© 2000 John Petroff
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© 2000 John Petroff |
A traditional decomposition procedure breaks down a data series into seasonal (St), trend (Tt) and cyclical (Ct) patterns, just as above. But. the traditional approach to times-series analysis is simpler than the Box-Jenkins approach, and it achieves similar results at a considerable saving of time. The traditional decomposition simplicity is sufficient for most business data because it allows several assumptions that immediately save steps and investigation. For instance, monthly data is used and seasonality is automatically assumed to be over 12 months. However, Box-Jenkins approach is more universal and is appropriate for any time-series.
The following description of classical decomposition is essentially
based on Gross and Peterson.
A decomposition can be additive (i.e. St + Tt
+ Ct ) or multiplicative (StxTtxCt).
The multiplicative decomposition is more common. Seasonal
indices are calculated by
- taking twelve month moving totals
- adding two consecutive twelve moving totals
- divide the sum of two consecutive twelve months moving totals
by 24 to obtain a centered monthly average
- divide each actual observation by the corresponding centered
moving average for that month to obtain seasonal unadjusted monthly
indices
- obtain averages of unadjusted monthly indices for each of the
twelve months
- sum the index averages and if not equal to 12 divide sum of
the monthly index averages to obtain an adjustment coefficient
- multiply each unadjusted monthly index by the adjustment coefficient
to obtain final indices for the twelve months.
One may wonder why the second step of the procedure is to add
two consecutive twelve months totals rather than simply taking
a twenty four month total. The reason is that a twenty four months
total would not be centered: there would be twelve months before
and eleven months after.
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Table T-5.30 below illustrates the classical decomposition method. Note that the adjusted seasonal indexes are shown in the months of January through December 1996. As required above, these adjusted seasonal indexes are obtained by multiplying the average monthly ratio of each month by total of monthly ratios12.016 and dividing by 12. The total of monthly ratios T=12.016 is placed in row for the month of January. Note also that the table contains averages of each column, which are used for control purposes. For instance, observe that the average of monthly actual sales is identical to the monthly seasonally adjusted sales, as it should be, naturally. This table is developed with a standard spreadsheet. The entries into a portion of the cells appears in Appendix 5B.
Graph G-5.3 shows the unadjusted and seasonally adjusted data from Table T-5.30.  |
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