will remove the scroll and unload the dialog. The scroll dialog will also unload if you change worksheets. To restore the dialog, click SigmaXL Chart Tools > Enable Scrolling.
 Click OK. A scroll bar
appears beneath the forecast chart. You can also change the
Start Subgroup and Window Width
and Update.
You can scroll through by clicking to the right or left, with the specified window width of 20. Click left once to view the chart as shown.
 Click Cancel to exit
the scroll dialog.
 Scroll down to view the Exponential
Smoothing Model header:
The model Simple Exponential Smoothing with Additive Errors (A, N, N) – Exponentially Weighted Moving Average (EWMA) is the user specified model. This is the same model information that was displayed in the Model Selection dialog.
If we had checked Specify Model Periods in the main dialog, the start, end or withhold selection would be summarized here as well.
 The Exponential Smoothing Model
Information is given as:
This is a summary of model information with Seasonal Frequency = 1 (nonseasonal); Model Selection Criterion = “Specified” because the model was user specified; and BoxCox Transformation = “N/A” because BoxCox Transformation was unchecked.
 The Parameter Estimates are:
The parameter estimates closely match the values obtained earlier in the demonstration using Solver:
The slight differences in parameter results are due to differences in optimization method.
 The Exponential Smoothing Model
Statistics are:
Degrees of freedom (DF) = n – 2 (terms in the model).
The results match those given in the demonstration using Solver:
The equations may be viewed by clicking on the Demo cells M5, M6, M7 or M8.
 The InSample Forecast Accuracy metrics
are:
MASE is less than one, so it is a better forecast than would be obtained from a naïve forecast (set all forecasts to be the value of the last observation). See Forecast Accuracy Metrics. The results closely match those given in the demonstration using Solver:
The equations may be viewed by clicking on the Demo cells P3 to P7.
 The Forecast Table is given as:
These are the same forecast and prediction interval values displayed in the Forecast Chart but provided for further analysis or charting. If Withhold Periods are specified, the Withhold Data will be displayed as well.
 Click on the Exp. Smooth. ACF
PACF LB sheet to view the ACF/PACF/LB Plots:
These match the plots that we obtained previously in the Demo. We can see that all of the autocorrelation has been removed by the exponential smoothing model (with the exception of lag 15 in the PACF), so this is a good fit to the time series data.
The LB plot is a plot of LjungBox test PValues for various lags and is used to determine if a group of autocorrelations are significant, (i.e., the autocorrelations do not come from a white noise series). The red PValues are significant (alpha=.05) and the blue PValues are not significant. It is desirable that all PValues be blue. The ACF/PACF plots indicated that almost all of the correlation has been accounted for in the model, but the LjungBox plot shows that some significant autocorrelation still remains  so the model can potentially be improved. This does not mean that the model is a bad model, it can still be very useful for prediction purposes, but the prediction intervals may not provide accurate coverage.
 Click on the Exp. Smoothing Residuals
sheet to view the Residual Plots:
These residual plots are the same as used in SigmaXL’s Multiple Regression. The histogram matches what we obtained in the Demo (the normal curve is not applied here). 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. Later, we will apply a control chart to the residuals to formally test for significant outliers or assignable causes.
 Now we will rerun Exponential Smoothing
on the Concentration data but use Automatic Model Selection.
Click Recall SigmaXL Dialog menu or press
F3 to
recall last dialog.
 Click Model Options.
Select Automatic Model Selection. We will use
the default Model Selection Criterion: AICc – Akaike
information criterion with small sample size correction.
 Click OK to return to
the Exponential Smoothing Forecast dialog. Click OK. The
exponential smoothing forecast report is given.
 Scroll down to view the Exponential
Smoothing Model header:
The model Simple Exponential Smoothing with Multiplicative Errors (M, N, N) was selected as the best fit for the Concentration data based on the AICc criterion. The point forecasts produced by the Multiplicative and Additive models are identical if they use the same smoothing parameter values. Multiplicative will, however, generate different prediction intervals to accommodate change in variance.
 The Exponential Smoothing Model
Information is given as:
This is a summary of model information with Seasonal Frequency = 1 (nonseasonal); Model Selection Criterion = “AICc” and BoxCox Transformation = “N/A” because BoxCox Transformation was unchecked.
 The Parameter Estimates are:
These are fairly close to the parameter estimates obtained above using the additive model. 
The Exponential Smoothing Model Statistics are:
The LogLikelihood, AICc, AIC and BIC are close to the values obtained above using the additive model, but the LogLikelihood is slightly higher giving a lower AICc, so multiplicative was selected as the best model. Note however that the StDev and Variance are very different. This difference is due to the multiplicative residuals being relative errors: e_t=(y_ty┴^_t)/y┴^_t 
The InSample Forecast Accuracy metrics, Forecast Table and
ACF/PACF/LB Plots for multiplicative are very close to the
additive model. The multiplicative residual plots look the same,
but note the different scale due to the relative errors.

