2. Time Series - A collection of measurements recorded at specific intervals of time. 1. Short term features Noise: Spike/Outlier: Minor variation about a general trend An obvious difference from the surrounding values e.g.

3. 2. Long term features Trend: Seasonal Variation: Often there is a trend for measurements to remain steady, or show a definite increase or decrease over time. Fairly regular up/down patterns (called cyclical movement if over very long periods) e.g. Long Term Trend: Over long term, sales are increasing overall Seasonal Variation: Sales peak in summer and are lowest in winter. Sales rise again in spring and these are higher than in autumn Ice Cream Sales (in 000’s)

4. Smoothing Techniques - Used when averaging out random variations to see if there is an overall trend. - Done by averaging all of the data over the period of any natural cycle. The number of values used to form a moving average is called ‘the order of the moving average’ (e.g. we use a 5 point moving mean (order of 5) if the natural cycle is a working week)

5. e.g. Use 4 point moving means to smooth the data for Elliot’s Fish and Chip shop. Then graph the raw data and the mean of means. SeasonQuarterly sales Moving mean Mean of means Seasonal Difference Sept.9040 Dec.8650 Mar. 968370 June9250 Sept.9033 Dec.8578 Mar. 978495 June9407 Sept.9209 Dec.8740 Mar. 988618 June9504 Sept.9246 Dec.8929 Mar. 998670 8828 8826 8808 8839 8878 8922 8963 8994 9018 9027 9074 9087 8827 8817 8824 8859 8900 8943 8979 9006 9023 9051 9081 Mean of means are used so there is a 1 to 1 correspondence between the raw and smoothed data.

7. Seasonal Effects Seasonal Difference =data value – moving mean Seasonal Effect =averaging off all of the seasonal differences Making Predictions - Extend the trend line to find the smoothed data value then add/subtract the average seasonal difference e.g. Using the data and graph of ‘Elliot’s Fish and Chip Shop’ a) Find the seasonal effects for June and December b)Use the long term trend line to predict the turnover in December 1999 and June 2000.

8. e.g. Use 4 point moving means to smooth the data for Elliot’s Fish and Chip shop. Then graph the raw data and the mean of means. SeasonQuarterly sales Moving mean Mean of means Seasonal Difference Sept.9040 Dec.8650 Mar. 968370 June9250 Sept.9033 Dec.8578 Mar. 978495 June9407 Sept.9209 Dec.8740 Mar. 988618 June9504 Sept.9246 Dec.8929 Mar. 998670 8828 8826 8808 8839 8878 8922 8963 8994 9018 9027 9074 9087 8827 8817 8824 8859 8900 8943 8979 9006 9023 9051 9081 -457 433 209 -281 -405 464 230 -266 -405 453 165

9. e.g. Using the data and graph of ‘Elliot’s Fish and Chip Shop’ a) Find the seasonal effects for June and December b)Use the long term trend line to predict the turnover in December 1999 and June 2000. Seasonal Effect for June =433 + 464 + 453 3 = 1350 = $450 3 Seasonal Effect for December =-281 + -266 2 = -547 = -$273.5 2

10. Dec ‘99 = 9200 June ‘00 = 9250

11. e.g. Using the data and graph of ‘Elliot’s Fish and Chip Shop’ a) Find the seasonal effects for June and December b)Use the long term trend line to predict the turnover in December 1999 and June 2000. Seasonal Effect for June =433 + 464 + 453 3 = 1350 = $450 3 Seasonal Effect for December =-281 + -266 2 = -547 = -$273.5 2 Prediction for December 1999 =9200 + -273.5 = $8926.50 Prediction for June 2000 =9250 + 450 =$9700