Seasonal index represents the extent of seasonal influence for a particular segment of the year. The calculation involves a comparison of the expected values of that period to the grand mean.
A seasonal index is how much the average for that particular period tends to be above (or below) the grand average. Therefore, to get an accurate estimate for the seasonal index, we compute the average of the first period of the cycle, and the second period, etc, and divide each by the overall average. The formula for computing seasonal factors is Si = Di/D,
where:
Si = the seasonal index for ith period,
Di = the average values of ith period,
D = grand average,
i = the ith seasonal period of the cycle.
A seasonal index of 1.00 for a particular month indicates that the expected value of that month is 1/12 of the overall average. A seasonal index of 1.25 indicates that the expected value for that month is 25% greater than 1/12 of the overall average. A seasonal index of 80 indicates that the expected value for that month is 20% less than 1/12 of the overall average. A summary of Vintage Restaurant food and beverage sales calculations are as shown below (See Appendix for detail of calculations):
As is clear, January, February and March have high seasonal indices, whereas June, September and October have low seasonal indices. Clearly, since the business is located in Florida, there is increased tourist traffic in the mild months of January, February and March. In the peak summer seasons in May, June, and the subsequent rains falling in June to October, the tourist traffic is declined. Hence the seasonal indices give quite an intuition into forecasting future predictions, as is evidenced by the following test of seasonality.
Test of Seasonality:
We conduct the test for the presence of any significant seasonal component in a given time series using its seasonal index vector.
H0: There is no significant seasonal component in the time series.
HA: The null hypothesis is false.
Chi Square Value: 69.72245
p-value = 0
Conclusion:
There exists very strong evidence against the null hypothesis; hence there is significant amount of seasonal component in the time series, as was evidenced by intuition.
FORECASTING
Incorporating seasonality in a forecast is useful when the time series has both trend and seasonal components. The final step in the forecast is to use the seasonal index to adjust the trend projection. One simple way to forecast using a seasonal adjustment is to use a seasonal factor in combination with an appropriate underlying trend of total value of cycles. To forecast January to December sales for the fourth year, we proceed as follows:
- We calculate the annual sales for each year:
Y = 146.5T + 1958.7
The main question is whether this equation represents the trend. Often fitting a straight line to the seasonal data is misleading. By constructing the scatter-diagram, we compare whether a straight line or a parabola would fit better. The estimated quadratic tend is:
Y = 2.5T2 + 136.5T + 1967
By observation, a linear trend might be fair enough to give a suitable prediction.
For T = 5, we get Y = 146.5*4 + 1958.7 = 2544.7. The average monthly sale during next year, therefore, is 2544.7/12 = 212.06
Finally, the forecast for each month of the month is calculated by multiplying the average monthly sales forecast by the seasonal index. The table is as below:
Now assuming that the January sales for the fourth year turn out to be $295,000, the forecast error would be: 42220.
Since a forecast error of $42,220 is quite large, Karen might be puzzled about the difference between your forecast and the actual sales value. The cycles can be easily studied if the trend itself is removed. This is done by expressing each actual value in the time series as a percentage of the calculated trend for the same date. The resulting time series has no trend, but oscillates around a central value of 100.
A variety of factors are likely influencing data. It is very important in the study that these different influences or components be separated or decomposed out of the 'raw' data levels. Decomposition Analysis is the pattern generated by the time series and not necessarily the individual data values that offers to the manager who is an observer, a planner, or a controller of the system. Therefore, the Decomposition Analysis is used to identify several patterns that appear simultaneously in a time series. In general, there are four types of components in time series analysis: Seasonality, Trend, Cycling and Irregularity.
Xt = St . Tt. Ct . I
The first three components are deterministic and are called "Signals", while the last component is a random variable, "Noise". To be able to make a proper forecast, we must know to what extent each component is present in the data. Hence, to understand and measure these components, the forecast procedure involves initially removing the component effects from the data (decomposition). After the effects are measured, making a forecast involves putting back the components on forecast estimates (recomposition). Pursuing this can help Karen take a proper judgment.
APPENDIX
The monthly forecasts for the 12 months of the fourth year are as shown below:
Suppose the actual January sales for the fourth year turn out to be $295,000. The forecasted January sales are $296,458.
Error between actual and forecasted sales = $296,458 - $295,000 = $1458
Percentage Error =
This is an extremely small percentage error. Karen does not have to worry about this error and she can be assured that her forecast model is extremely good.
REFERENCES
Cooper, D. & Schindler, S. (2003). Business Research Methods. Boston: McGraw-Hill Irwin.
