Sumber : http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc4.htm
What are Moving Average or Smoothing Techniques?
Smoothing data removes random variation and shows trends and cyclic components
Inherent in the collection of data taken over time is some form of random variation. There
exist methods for reducing of canceling the effect due to random variation. An often-used
technique in industry is "smoothing". This technique, when properly applied, reveals
more clearly the underlying trend, seasonal and cyclic components.
There are two distinct groups of smoothing methods


Averaging Methods
Exponential Smoothing Methods
Taking averages is the simplest way to smooth data
Mean squared error is a way to judge how good a model is
Average weighs all past observations equally
In summary, we state that
1. The "simple" average or mean of all past observations is only a useful estimate for
forecasting when there are no trends. If there are trends, use different estimates
that take the trend into account.
2. The average "weighs" all past observations equally. For example, the average of
the values 3, 4, 5 is 4. We know, of course, that an average is computed by adding
all the values and dividing the sum by the number of values.
Single Moving Average
Taking a moving average is a smoothing process
An alternative way to summarize the past data is to compute the mean of successive
smaller sets of numbers of past data as follows:
Recall the set of numbers 9, 8, 9, 12, 9, 12, 11, 7, 13, 9, 11, 10 which were the dollar
amount of 12 suppliers selected at random. Let us set M, the size of the "smaller set"
equal to 3. Then the average of the first 3 numbers is: (9 + 8 + 9) / 3 = 8.667.
This is called "smoothing" (i.e., some form of averaging). This smoothing process is
continued by advancing one period and calculating the next average of three numbers,
dropping the first number.
The next table summarizes the process, which is referred to as Moving Averaging. The
general expression for the moving average is
Mt = [ Xt + Xt-1 + ... + Xt-N+1] / N
Results of Moving Average
Supplier $
1
2
3
4
5
6
7
8
9
10
11
12
MA Error Error squared
9
8
9 8.667 0.333
12 9.667 2.333
9 10.000 -1.000
12 11.000 1.000
11 10.667 0.333
7 10.000 -3.000
13 10.333 2.667
9 9.667 -0.667
11 11.000 0
10 10.000 0
0.111
5.444
1.000
1.000
0.111
9.000
7.111
0.444
0
0
The MSE = 2.018 as compared to 3 in the previous case.
Centered Moving Average
When computing a running moving average, placing the average in the middle time
period makes sense
In the previous example we computed the average of the first 3 time periods and placed it
next to period 3. We could have placed the average in the middle of the time interval of
three periods, that is, next to period 2. This works well with odd time periods, but not so
good for even time periods. So where would we place the first moving average when M =
4?
Technically, the Moving Average would fall at t = 2.5, 3.5, ...
To avoid this problem we smooth the MA's using M = 2. Thus we smooth the smoothed
values!
If we average an even number of terms, we need to smooth the smoothed values
The following table shows the results using M = 4.
Interim Steps
Period Value MA Centered
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
9
8
9.5
9
9.5
9.5
12
10.0
10.5
9
10.750
11.0
12
9
This is the final table:
Period Value Centered MA
1
2
3
4
5
6
7
9
8
9
12
9
12
11
9.5
10.0
10.75