Exponential Smoothing Analysis

Exponential smoothing is particular type of moving average technique applied to time series data, used to produce smoothed data for presentation, or to make forecasts. The Exponential smoothing method weights past observations by exponentially decreasing weights to forecast future values.

There are some categories of this method:

- Single exponential smoothing
- Browns Double exponential smoothing method
- Holts Double exponential smoothing method
- Winters Triple exponential smoothing method

Single Exponential Smoothing is a procedure that repeats enumeration continually by using the newest data. This method can be used if the data is not significantly influenced by trend and seasonal factor.

To smooth the data with single exponential smoothing requires a parameter called the smoothing constant (). Each data point is given a certain weighting, for the newest data, (1-) for older data and etc. The value of must be between 0 and 1. The following is the equation of smoothed value:

By doing a simple substitution, the equation above can be written as:

Forecasting with single exponential smoothing can be done by substituting this equation:

The equation above also can be written in the following way:

where is the forecasting error for n period. From this equation, we can see that the forecasting resulted with this method is the last forecasted value added with an adjustment for error in the last forecasted value.

Practically, to calculate the smoothing statistic at the first observation , we can use the equation . Then it is substituted into the smoothing statistic equation to calculate , and the smoothing process is continued until we get value. To calculate the equation above, a starting value is needed. can be calculated from the average of several observations. The first several observations can be chosen to determine .

This smoothing method can be used for data which contains linear trend. This method is often called as Brownâ€™s one-parameter linear method.

The following equations are used in double exponential smoothing with Browns method:

Single smoothing statistic equation:

Double smoothing statistic equation:

The procedure to calculate forecasting m forward period with double exponential smoothing with Brown method can be calculated from this equation:

This equation is similar to linear trend method, where:

The smoothing statistic equation above can be solved if the estimation value for is defined. Starting value is defined as:

We can use linear trend model constant calculated with the least squares estimation method to estimate the coefficient of , and .

This method is similar to Browns method, but Holts Method uses different parameters than the one used in original series to smooth the trend value.

The prediction of exponential smoothing can be obtained by using two smoothing constants (with values between 0 and 1) and three equations as follows:

( 1 )

( 2 )

( 3 )

Equation (1) calculates smoothing value from the trend of the previous period added by the last smoothing value . Equation (2) calculates trend value from , , and . Finally, equation (3) (forward prediction) is obtained from trend, , multiplied with the amount of next period forecasted, m, and added to basic value .

There are two parameters needed to estimate exponential smoothing with Holts method, the smoothing value and the trend . To find these parameters, the least squares method is used. The estimation value for is the intercept value of linear estimation, while is the slope value.

If a time series is stationary, the moving average method or single exponential smoothing can be used to analyze it. If a time series data has a trend component, then double exponential smoothing with Holts method can be used. However, if the time series data contains a seasonal component, then the Triple Exponential Smoothing (Winters) method can be used to handle it.

This method is based on three smoothing equations, Stationary Component, Trend and Seasonal. Both Seasonal component and Trend can be additive or multiplicative.

The whole smoothing equation

Trend smoothing

Seasonal smoothing

Forecasted value

The whole smoothing equation

Trend smoothing

Seasonal smoothing

Forecasted value

Where l is seasonal length (for example, amount of month, or quartile in a year), T is trend component, S is seasonal adjustment factor, and is forecasted value for m next period.

The starting values for and can be obtained from regression equations which have actual variables as dependent variables and time variables as independent variables. This equation constant is a starting value estimation for and slope of regression coefficient is a starting value estimation for the trend component . Whereas the starting value for the seasonal component is calculated by using dummy-variable regression on detrended data (without trend).

**Zaitun Time** performs exponential smoothing analysis of time series data, including single exponential smoothing, double exponential smoothing Brown, double exponential smoothing (Holts), and triple exponential smoothing (Winters). To perform exponential smoothing analysis on a time series variable:

- Click Analysis -> Exponential Smoothing
- Select Variable Dialog appears. Choose a variable you want to analyze with exponential smoothing analysis, and then click OK.
- The Exponential Smoothing form will appear. Choose the Exponential Smoothing method you want to apply to your variable, and determine the smoothing constant (alpha, beta, and gamma). For Triple Exponential Smoothing, determine its type, multiplicative or additive, and seasonal length.
**Zaitun Time Series**also provides Grid Search facility to facilitate user in searching smoothing constant values in yielding the least MSE value. You can search smoothing constant value by determining minimum and maximum boundary and increment interval. Application will search the combination of smoothing constant value in interval above which has the least MSE. N combination (default =10) will be shown. To choose the best value of smoothing constant click the value in list and click Select This button.- To select analysis result that will be shown on Result View, click Results button. You can select some result views by clicking the checkbox of every selection. For Forecasted selection, you have to enter the data step you wish to forecast.
- To save the residual, predicted or smoothed data from the model as a new variable, click Storage button. Check the items you want to save as new variables, and then type the new variable names.
- After selecting the result views and determining whether you want to save the new variables or not, the software will show the Exponential Smoothing form again. Click the OK button to finish your analysis and to show the result views.
- The selected result views on previous step will be viewed as several tabs on Result View panel.

The result views of exponential smoothing analysis in Zaitun Time Series are grouped into two categories, tables and graphics. The details of them are described here:

- Tables
- Model Summary

Shows the summary of exponential smoothing model - Exponential Smoothing Table

Shows actual, smoothed, trend, seasonal, predicted and residual values of exponential smoothing model - Forecasted

Shows forecasted values from exponential smoothing model, as many steps of data you want to forecast

- Model Summary
- Graphics
- Actual and Predicted

Shows a line plot for actual and predicted values of exponential smoothing model - Actual and Smoothed

Shows a line plot for actual and smoothed values of exponential smoothing model - Actual and Forecasted

Shows a line plot for actual and forecasted values of exponential smoothing model - Actual vs. Predicted

Shows a scatter plot between actual and predicted values - Residual

Shows a line plot for residual values of exponential smoothing model - Residual vs. Actual

Show a scatter plot between residual and actual values - Residual vs. Predicted

Shows a scatter plot between residual and predicted values

- Actual and Predicted