1. 19-1 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e Chapter 19 Time Series and Trend Analysis Introductory Mathematics & Statistics

2. 19-2 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e Learning Objectives Describe a time series and explain its use Identify and interpret the four basic measures of variation that appear in a time series analysis (secular trend, seasonal variation, cyclical variation and random or irregular variation) Identify and use common methods of fitting secular trend lines to time series (including freehand drawing, least-squares linear regression, semi-averages, moving averages, exponential smoothing and growth models) Make forecasts

3. 19-3 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 19.1 Introduction A time series consists of a set of observations that are measured at specified (usually equal) time intervals Time series analysis attempts to identify those factors that exert an influence on the values in the series Once these factors are identified, the time series may be projected into the future and used for both short term and long-term forecasting Some examples of time series include: 1. sales figures for individual businesses or complete industries 2. gross national product and other macroeconomic measures 3. share prices 4. the value of the Australian dollar relative to the US dollar

4. 19-4 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 19.2 Time series components The four components usually identified are: 1. secular trend 2. seasonal variation 3. cyclical variation 4. random or irregular variation Secular trend –The secular trend is the long-term growth or decline of a series –Secular trends allow us to look at past patterns or trends and use these to make some prediction about the future –It is possible to isolate the effect of secular trends from the time series and hence make studies of the other components easier

5. 19-5 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 19.2 Time series components (cont…) Seasonal variation –The seasonal variation of a time series is a pattern of change that recurs regularly over time –Seasonal patterns typically are one year long; that is, the pattern starts repeating itself at a fixed time each year –While variations may recur every year, the concept of seasonal variation also extends to those patterns that occur monthly, weekly, daily or even hourly –Examples include: 1. Air conditioner sales are greater in the summer months 2. Heater sales are greater in the winter months 3. The total number of people seeking work is large at the end of each year when students leave school

6. 19-6 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 19.2 Time series components (cont…) Seasonal variation (cont…) –Time series graphs may be prepared using an adjustment for seasonal variations, although the other components of the series are not affected –Such graphs are said to be seasonally adjusted or deseasonalised –The purpose of a seasonally adjusted series is to obtain a true picture of genuine movements in the time series after the seasonal effects have been removed

7. 19-7 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 19.2 Time series components (cont…) Cyclical variation –Cyclical variations have recurring patterns, but have a longer and more erratic time scale –Examples of causes of cyclical variation are: 1. natural disasters, such as floods, earthquakes, cyclones and droughts 2. wars 3. changes in interest rates 4. major increases or decreases in the population 5. the opening of a new shopping complex 6. the building of a new airport –Cycles can be far from regular and it is usually impossible to predict just how long these periods of expansion and contraction will be

8. 19-8 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 19.2 Time series components (cont…) Random variation –Random variation or irregular variation in a time series occurs over varying (usually short) periods –It follows no regular pattern and is by nature unpredictable –Examples of events that might cause irregular variation are: 1. the assassination (or disappearance) of a country’s leader 2. short-term variations in the weather, such as unseasonably warm winters (which may affect sales of certain products) 3. sudden changes in interest rates 4. the collapse of large (or even small) companies –In general, if the variation in a time series cannot be accounted for by secular trend, or by seasonal or cyclical variation, it is usually attributed to random variation

9. 19-9 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 19.2 Time series components (cont…) Interaction of components –A further question to be decided is just how individual components interact to form a time series –If they are assumed to multiply together to obtain a forecast, the model is called a multiplicative model –This type of model is more appropriate when the amount of the increase in the time series data points gets larger as time passes –If the components are assumed to add together to obtain a forecast, the model is called an additive model –This type of model is more appropriate when the amount of the increase in the time series data points stays much the same as time passes

10. 19-10 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 19.3 Measurement of secular trend The basic aim of estimating the trend of a time series is to describe the underlying movement of the series When measuring a secular trend, a good starting point is to graph the data The trend itself may be represented by a smooth curve or a straight line Measurement of secular trend can be somewhat subjective, depending on the method used to measure it

11. 19-11 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 19.4 Freehand drawing Freehand drawing is a quick and simple method of measuring secular trend It involves first plotting the data on a scatter diagram and joining successive points with straight lines A look at the overall graph will enable you to get a ‘feel’ for the direction of the trend The process is somewhat subjective and does not require any mathematical calculation

12. 19-12 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 19.5 Semi-averages Of all the scientific methods of fitting a straight line to describe secular trend, the method of semi-averages is the easiest It can be affected by outlying or unusual points in the series In many cases it does provide a reasonable approximation of the trend of a time series if a straight line is to be used The technique (assuming that the measurements are yearly) is as follows: 1.Divide the data are into two equal time ranges. (If the number of years in the time series is odd, disregard the middle year.) 2.Calculate the average (arithmetic mean) of the observations in each of the two time ranges. Plot these two averages on the graph above the points that represent the respective midpoints of the years in each of the two time ranges.

