Time series. VCEcoverage. Area of study. Units 3 & 4 Data analysis

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1 Time series 4 VCEcoverage Area of study Units & 4 Data analysis In this cha chapter 4A Time series and trend lines 4B Fitting trend lines: the 2-mean and -median methods 4C Least-squares trend lines 4D Smoothing time series 4E Smoothing with an even number of points 4F Median smoothing 4G Seasonal adjustment

2 46 Further Mathematics Time series and trend lines In previous chapters we looked at bivariate, or (x, y), data where both x and y could vary independently. In this chapter we shall consider cases where the x-variable is time and, generally, where time goes up in even increments such as hours, days, weeks or even years. In these cases we have what is called a time series. The main purpose of a time series is to see how some quantity varies with time. For example, a company may wish to record its daily sales figures over a 0-day period. Time Day Day 2 Day Day 4 Day Day 6 Day 7 Day 8 Day 9 Day 0 Sales ($) We could also make a graph of this time series as shown in the figure. As can be seen from this graph, there seems to be a trend upwards clearly, this company is increasing its revenues! Types of trend Although many types of trend exist, in Further Mathematics we shall be looking at trends that are classified as secular, seasonal, cyclic and random. Secular trends If over a reasonably long period of time a trend appears to be either increasing or decreasing steadily, with no major changes of direction, then it is called a secular trend. It is important to look at the data over a long period. If the trend in the figure above continued for, say, 0 days, then we could safely conclude that the company was indeed becoming more profitable. What appears to be a steady increase over a short term say, stock market share prices can turn out to be something quite different over the long run. Seasonal trends Certain data seem to fluctuate during the year, as the seasons change. Consequently, this is termed a seasonal trend. The most obvious example of a seasonal trend would be the average monthly maximum temperatures over a year. A sample of this type of trend is shown in the figure. Note that the months have been coded, so that = Jan., 2 = Feb., and so on. From which hemisphere of the world would these data come? Cyclic trends Temp. ( C) Sales ($) t Days Months Like seasonal trends, cyclic trends show fluctuations upwards and downwards, but not according to season. Businesses often have cycles where at times profits increase, then decline, then increase again. A good example of this would be the sales of a new major software product, such as a word processor. At first, sales are slow; then they pick up as the product becomes popular. When enough people have bought the product, sales may fall off until a new version of the product comes on the market, causing sales to increase again. This cycle can be repeated many times, which is why there are many versions of some software products t

3 Chapter 4 Time series 47 Random trends Trends may seem to occur at random. This can be caused by external events such as floods, wars, new technologies or inventions, or anything else that results from random causes. There is no obvious way to predict the direction of the trend or even when it changes direction. In the figure, there are a couple of minor fluctuations at t = 4 and t = 8, and a major one at t =. The major fluctuation could have been caused by a change in government which positively affected sales. The trend line If we want to predict the future values of a trend, it is important to be able to fit a straight line to the data that we already have. There are a number of techniques which can be used to determine the trend line. As discussed in chapter, you could fit the line by eye or use the equal number of points technique. In this chapter, two more mathematical methods are shown. Fit a straight line to the following time series data, which represent the body temperature of a patient with appendicitis, taken every hour. THINK 2 WORKED Example Attempt to fit a line using your eye. By trial and error, a line such as the one at right could be the trend line. Evaluate the trend. remember remember WRITE Temp. ( C) Temp. ( C) Profits t t Hours t Hours It is unlikely that the temperature will continue to rise indefinitely, but the line may be significant over the short term.. Time series are a set of measurements taken over (usually) equally spaced time intervals, such as hourly, daily, weekly, monthly or annually. 2. There are 4 basic types of trend: (a) secular: increasing or decreasing steadily (b) seasonal: varying from season to season (c) cyclic: similar to seasonal but not tied to a calendar cycle (d) random: varying from external causes happening at random.