Now we will rerun Exponential Smoothing on the Concentration
data with Automatic Model Selection, but will use Specify
Withhold Periods. Click Recall SigmaXL Dialog menu or press F3
to recall last dialog. Check Specify Model Periods. Set Withhold
Periods = 24 (i.e., we will forecast 24 periods and compare
against the withheld actual). We will use the default Withhold
Forecast Type: OneStepAhead with Prediction Interval at:
Start
of Withhold.
• Specify Model Periods option allows you to specify the start and end periods used in automatic model identification and parameter estimation. Typically, Start Model at Period is kept = 1 and Withhold Periods specifies the number of periods to be withheld for outofsample testing. End Model at Period specifies the end period, so the withhold sample size would be: total number of observations – end period.
• Withhold Forecast Type: OneStepAhead will exclude the withhold sample from automatic model identification and parameter estimation, but uses the withhold data to update the predicted onestep ahead forecast. This is useful to assess forecast error when you only care about the shortterm onestep ahead prediction.
• Withhold Forecast Type: OneStepAhead with Prediction Interval at: Start of Withhold will display the prediction interval for the duration of the withhold sample. Note that the length of the prediction interval is determined by the number of withhold periods, so overrides the specified No. of Forecast Periods.
• Withhold Forecast Type: OneStepAhead with Prediction Interval at: End of Withhold will display the prediction interval at the end of the withhold sample. The length of the prediction interval is determined by the specified No. of Forecast Periods.
• Include in Residuals will treat the onestepahead forecast errors as residuals (even though they were not part of the model estimation process) and will be included in the ACF/PACF/LB Residual Plots along with the Residuals report and graphs. Typically, this is kept unchecked.
• Withhold Forecast Type: MultiStepAhead with Prediction Interval at Start of Withhold will exclude the withhold sample from automatic model identification and parameter estimation and does not use the withhold data to update the predicted onestep ahead forecast. This is useful to assess forecast error when you are interested in a longterm forecast window (horizon). The prediction interval will be displayed for the duration of the withhold sample. Note that the length of the prediction interval is determined by the number of withhold periods, so overrides the specified No. of Forecast Periods. These forecast errors are not included in ACF/PACF/LB Residual Plots or the Residuals report and graphs.

Click Model Options. Select Automatic Model Selection. We will
use the default Model Selection Criterion: AICc – Akaike
information criterion with small sample size correction.
Tip: When using Recall SigmaXL Dialog, and if there are no changes to the Model Option settings, the previous settings will be used. It is not necessary to repeat this step. 
Click OK to return to the Exponential Smoothing
Forecast dialog. Click OK. The exponential
smoothing forecast report is given:
The blank dots are the data values in the withhold sample with a onestepahead forecast and prediction intervals displayed at the start of the withhold sample.

Scroll down to view the Exponential Smoothing Model header:
As with the complete data, the model Simple Exponential Smoothing with Multiplicative Errors (M, N, N) was selected as the best fit for the Concentration data based on the AICc criterion. The header also includes the number of specified withhold periods. 
The Parameter Estimates are:
These are close to the parameter estimates obtained above (which used all of the data with the multiplicative model). 
The Exponential Smoothing Model Statistics are:
These are fairly close to the model statistics obtained above (which used all of the data with the multiplicative model). Here we are using only 173 of the 197 observations. 
The Forecast Accuracy metrics are:
As expected, the OutofSample (Withhold) OneStepAhead Forecast errors are larger than the InSample (Estimation) OneStepAhead Forecast errors. 
The Forecast Table is given as:
These are the same forecast and prediction interval values displayed in the Forecast Chart, but provided for further analysis or charting. The Withhold Data is also displayed. 
The ACF/PACF/LB Residual Plots and Residual Plots are based on
the insample data. The plots look similar to the complete data
above, except for the LjungBox PValues:
The Simple Exponential Smoothing with Multiplicative Errors (M, N, N) model is a better fit to the subset than the complete data, with all PValues being blue (> .05).The Simple Exponential Smoothing with Multiplicative Errors (M, N, N) model is a better fit to the subset than the complete data, with all PValues being blue (> .05).  If Include in Residuals was checked then the residuals would also include the OutofSample (Withhold) OneStepAhead Forecast errors.

Now we will rerun Exponential Smoothing on the Concentration
data, but use MultiStepAhead for the Withhold Forecast. Click
Recall SigmaXL Dialog menu or press F3 to recall last dialog.
Select Withhold Forecast Type: MultiStepAhead with Prediction
Interval at Start of Withhold.

Click OK. The exponential smoothing forecast
report is given:
The blank dots are the data values in the withhold sample with a multistep forecast and prediction intervals displayed at the start of the withhold sample. 
The Forecast Accuracy metrics are:
As expected, the OutofSample (Withhold) MultiStepAhead Forecast errors are larger than the InSample (Estimation) OneStepAhead Forecast errors and the OutofSample (Withhold) OneStepAhead Forecast errors above.
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