Ideas
Predictions
The purpose of smoothing time series data using a moving average or deseasonalising the data is to take out the 'peaks' and 'troughs' and view the underlying trend. Often, but not always, the smoothed data will appear to be linear in nature. If it is linear in nature we can calculate the  least-squares regression line for the smoothed data and use this line to make future predictions. Making future predictions with time series data is also called forecasting.
Once we have a predicted value from the underlying trend line, we will need to factor back in the season it's from. If it's from a 'peak' season then our prediction should be adjusted upwards. If it's from a 'trough' season then it should be adjusted downwards. We multiply the predicted score from the regression line by the appropriate  seasonal index to adjust the predicted value.
Steps to predict from time series data:
Smooth the data using the appropriate moving average or by deseasonalising the data.
Give each time period a number eg: Mon Week 1 becomes t=1, Tues Week 1 becomes t=2 etc.
Calculate the equation of the least-squares regression line using the time period in list 1 and the smoothed data in list 2 of your calculator. Round numbers to four decimal places unless instructed otherwise.
Substitute a future time value into the equation of the least-squares regression line to obtain a predicted score.
Multiply the predicted score by the appropriate seasonal index to factor back in the seasonality of the data.
If asked to comment on the reliability of the prediction, consider whether the future time value is close to the data. Generally predicting within one cycle of the existing data is considered reliable.
Note:
Moving average data or deseasonalised data can be used to find the regression line. Read the question carefully.
Correlation has little meaning with time series data as the smoothed line will almost always have a strong, linear correlation for an underlying linear trend.
Time series predictions are almost always future predictions so we don't tend to use the words 'extrapolation' or 'interpolation' with these problems. Instead, we consider how close the predicted value is to the original data set and use the words 'within one cycle'.
Showing working is essential in these type of problems. You should always state the value of t that you are using, the equation of the regression line and the seasonal index you are multiplying with.
Time is always the explanatory variable (plotted on the horizontal axis).
When using time series data our prediction will almost always be an extrapolation. As such, we know our predictions might be somewhat unreliable due to our inability to accurately predict future events. However, just as when making predictions in Chapter 2, we will generally consider a prediction made close to the existing data range to be reliable. For the purpose of this course, we will view predictions that have been made within one cycle of the available data to be close and hence, reliable.
Examples
Example 1
The following data shows the sales of air conditioners at a leading retailer over four quarters of three consecutive years.
| \text{Month} | \text{Time }(t) | \text{Number of}\\ \text{ air conditioners sold} | \text{Proportion}\\ \text{ of yearly mean} | |
|---|---|---|---|---|
| Year 1 | \text{March} | 1 | 1042 | 0.8529 |
| \text{June} | 2 | 486 | 0.3978 | |
| \text{Sept} | 3 | 613 | 0.5017 | |
| \text{Dec} | 4 | 2746 | 2.2476 | |
| Year 2 | \text{March} | 5 | 1160 | 0.8183 |
| \text{June} | 6 | 609 | 0.4296 | |
| \text{Sept} | 7 | 1139 | 0.8035 | |
| \text{Dec} | 8 | 2762 | 1.9485 | |
| Year 3 | \text{March} | 9 | 1795 | 0.9638 |
| \text{June} | 10 | 1181 | 0.6341 | |
| \text{Sept} | 11 | 1094 | 0.5874 | |
| \text{Dec} | 12 | 3380 | 1.8148 |
Calculate the seasonal component for the quarters ending in March, June, September and December, rounding to four decimal places if necessary.
| March | June | September | December |
|---|---|---|---|
| \text{ } |
The data is smoothed using a 4 point centred moving average as shown in the table below. Calculate the missing values.
| \text{Month} | \text{Time }(t) | \text{Number of}\\ \text{ air conditioners sold} | \text{4CMA} | |
|---|---|---|---|---|
| Year 1 | \text{March} | 1 | 1042 | |
| \text{June} | 2 | 486 | ||
| \text{Sept} | 3 | 613 | 1236.5 | |
| \text{Dec} | 4 | 2746 | 1266.625 | |
| Year 2 | \text{March} | 5 | 1160 | 1347.75 |
| \text{June} | 6 | 609 | A | |
| \text{Sept} | 7 | 1139 | 1496.875 | |
| \text{Dec} | 8 | 2762 | 1647.75 | |
| Year 3 | \text{March} | 9 | 1795 | 1713.625 |
| \text{June} | 10 | 1181 | B | |
| \text{Sept} | 11 | 1094 | ||
| \text{Dec} | 12 | 3380 |
Use your calculator to calculate the equation of the least squares regression line that fits the 4CMA data. Give the equation of the line in the form y=at + b. Round a and b to four decimal places.
Predict the number of air conditioners sold in the quarter ending December year 4. Round your answer to the nearest whole air conditioner sold.
Comment on the reliability of your prediction.
Example 2
A new pop up ice-cream shop records their sales over their first month. The data is tabulated below. The shop is only open from Friday to Sunday.
| \text{Day} | \text{Time }(t) | \text{Sales (dollars)} | \text{Deseasonalised data} | |
|---|---|---|---|---|
| Week 1 | \text{Fri} | 1 | 2036 | 2101.14 |
| \text{Sat} | 2 | 2257 | 2040.87 | |
| \text{Sun} | 3 | 1936 | 2092.75 | |
| Week 2 | \text{Fri} | 4 | 2224 | X |
| \text{Sat} | 5 | 2547 | 2303.10 | |
| \text{Sun} | 6 | 2060 | 2226.79 | |
| Week 3 | \text{Fri} | 7 | 2349 | 2424.15 |
| \text{Sat} | 8 | 2706 | 2446.88 | |
| \text{Sun} | 9 | Y | 2431.09 | |
| Week 4 | \text{Fri} | 10 | 2435 | 2512.90 |
| \text{Sat} | 11 | 2824 | 2553.58 | |
| \text{Sun} | 12 | 2398 | 2592.15 |
Seasonal indices:
| Fri | Sat | Sun |
|---|---|---|
| 0.9690 | 1.1059 | 0.9251 |
On which day will shop be most likely to need extra help?
Calculate the value of X in the table. Round the value off to two decimal places if necessary.
Calculate the value of Y in the table. Round the value off to a single decimal place if necessary.
Using your calculator, determine the equation of least squares regression line for the deseasonalised data, where t=1 is Friday of Week 1.
Give the equation of the line in the form y=at + b. Round a and b to four decimal places. You can make use of a and b in your working.
Predict the sales for Friday of the sixth week. Give your answer in dollars and round off any figures to two decimal places if needed.
Will the Friday in the sixth week be within one seasonal cycle of the data?
Steps to predict from time series data:
Smooth the data using the appropriate moving average or by deseasonalising the data.
Give each time period a number.
Calculate the equation of the least-squares regression line.
Substitute a future time value into the equation of the least-squares regression line to predict score.
Multiply the predicted score by the appropriate seasonal index to factor back in the seasonality of the data.
If asked to comment on the reliability of the prediction, consider whether the future time value is close to the data.