4 48 Further Mathematics 4A Time series and trend lines EXCEL Spreadsheet Trend lines WORKED Example For the first questions, identify whether the trends are likely to be secular, seasonal, cyclic or random: the amount of rainfall, per month, in Western Victoria 2 the number of soldiers in the United States army, measured annually the number of people living in Australia, measured annually 4 the share price of BHP, measured monthly the number of seats held by the Liberal Party in Federal Parliament 6 Fit a trend line to the data in the graph at right. 7 A wildlife park ranger is travelling on safari towards the centre of a wildlife park. Each day (t), he records the number of sightings (y) of zebra that he notes. He draws up the following table. Temp. ( C) t Days Mathcad Trend lines t y Fit a trend line to the data. What type of trend is best reflected by these data?

5 Chapter 4 Time series 49 8 The monthly share prices of a recently privatised telephone company were recorded as follows. Date Jan. 0 Feb. 0 Mar. 0 Apr. 0 May 0 Jun. 0 Jul. 0 Aug. 0 Price ($) Graph the data (let = Jan., 2 = Feb.... and so on) and fit a trend line to the data. Use this line to predict the share price in January of the following year. Comment on the feasibility of the predicted share price. 9 Plot the following monthly sales data for umbrellas. Fit a trend line. Discuss the type of trend best reflected by the data and the limitations of your trend line. Month Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec. Sales Consider the data in the figure, which represent the price of oranges over a 9-week period. a Fit a straight trend line to the data. b Predict the price in week t Weeks The following table represents the quarterly sales figures (in thousands) of a popular software product. Plot the data and fit a trend line using the equal number of points method. Discuss the type of trend best reflected by these data. Quarter Q-96 Q2-96 Q-96 Q4-96 Q-97 Q2-97 Q-97 Q4-97 Q-98 Q2-98 Q-98 Q4-98 Sales Price (cents) 2 The number of employees at the Comnatpac Bank was recorded over a 0-month period. Plot and fit a trend line to the data. What would you say about the trend? Month Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec. Employees

6 0 Further Mathematics Fitting trend lines: the 2-mean and -median methods As we saw in chapter, using our eyes to fit straight lines to a set of data is an unsatisfactory mathematical technique. In this section we shall look at two methods which are more precise because specific mathematical processes are used to calculate the equation of the trend line. These methods are:. the 2-mean method 2. the -median method. Both methods take advantage of the fact that the independent variable, t, increases uniformly, usually in the sequence, 2,,... and so on. The 2-mean method This method is the simplest to use. Step. Divide the data into lower and upper halves, according to the value of t. Step 2. Compute, separately, the means of both halves, for both t and y values. Call these means t L, yl, for the lower half, and t U, y U, for the upper half. Step. The gradient of the line you want to fit is given by: y U y L m = t U t L rise This is nothing more than the form for the gradient. run Step 4. By transposing the linear equation, y = mx + b, the y-intercept (b) is computed from the equation: b = y L m t L. WORKED Example 2 Find the straight trend line for the following data set, using the 2-mean method. t y THINK WRITE Split the data into halves. The first half has t =, 2,, 4,. The second half has t = 6, 7, 8, 9, 0. There are points in each group. 2 Calculate the means of each half y L = = t L = = y U = = t U = = 8

7 Chapter 4 Time series THINK WRITE Calculate the gradient using the formula: m = y L y U t U t L m = = = Calculate the y-intercept using the formula: b = b = y =.08 L m t L 6 7 State the equation of the straight line. Plot the data points and the fitted straight line. Note: With the equation we need only 2 points to graph the line. Interpret results. y = mt + b = 2.04t +.08 When t = 0, y = (0,.08) When t = 0, y = 2.08(0) +.08 = 2.88 (0, 2.88) y The line is an excellent fit to what appear to be linear data. t Because of the excellent fit of the equation to the data, the equation could be used to predict values of y for values of t greater than 0. The -median method As you may recall from chapter, the -median method can be used to fit a straight line to a set of data points. Let us apply this method to the data set from worked example 2. Recall the basic steps of the -median method: Step. Divide the data into groups: lower, middle and upper. Step 2. Find the medians of each of the groups. Step. Calculate the gradient using the formula: m = y U t U y L t L Step 4. Calculate y-intercept using formula b = -- [y L + y M + y U ) m(t L + t M + t U )] (Note that we are using t rather than x for time series data.)

