Method to Identify the Component of Seasonal Variation in a Time Series
This technique is called as Ratio to Moving Average Method. In this technique, we construct an index which has a base of 100. The magnitude of seasonal variations is measured by the individual deviations from the base of 100. The six steps employed in the construction of the seasonal index are explained with the help of an example.
Moving Totals and Moving Averages
Before we look at the six steps constituting the construction process, we look at moving totals and moving averages. We will come across them in the calculation of the seasonal index. Consider 4, 6, 2, 7, 9, 8 and 4 which is a set (collection) of numbers. For this set of numbers, if we compute the moving totals of order 3, they will be like 4 + 6 + 2, 6 + 2 + 7, 2 + 7 + 9, 7 + 9 + 8 and 9 + 8 + 4. From this, the meaning of order should be clear. It refers to the number of elements, we ought to treat as a single group every time we calculate the moving total. We also note that in each subsequent calculation, we exclude the first number and include the number which comes immediately after the last number of the first group. That is the moving total for the second group includes 7, which immediately succeeds 2. This is why precisely, it is called as moving total. The moving totals for the above set of numbers will be 12, 15, 18, 24 and 21.
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The centered averages then can be shown as below.
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Original data
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4,
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6,
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2,
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7,
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9,
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8,
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Moving average
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4.75
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6,
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6.5
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Centered moving average
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5.375
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6.25
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Thus whenever even number of data points are given the moving averages are required to be centered. In case of odd number of data points, the moving averages are already centered and hence no longer need to be centered further.
Primarily, the concept of moving averages is used to smoothen out the fluctuations inherent in the time series data.
The order of the moving average can be expressed in terms of days, weeks, months and years depending on the context.
Now, we take up an example to understand how the component of seasonal variation is computed.
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Then the moving averages for this set of numbers refer to arithmetic mean of the above moving totals. That is, each value in the set of moving totals should be divided by the order. Therefore, the moving averages will be 4, 5, 6, 8 and 7 respectively. Now observe the relative position of the moving average with respect to the original data.
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Original data
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4, 6, 2, 7, 9, 8, 4
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Moving average
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4, 5, 6, 8, 7
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The objective of expressing the data in this form is to point out that the values in the moving averages row is the mean of the three numbers immediately above it. In other words, they are already centered. We do not have to center them further. Now is this the case if we have even number of data points? No. Whenever, even number of data points are present we have to center the moving average. To understand this, consider the same set of numbers we considered above except that we leave out the last number. That is, the set now consists of 4, 6, 2, 7, 9, 8. The mid point for this set is between the third and the fourth values. Now we compare the moving averages of order four and the original data.
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Original data
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4,
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6,
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2,
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7,
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9,
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8
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Moving average
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4.75,
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6,
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6.5
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We observe that 4.75 falls between the second and the third data points. In this case, centering would be to associate the moving average with either the second or the third data point. For the given number set we can associate the moving average with the third data point. This we do by taking the average of the moving averages 4.75 and 6, which is 5.375. Therefore 5.375 and 6.25 are associated with 2 and 7 respectively.