13. 19-13 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 19.5 Semi-averages (cont...) 3. Draw a straight line through the two points plotted in Step 2 4. Make forecasts for future years by projecting upwards from the horizontal (year) axis onto the line, and then horizontally across to the vertical axis for the predicted value The method of semi-averages is simplistic, but it is at least a starting point to fitting a straight line to the data If the resulting straight line does appear to be a reasonable fit, it could be used in its own right; or perhaps a more sophisticated least-squares regression line could be tried

14. 19-14 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 19.6 Least-squares linear regression A more advanced method of fitting a straight line to a time series is to use the method of least-squares linear regression In this case of a time series, the observations are the (dependent) y-variables and time is the (independent) x-variable The calculations may be simplified as follows: Case 1 Suppose that there are n time observations (x) where n is an odd number. 1. Let the middle time observation have the value x = 0 2. Number each successive time observation greater than the middle one (counting upwards) with the values x = 1, x = 2, x = 3 and so on 3. Number each successive time observation less than the middle one (counting downwards) with the values x = –1, x = –2, x = –3, and so on

15. 19-15 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 19.6 Least-squares linear regression (cont…) Case 2 Suppose that there are n time observations (x) where n is an even number 1. Let the n/2th observation have the value x = –1 and the (n/2+1)th observation have the value x = +1 2. Number each successive time observation greater than the (n/2+1)th observation (counting upwards by 2) with the values x = 3, x = 5, x = 7 and so on 3. Number each successive time observation less than the (n/2+1)th observation (counting downwards by 2) with the values x = –3, x = –5, x = –7 and so on

16. 19-16 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 19.6 Least-squares linear regression (cont…) In both cases, the mean of the x-values will be zero The least-squares regression line of y on x will still have the same form as However, the time period will be represented by its corresponding x-value The formulae for finding the values of a and b are:

17. 19-17 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 19.7 Moving averages The method of moving averages is based on the premise that, if the values in a time series are averaged over a sufficient period, the effect of short-term variations will be reduced Moving averages are generally not used for predictive purposes but rather to simply provide an indication of the trend There are several different techniques for calculating a moving average of a data point Method 1 For a moving average, calculate the average of the m observations before the observation, the observation itself and the m observations after the observation

18. 19-18 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 19.7 Moving averages (cont…) Method 1 (cont…) Thus, a total of (2m + 1) observations must be averaged each time a moving average is calculated Method 2 To calculate the n-year (or n-month or, n-quarter and so on) moving average for a data point in a time series, calculate the average (or mean) of the data point itself and n – 1 data points before it. This means that the first n – 1 values in the time series will not have a smoothed value

19. 19-19 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 19.8 Single exponential smoothing Many of the objections about moving averages can be overcome if we use a kind of continuous moving average In this case, the smoothed value for an observation x is an average weighted to two quantities, namely: 1. the actual value of observation x 2. a value derived from the history of the time series This technique is known as single exponential smoothing and is a method for continually revising an estimate in the light of more recent trends

20. 19-20 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 19.8 Single exponential smoothing (cont...) The smoothed value for an observation x is obtained from Where S x = the smoothed value for observation x y x = the actual value of observation x S x – 1 = the smoothed value previously calculated for observation (x – 1)  = the smoothing constant (where 0 ≤  ≤ 1)

21. 19-21 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 19.8 Single exponential smoothing (cont..) Thus, the smoothed value of each observation is a function of the smoothed value of the observation immediately before it This iterative process means that errors made in calculating the smoothed values will carry on for each successive time period The method of exponential smoothing with a single smoothing constant can be very effective when a series has no real upward or downward trend but several components with wide variations

22. 19-22 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 19.9 Using a growth model Suppose that we note from a graph of the data that the trend appears to be exponential A growth model may be appropriate, since while it takes account of the exponential nature of the trend, it also allows us to create an equation for the model that may be used to make forecasts for several periods into the future Then we may fit the model: Where a and b are constants and is the predicted value of y

23. 19-23 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e 19.9 Using a growth model (cont…) The aim now is to find the most appropriate values of a and b. The steps are: 1. Take the natural logarithms (i.e. logarithms to the base e) of the y-values to form a variable z such that: 2. Find the least-squares regression line of z on x, say: 3. Then the values of a and b in Equation 19.5 are:

24. 19-24 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e Summary We discussed a time series and explained its use We also identified and interpreted the four basic measures of variation that appear in a time series analysis (secular trend, seasonal variation, cyclical variation and random or irregular variation) We identified and used common methods of fitting secular trend lines to time series (including freehand drawing, least-squares linear regression, semi- averages, moving averages, exponential smoothing and growth models) Lastly we looked at making forecasts