8 2 Further Mathematics WORKED Example Find the straight trend line for the following data set, using the -median method. t y THINK WRITE 2 Split the data into groups. Since there are 0 points, we divide using the 4 pattern, as shown in chapter. Calculate the median of each group. t L = 2; y L = 9 t M =.; y M = 7 t U = 9; y U = 2 The groupings are shown as shading in the table above. Calculate the gradient using the formula: m = y U t U y L t L m 2 9 = = = 2 4 Calculate the y-intercept using the formula: b = -- [( ) 2( )] b = -- [(y L + y M + y U ) m(t L + t M + t U )] = -- [49 2(6.)] = -- 6 =. State the equation of the straight line. y = mt + b = 2t +. In comparing the results obtained in worked examples 2 and, we find that the gradients are similar (2 compared to 2.04) and the y-intercepts are also close (. compared to.08). Either line would be a satisfactory trend line. Once you have an equation it can be used to predict or extrapolate future values as well as interpolate values within the table. For instance, we calculated a trend line of y = 2t +.. We could use this equation to find y at various values: y(.) = 2(.) +. = 6. (interpolation) y(2) = 2(2) +. = 29.2 (prediction or extrapolation) However, the further into the future you try to predict, the less likely it is that your answer will be accurate. Limitations to fitting trend lines It is important to note that these techniques are limited to the case where the trend is clearly linear; they cannot be applied effectively to cyclical or seasonal trends. As an example, consider the following data set which represents sales of swimsuits at a popular clothing store.

9 Chapter 4 Time series WORKED Example Use a 2-mean and 2-mean method 4 b -median methods to fit a trend line to the swimsuit data below. Month (t) Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec. Sales (y) THINK WRITE a 2 Split the data into halves. Note = Jan., 2 = Feb.,... 2 = Dec. Calculate the means of each half. Note: In this case, and are easily calculated. a The first half has t =, 2,, 4,, 6. The second half has t = 7, 8, 9, 0,, 2. There are 6 points in each group. t L t U yl = = 78.8 t L =. 4 Calculate the gradient using the formula: m = y U t U y L y L Calculate the y-intercept using the formula: b = y L m t L State the equation of the straight line. Plot the data points and the fitted trend line. Note: The trend line is a poor predictor, due to the cyclic nature of the trend. 6 y y U = = = 9. b = = 7.94 y = mt + b =.97t median method Month (x) Sales (y) t U m = = = t Continued over page

10 4 Further Mathematics THINK WRITE b Split the data into groups. Note: Change the variable from t to x to match formulas for the -median method. b The groups are shown with shading in the preceding table. 2 Calculate the medians of each group. When calculating medians for 4 data points, choose the two middle t-values and the two middle y-values independently (see chapter ). The middle two t-values in the first group are 2 and. The middle two y-values in this group are 8 and 96. t L = 2.; y L = 89. (using 8 and 96) t M = 6.; y M = 64. (using 62 and 67) t U = 0.; y U = 99 (using 96 and 02) Calculate the gradient using the formula: m = y U t U y L t L m = = =.9 4 Calculate the y-intercept using the formula: b = -- [(y L + y M + y U ) m(t L + t M + t U )] State the equation of the straight line. b = -- [( ).9( )] = -- [2.9(9.)] = = 76.6 y = mt + b =.9t Plot the data and the trend line. y 20 Note: Although this line is slightly 0 00 different from that in part a, it is no 90 better as a predictor. No method will effectively fit a straight line when the 60 data are not linear! x Note: Care must be taken when fitting trend lines. Look at the data first, and if they have only a secular or random trend (not cyclic or seasonal), then you can probably fit a straight line using either the 2-mean or -median method. Which method you use is up to you they essentially yield equally effective trend lines.

