1. A.S. 3.8 INTERNAL 4 CREDITS Time Series

2. Time Series Overview Investigate Time Series Data A.S. 3.8 AS91580 Achieve Students need to tell the story of one data series by describing seasonal effect, trend & residual/other effects. Produce plots of raw data, seasonal effects, residuals, fitted values and predictions including prediction intervals. Comment on trend, seasonal pattern, residuals and any unusual features. MeritStudents need to investigate aspects of the series and justify their comments. Excellence Students need to incorporate independently gathered contextual information to their time series analysis and possibly combine series to investigate other patterns in the data.

3. Using the statistical enquiry cycle to investigate time series data involves: using existing data sets selecting a variable to investigate selecting and using appropriate display(s) identifying features in the data and relating this to the context finding an appropriate model using the model to make a forecast communicating findings in a conclusion

4. What is a Time Series? A time series consists of data collected, recorded or observed over successive intervals of time Usually the time intervals are equal; data is recorded for every hour, every month, quarterly, every year etc The variable being measured changes with time

5. Aims in Time Series... – why do we do it? Understanding the data: are there patterns in the variation? Is it seasonal? Forecasting or predicting future values Best displayed on a scatterplot with time on the horizontal axis and the series value (variable changing over time) on vertical axis

6. Features/Components of a Time Series Long term trend Cyclical effect Seasonal effect Random or irregular events Shifts or ramps

7. Long term Trend The most slowly changing component Underlying trend seen changing smoothly over a long period of time Often linear (increasing or decreasing trend in straight line) but can be non-linear trend (curve). Due to long term changes such as population growth or change, technological change etc Made clearer by smoothing techniques.

9. smoothing shows the long term decreasing trend

10. Cyclical Variation The next most slowly changing component Recurrent, wave-like changes in the series The period and amplitude of the wave are not predictable

11. Seasonal Variation More rapid change than cycle, period of less than one year Regular period based on calendar or clock Could be literally seasonal but may be monthly, daily etc Seasonal variation can be removed to make the long term trend clearer by taking averages over a certain period and smoothing out waves.

12. Seasonal variation – high production in Summer each year and low production in winter (Mar-Sept)

13. Random Irregular Component Most rapidly changing component Totally unpredictable What is left over after other components have been taken out Special events may cause a spike or outlier on the graph

14. Possible spike/outlier – Dec 92 had slightly higher production than expected. Why??

15. Shifts in a time series Sometimes new technology or events such as new legislation will suddenly change the magnitude of the data in a time series Trend line can be displaced in ramps or steps eg. In 1980, divorce laws relaxed in NZ

17. The two types of model Time series where the periodic variation is of constant magnitude are called ADDITIVE models If the peaks and troughs in the raw data increase in magnitude as time increases the model is said to be MULTIPLICATIVE

18. A possible multiplicative model