Process or Product Monitoring and Control
6.4. Introduction to Time Series Analysis
6.4.3. What is Exponential Smoothing?
|What happens if the data show trend and seasonality?|
|To handle seasonality, we have to add a third parameter||
In this case double smoothing will not work. We now introduce a
third equation to take care of seasonality (sometimes called
periodicity). The resulting set of equations is called the
"Holt-Winters" (HW) method after the names of the inventors.
The basic equations for their method are given by:
|Complete season needed||To initialize the HW method we need at least one complete season's data to determine initial estimates of the seasonal indices I t-L.|
|L periods in a season||A complete season's data consists of L periods. And we need to estimate the trend factor from one period to the next. To accomplish this, it is advisable to use two complete seasons; that is, 2L periods.|
|Initial values for the trend factor|
|How to get initial estimates for trend and seasonality parameters||
The general formula to estimate the initial trend is given by
|Initial values for the Seasonal Indices|
|As we will see in the example, we work with data that consist of 6 years with 4 periods (that is, 4 quarters) per year. Then|
|Step 1: compute yearly averages||
Compute the averages of each of the 6 years
|Step 2: divide by yearly averages||
Divide the observations by the appropriate yearly mean
|Step 3: form seasonal indices||
Now the seasonal indices are formed by computing the average
of each row. Thus the initial seasonal indices (symbolically) are:
I2 = ( y2/A1 + y6/A2 + y10/A3 + y14/A4 + y18/A5 + y22/A6)/6
I3 = ( y3/A1 + y7/A2 + y11/A3 + y15/A4 + y19/A5 + y22/A6)/6
I4 = ( y4/A1 + y8/A2 + y12/A3 + y16/A4 + y20/A5 + y24/A6)/6
The next page contains an example of triple exponential smoothing.
|The case of the Zero Coefficients|
|Zero coefficients for trend and seasonality parameters||
Sometimes it happens that a computer program for triple exponential
smoothing outputs a final coefficient for trend
or for seasonality
() of zero.
Or worse, both are outputted as zero!
Does this indicate that there is no trend and/or no seasonality?
Of course not! It only means that the initial values for trend and/or seasonality were right on the money. No updating was necessary in order to arrive at the lowest possible MSE. We should inspect the updating formulas to verify this.