11 remember remember Chapter 4 Time series. The trend line is a straight line that can be used to represent the entire time series and could be used for predicting the future values of the time series. The line can be found in several ways. (a) Fit by eye: try to fit a straight line using methods of chapter. (b) Calculate using the 2-mean method. (i) Divide the time series into lower and upper halves. (ii) Compute the means of the t and y values for each half. These are called t L, y L, for the lower half, and t U, y U, for the upper half. y U y L (iii) Compute the gradient using the formula: m = (iv) Compute the y-intercept using the formula: b = y L m t L. (v) Write the equation of the straight line: y = mt + b. (c) Calculate using the -median method. (i) Divide the data into groups: lower, middle and upper. (ii) Find the median of each of the groups. y U y (iii) Calculate the gradient using the formula: m = L. t U t L (iv) Calculate the y-intercept using the formula: b = -- [(y L + y M + y U ) m(t L + t M + t U )] (v) Write the equation of the straight line: y = mt + b. t U t L 4B Fitting trend lines: the 2-mean and -median methods WORKED Example 2 The following table represents the number of cars remaining to be completed on an assembly line. Fit a straight line to the following data using the 2-mean method. Put the first points in the first half. SkillSHEET 4. Time (hours) Cars remaining WORKED Example 2 Repeat question, putting the first 4 points in the first half. Comment on the difference in the answers. Repeat question using the -median method and comment on the difference in the trend line equation. 4 From the equations of the trend lines, it should be possible to predict when there are no cars left on the assembly line. This is done by finding the value of t which makes y = 0. Using the equations from questions, find the times when there will be no cars left on the assembly line. EXCEL -median method Spreadsheet

12 6 Further Mathematics WORKED Example 4 When the MicroHard Company first started, it employed only one person. Each month the company has grown, so that after 2 months there are 4 people working there. The time series data are shown by the graph at right. a b Fit a -median line to the data. Predict the number of employees after a further 2 months. 6 The table below shows the share price of MicroHard during a volatile period in the stock market. Fit: a 2-mean and b -median lines and comment on your result. What type of trend is this? Number of staff y Months t Day Price ($) The following time series shows the number of Internet Web Sites over a 9-month period. Plot the data and fit a -median trend line. Comment on this line as a predictor of further growth. Time (months) Sites (millions) EXCEL Spreadsheet Twomean method 8 For the following 4 tables, identify the type of trend. Use the 2-mean method to calculate the trend line if the trend appears to be secular or random. If the trend is cyclic or seasonal, do not calculate a line. a x y b x y c x y d x y Repeat question 7 using the 2-mean method. Was there much difference between the 2 equations?

13 Chapter 4 Time series 7 Least-squares trend lines As we saw in chapter, a better technique for fitting straight lines is least squares regression. Again, this method will be effective only if the data given are reasonably linear, so it is a good idea to plot the data first. In fact, because we are now dealing with time series, the linear regression can be simplified greatly. This simplification relies on a technique called centring. Centring the time scale With most (if not all) time series, we are given the t data in a constant sequence, usually, 2,, 4... and so on. Because these data represent time and not any real set of numbers, it really does not matter what values we use for time we could equally use 2, 4, 6, or even 0, 9, 8. Centring involves transforming the data so that the middle record has a value of 0, the smaller time values are negative and the larger ones are positive. We can then take advantage of this pattern and simplify the least-squares regression formulas. How this is accomplished depends on whether we have an even number or an odd number of data points. Centring with an odd number of points Consider the following table. The first row is the traditional time data, 2,, 4... while the second row contains the centred data. T t centred Note: Centring is extremely simple find the middle time value and set it to 0, and adjust all the other points accordingly. Centring with an even number of points In this case there are only 8 points; note the difference in the centring method. T t centred Here there are two middle values; the larger one becomes 0., the smaller one 0.. The remaining points are adjusted accordingly. Why bother to centre the data? The reason is that the mean or average value of t centred becomes 0. Also note, as a check of your centring, that the sum of all the t values is 0. Recall the regression formulas from chapter 2, where x has been replaced by t. m Σty nty = and b = y m t Σt 2 nt 2 Now, if we used t centred in place of t, then t = 0, so the formulas are simplified to: m Σty = and b = y. Σt 2 This makes the calculation of m and b extremely simple and fast, especially if you use a calculator. Note also that neither m nor b depend upon n, the number of points.

14 8 Further Mathematics WORKED Example Use the following data from worked example. Centre the time data where Jan. =, Feb. = 2,..., and calculate the trend line for the centred data using the least-squares method. t y THINK Centre the time data. Compute Σt 2 for m by squaring the t values. Compute Σty for m by first writing the y-values. Then multiply by t centred. Now add up row 2 and row 4. Compute m and b from formulas and state the equation for the trend line. WRITE The centred t-values are: t t y ty Σt 2 = = 82. Σty = = 7. Σty m = Σt 2 7. = = 2.08 b = y = = 6. so y = 2.08t + 6. Plotting the regression line To compare the equation obtained by the least-squares method with the other methods already discussed, we need to transform the regression line back to the original t scale. Step. Transform the time values from t centred back to t. In worked example we subtracted. from each original t value to get t centred. In other words t centred = t.. Step 2. Substitute into the regression equation. In worked example : y = 2.08(t.) + 6. y = 2.08t y = 2.08t Step. Plot this equation along with the original time series. This equation can be used to predict future trends. y t

15 Chapter 4 Time series 9 Can least-squares regression handle cyclic time series? Let us consider the following cyclical data of average monthly maximum temperatures in an Australian city. To see how an odd number of points are centred, we shall look at an -month cycle only. Month Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Temperature To analyse these data we use the following spreadsheet solution. Month t t centred temp (y) t 2 ty Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov These formulas are used for the spreadsheet. B C D E F G 4 Month t t centred temp (y) t 2 ty Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov Now, look at the regression line obtained from this spreadsheet: y = 0.49t To transform back to the original time scale: t centred = t 6 y = 0.49(t 6) y = 0.49t Totals m b =D^2 =D6^2 =D7^2 =D8^2 =D9^2 =D0^2 =D^2 =D2^2 =D^2 =D4^2 =D^2 Totals =SUM(E:E) =SUM(F:F) m b y =D*E =D6*E6 =D7*E7 =D8*E8 =D9*E9 =D0*E0 =D*E =D2*E2 =D*E =D4*E4 =D*E =SUM(G:G) =G7/F7 =E7/ t

16 60 Further Mathematics This line is plotted along with the original temperature data. This equation could be used to predict future trends. The graph demonstrates that the trend line equation would be a poor predictor. Consider the observation for May (t = ). The equation would predict y = 0.49() = 2.8, while the actual observation was only 8. For most months, the data points are a long way from the straight line. This also indicates a poor prediction line. In fact, from chapter, it was found that the correlation coefficient was 0.4, which indicates that time (t) has only a moderately weak effect on temperature. So least squares is no better than the earlier methods in fitting trend lines for data that are not linear. remember remember To use the least-squares regression technique to find a trend line:. Centre the data so that the sum of all t values = 0. There are two methods, depending on whether there are even or odd number of points in the time series. Σty 2. The gradient can be calculated simply from the formula: m = Σt 2. The y-intercept can be calculated from: b = y. 4C Least-squares trend lines EXCEL Spreadsheet Least squares trend lines WORKED Example The Teeny-Tiny-Tot Company has started to make prams. Its sales figures for the first 8 months are given in the table below. Date Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sales a b c d Centre the time data, using the sequence Jan. =, Feb. = 2,..., and calculate the trend line for the centred data using the least-squares method. Plot the data points and the trend line on the same set of axes. Use the trend line equation to predict the company s sales for December. Comment on the suitability of the trend line as a predictor of future trends, supporting your arguments with mathematical statements. 2 The sales figures of Harold Courtenay s latest novel (in thousands of units) are given in the table below. The book was released a week before the first figures were collected. Time (weeks) Sales ( 000) a Centre the time data and calculate the trend line for these data using the leastsquares method. b Plot the data points and the trend line on the same set of axes. c Use the trend line equation to predict the sales for weeks 0, 2 and 4. d Comment on the suitability of the trend line as a predictor of future trends, supporting your arguments with mathematical statements.

