Time Series

1. AS 90641 (internal) 3 credits Time Series

2. Time Series • A time series is a set of values of a variable measured at regular intervals of time • E.g. temperature measured daily, monthly sales figures • Analysis looks for patterns of change over time • Try to predict future movements

3. Features of Time Series • Long term trends • Random fluctuations (noise) • Periodic movements • Seasonal effects • Cyclical effects • Spikes/troughs (outliers) • Shifts (ramps)

4. Long term trend • The tendency of the variable to increase or decrease (or stay constant) over a long period of time

5. Long Term Trend (increasing)

6. Random fluctuations (noise) • Unpredictable minor rises and falls which occur throughout the series • A value may rise or fall from one measurement to the other without necessarily indicating a long term change in the trend

7. Random Fluctuations

8. Seasonal effects • A regular fluctuation which results from changes of seasons, time of day, month of year, etc. • E.g. Hourly temperature higher at midday than at midnight

9. Seasonal effects

10. Cyclical effects • Longer term patterns such as weather patterns (e.g. El Nino) or changes due to the expansion/contraction of economies • Typically less frequent than seasonal effects

11. Spike/trough (outlier) • A value which is significantly higher or lower than what would be expected • Caused by a special event like extreme weather conditions, natural disasters, stock market crash, etc.

12. Spike/trough (outlier)

13. Shift (Ramp) • New technology or an event causes a sudden, permanent step up or down in the pattern

14. Shift (Ramp)

15. Describe the long term trend

16. The long term trend is a slight increase in sales

17. Describe the long term trend

18. Stable solve rate of around 90% until about 1940, decline until about 1990, small improvement since then

19. Describe the long term trend

20. Crime rate was low and stable until about 1950, then dramatically increased until 1990, since then has dropped to about 10%

21. Textbook • Pages 8-12 Exercise 1.01 • Questions 1ab, 2, 4 • Homework: page 5

22. Moving Mean • Used to smooth out random and periodic variations in order to show the long term trend more clearly • Use a moving mean order “n” where “n” is the number of data in one cycle (e.g. 4 for quarterly, 12 for monthly) • Find the average of the first “n” terms and write result beside middle term

23. Moving Mean • Calculate the moving means of order 3 for this data

24. Moving Mean • Calculate the moving means of order 3 for this data

25. Moving Mean • Calculate the moving means of order 3 for this data

26. Centred Moving Mean • For odd order a moving mean is sufficient as you can place the MM in the middle of the periods you have averaged • For even order, you must do a second moving mean called a centred moving mean

27. Centred Moving Mean • Calculate the CMM for this data

28. Centred Moving Mean • Calculate the CMM for this data

29. Centred Moving Mean • Calculate the CMM for this data

30. Centred Moving Mean • Calculate the CMM for this data

31. Centred Moving Mean • Calculate the CMM for this data

32. Centred Moving Mean • Calculate the CMM for this data