6.
Process or Product Monitoring and Control
6.4. Introduction to Time Series Analysis 6.4.3. What is Exponential Smoothing?
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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:
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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
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A complete season's data consists of |
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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
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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. | |||||||||||||||||||||||||||||||||||||
Step 1: compute yearly averages |
Step 1:
Compute the averages of each of the 6 years.
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Step 2: divide by yearly averages |
Step 2:
Divide the observations by the appropriate yearly mean.
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Step 3: form seasonal indices |
Step 3:
Now the seasonal indices are formed by computing the average
of each row. Thus the initial seasonal indices (symbolically) are:
The next page contains an example of triple exponential smoothing. |
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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 ( 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. |