17 Chapter 4 Time series 6 Farmer Brown s yield of soybeans per hectare has been monitored over the last 8 years. By using modern farming methods she has increased her yield most years. There was a drought in 998, causing a bad yield. Yields are measured in tonnes per hectare. a Centre the time data and calculate the trend line for these data using the leastsquares method. b Plot the data points and the trend line on the same set of axes. c Use the trend line equation to predict the yield for 200. d Comment on the suitability of the trend line as a predictor of future trends, supporting your arguments with mathematical statements. 4 The price of long-distance telephone calls has been changing recently. The cost of a -minute call to Paris from Melbourne has been monitored over a 7-week period. a b c d Year Yield Time period Cost ($) Centre the time data and calculate the trend line for these data using the leastsquares method. Plot the data points and the trend line on the same set of axes. Use the trend line equation to predict the next data points. Comment on the suitability of the trend line as a predictor of future trends, supporting your arguments with mathematical statements.

18 62 Further Mathematics Since management instituted new policies, the productivity at the DROF car plant has been improving slowly but steadily. The following table records the number of cars produced each week over a 0-week period. a Centre the time data and calculate the trend line for these data using the leastsquares method. b Plot the data points and the trend line on the same set of axes. c Use the trend line equation to predict car production for weeks 0, 2 and 4. d Comment on the suitability of the trend line as a predictor of future trends, supporting your arguments with mathematical statements. 6 The following data represent the share price for the first few days of a new software company. a b c d Centre the time data and calculate the trend line for these data using the leastsquares method. Plot the data points and the trend line on the same set of axes. Use the trend line equation to predict the price for the next 4 days. Comment on the suitability of the trend line as a predictor of future trends, supporting your arguments with mathematical statements. 7 The following table represents the daily water level in a reservoir during a drought. a b c d Time (weeks) Cars produced Day Price ($) Day Level (m) Centre the time data and calculate the trend line for these data using the leastsquares method. Plot the data points and the trend line on the same set of axes. Use the trend line equation to predict the price for the next 4 days. Comment on the suitability of the trend line as a predictor of future trends, supporting your arguments with mathematical statements. 8 The following table represents the height of James, measured each year. Year Height (cm) a b c d Centre the time data and calculate the trend line for these data using the leastsquares method. Plot the data points and the trend line on the same set of axes. Use the trend line equation to predict the height for the next 4 years. Comment on the suitability of the trend line as a predictor of future trends, supporting your arguments with mathematical statements.

19 Chapter 4 Time series 6 9 The average quarterly price of coffee (per 00 kg) has been recorded for years. Quarter Q-96 Q2-96 Q-96 Q4-96 Q-97 Q2-97 Q-97 Q4-97 Q-98 Q2-98 Q-98 Q4-98 Price ($) a Centre the time data and calculate the trend line for these data using the leastsquares method. b Plot the data points and the trend line on the same set of axes. c Use the trend line equation to predict the price for the next quarter. d Comment on the suitability of the trend line as a predictor of future trends, supporting your arguments with mathematical statements. 0 A mathematics teacher gives her students a test each month for 0 months, and the class average is recorded. These tests are carefully designed to be of similar difficulty. a Centre the time data and calculate the trend line for these data using the least-squares method. b Plot the data points and the trend line on the same set of axes. c Use the trend line equation to predict the results for the last exam in December. d Comment on the suitability of the trend line as a predictor of future trends, supporting your arguments with mathematical statements. Test Mark (%) Feb. 7 Mar. 6 Apr. 62 May 67 Jun. 6 Jul. 68 Aug. 70 Sep. 72 Oct. 74 Nov. 77 To make an improvement on the spreadsheet given in the text on page 9, can you modify it so that it automatically centres the t values? (Hint: If n = the number of data n + points, then you need to subtract to centre values.) 2 WorkSHEET 4.

