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 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 Brown’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
The whole smoothing equation
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:
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: