1. 1 Time Series Analysis Thanks to Kay Smith for making these slides available to me!

2. 2 Objectives Draw a time series on to a graph Use a moving average to calculate the trend. Calculate the residuals and seasonal factor.

3. 3 Time series analysis forms the basis of forecasting on the assumption that the near future will depend on the past and that any past patterns will continue. Time series analysis is primarily concerned with the identification of the model which best fits the past data. Introduction

4. 4 The first step is therefore to produce a scatterplot against time leaving room for forecasts in the near future. Scatterplot

5. 5 Methods of analysis depend on the appearance of the scatterplot Time series of past data General description Some suitable methods Linear, seasonalSeasonal decomposition: additive and multiplicative Linear, nonseasonal? Linear or nonlinear regression Nonlinear, nonseasonal Nonlinear regression Exponential smoothing

6. 6 A small ice-cream manufacturer reports sales figures of £50 000 for the last quarter. Is this good or bad? –Can this single figure be used to forecast future sales? No. A single figure never gives enough information. We also need to compare this figure with: –that for the previous quarter; –that for the same quarter of the previous year; –that expected from the current trend. Example

7. 7 We need to identify any past pattern and use it for future sales predictions. We also need to estimate how accurate those predictions are likely to be. Quarterly sales figures for the four years preceding the figure of £50 000 quoted above need to be obtained.

8. 8 Ice cream sales (£’000) Year Quarter 1 Quarter 2 Quarter 3 Quarter 4 200240608035 200330506030 200435608040 2005507010050

9. 9 Sales seem to show quarterly seasonal effects.

10. 10 The Additive Model An additive model identifies how much has been 'added' to the sales because of a particular season. Trend value has the seasonal effect 'averaged out'. Seasonal factor is the average effect for a quarter. Random factor is the variation due to neither of these. Additive model Observed sales value = Trend value + Seasonal Factor + Random factor A=T+S+R

11. 11 Calculate the cycle averages: (Q1+Q2+Q3+Q4)/4, etc The first cycle average: The second cycle average:

12. 12 Calculate the trend (moving average) figures: (Cycle average 1 + cycle average 2)/2, The first moving average trend figure: The second moving average trend figure : Date Sales (A) (£'000) Cycle Average Trend (T) Mov. aver. First Resid. R 1 = A- T Fitted value F = ( T+ S.F.) 2002 Q 1 40 2002 Q 2 60 53.75 2002 Q 3 80 52.50 51.25 2002 Q 4 35 50.00 48.75 2003 Q 1 30

13. 13 Trend plotted through Sales

14. 14 Next find First Residuals (Sales - Trend) for the last two quarters. The first residual figure is 80 - 52.5 = 27.5 The second residual figure is 35 - 50 =15.0 Date Sales (A) (£'000) Cycle Average Trend (T) Mov. aver. First Resid. R 1 = A- T Fitted value F = ( T+ S.F.) 2002 Q 1 40 2002 Q 2 60 53.75 2002 Q 3 80 52.50 +27.50 51.25 2002 Q 4 35 50.00 -15.00 60.00 2003 Q 1 30

15. 15 From all the First residuals calculate the Average Seasonal factor for each of the four quarters.

16. 16 Summary We have calculated: (1) Trend T for given time series; (2) Seasonal factors for quarterly data. This information aids in forecasting.