20 64 Further Mathematics Smoothing time series By now, you should appreciate the fact that fitting linear trend lines to time series that are not really linear is both bad mathematics and bad policy it doesn t work! So how can we have a method that generates trend lines for such time series? If the non-linear nature of the data is random we can use a technique called smoothing. If the non-linear nature is seasonal, we use a method called seasonal adjustment. Moving-average smoothing This technique relies on the principle that averages of data can be used to represent the original data. When applied to time series, a number of data points are averaged, then we move on to another group of data points in a systematic fashion and average them, and so on. It is generally quite simple. Consider the following example: Notice how the third column is computed from the first two.. Take the first three t points (, 2, ) and find their average (2); take the first three y points in the table (2, 0, ) and find their average (2.). 2. Take the next three t points (2,, 4) and find their average (); take the next three y points in the table (0,, ) and find their average (2.7).. Repeat until you reach the last three t points. 4. Take the last three t points (7, 8, 9) and find their average (8); take the last three y points in the table (8, 2, 9) and find their average (9.). As we use three points to average, moving along the table from left to right, this is called a -point moving average smoothing. Note: The calculation of the t values is quite simple. Time (t) Data (y) Moving average = = = = = = = We are free to choose any number of points for our smoothed graph; we could have a 4-point smoothing, a -point smoothing or even an -point smoothing. Although it is preferable to choose an odd number, such as or, it is possible to choose even numbers as well, with a slight change in the method. In either case it does not matter how many points are in our time series. Moving average smoothing with odd numbers of points As seen above, the method for smoothing with an odd number (,,...) is quite simple, and can be done in a vertical tabular form. It is crucial that the time values be equally spaced, but they don t have to be in the sequence, 2,. Note: There are fewer smoothed points than original ones. For a -point smooth, point at either end is lost, while for a -point smooth 2 points at either end are lost.

21 Chapter 4 Time series 6 WORKED Example 6 The temperature of a sick patient is measured every 2 hours and the results are recorded. a Create a -point moving average smoothing of the data. b Plot both original and smoothed data. c Predict the temperature for 8 hours using the last smoothed value. Time (hours) Temp. ( C) THINK WRITE a Put the data in a table. a 2 Calculate a -point moving average for each data point. Note: The lost values are at t = 2 and t = 6. Therefore, the first point plotted is (4, 6.87). Time Temp. Smoothed temp ( ) = ( ) = ( ) = ( ) = ( ) = ( ) = b 2 Plot the data. The smoothed line is the thicker, red one. Note: The smoothed data start at the 2nd time point and finish at the 7th point. Interpret the result. c Last smoothed data point is 7.0. b Temp. ( C) Number of hours The smoothed line has removed much of the fluctuation of the original time series and, in fact, clearly exposes the secular trend (upwards) in temperature. c The temperature at 8 hours is predicted to be The main reason for using a smoothing technique is to remove irregularities or wild variations in our time series.

22 66 Further Mathematics Prediction using moving averages Because the moving average does not generate a single linear equation, there are limited possibilities for using the resultant smoothed data for prediction. However, there are two things that can be done:. Predict the next value use the last smoothed value to predict the next time point. In the above example, our prediction for t = 8 would be temperature = 7.0. This is not necessarily an accurate prediction but it is the best we can do without a linear trend equation. 2. Fit a single straight line to the smoothed data using one of the 2-mean, -median or least-squares techniques, one could find a single equation for the smoothed data points. This is often the preferred technique. How many points should be in the moving average? When smoothing data, it is important to decide on the number of points to be used. Should a -point, -point or even an -point moving average be used? This is an important but complicated question. Here are some basic hints. (Let p = for points, p = for points... and let n = the number of points in the time series.). The value of p should be considerably smaller than n. For example, if n = 7, p should be no more than about If there is a cyclic or seasonal variation that we want to remove, let p length of cycle. Otherwise p should be less than the length of the cycle. However, movingaverage smoothing may not be effective at removing the variation. For example, if there are quarterly data with seasonal variation, use p = 4.. It is always preferable to use an odd value of p, regardless of whether n is even or odd. 4. The larger the value of p, the smoother the trend line of the resulting data becomes. More of the fluctuations will be removed. However, you can go too far. Moving average smoothing using a spreadsheet A spreadsheet can be devised to calculate the average data values and then the new set of smoothed points plotted on a graph. Below is the spreadsheet for worked example 6. The graph is shown in worked example 6. -point t temp. smooth Below are the formulas used. Note the row and column numbers carefully. Add or delete rows from the middle of the sheet (around row 9) rather than at the ends. There is no need to calculate the first (E6) and last (E) average, as these are the lost values. It should be clear how to turn this into a -point, or 7-point smooth. Why wouldn t we go any further?

23 Chapter 4 Time series 67 B C D E F 4 -point t temp. smooth =SUM(D6:D8)/ =SUM(D7:D9)/ =SUM(D8:D0)/ =SUM(D9:D)/ =SUM(D0:D2)/ =SUM(D:D)/ remember remember. Smoothing involves replacing the original time series with another one where most of the variation has been removed to see if there is a secular trend. Points are lost at the start and end of the time series. Refer to the text for detailed descriptions of the techniques involved. 2. Moving average smoothing works best with an odd number of points. For a -point smoothing, point is lost at either end of the time series. 4D Smoothing time series WORKED Example 6 The following table represents sales of a textbook. Year (t) Sales (y) a b c Create a -point moving average of the data. Plot both the original and smoothed data. Predict the sales for 998 using the last smoothed value. 2 The sales of a certain car seem to have been declining in recent months. The management wishes to find out if this is the case. Using a -point moving average, smooth the data and comment on the result. Use Jan. =, Feb. = 2... Month Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec. Sales EXCEL Moving average Spreadsheet

24 68 Further Mathematics Management is still not satisfied perform a -point moving average on the data from question 2 and discuss the result. 4 Consider the quarterly rainfall data below. Rainfall has been measured over a -year period. Since the data are seasonal, perform a -point moving average and comment on whether there is a trend other than the cyclical one. Time (t) Spring 998 Summer 998 Autumn 998 Winter 998 Spring 999 Summer 999 Autumn 999 Winter 999 Spring 2000 Summer 2000 Autumn 2000 Winter 2000 Rainfall (mm) The attendance at the Woop-Woop Football Club games was recorded over 0 years. Management wishes to see if there is a trend. Perform a -point moving average on the data and comment on the result. Year Attendance ( 000) Use a spreadsheet solution to complete a -point moving average on the following data which represent sales figures for a 2-week period. Week Sales Smoothed data Week Sales Smoothed data 7 Coffee price data are reproduced below. To remove the seasonal variation, perform a -point moving average to smooth the data. Plot the smoothed and original data and comment on your result. Quarter Q-96 Q2-96 Q-96 Q4-96 Q-97 Q2-97 Q-97 Q4-97 Q-98 Q2-98 Q-98 Q4-98 Price ($)

25 Chapter 4 Time series 69 8 The sales of a new car can vary due to the effect of advertising and promotion. The sales figures for Nassin Motor Company s new sedan are shown in the table. Perform a -point moving average to smooth the data. Plot the data, and use the last smoothed value to predict sales for the next month. Month Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec. Sales The attendance at weekly Trash n Treasure sales can vary due to weather. Perform a -point moving average to smooth the data in the table. Can you determine on which days the weather was poor? Week Attendance A large building site requries varying numbers of workers. The weekly employment figures over the last 7 weeks have been recorded. By performing a -point moving average, predict the number of people required for the next week. Week Employees

26 70 Further Mathematics Smoothing with an even number of points As mentioned in the previous section, it is usually preferable to use an odd number of points. However, there are times when an even number of points can be used that is, a 4-point, 6-point or even 0-point moving average. When we used an odd number of points, the result was automatically centred; that is, the y-data had the same t-values as the original (except at the first and last lost points). This does not occur with an even-point smoothing, as shown in the following example of a 4-point moving average. Time y-value 4-point average (smoothed value) Calculation Result ( ) ( ) ( ) ( ) Observe that the first average (0.) is not aligned with any particular year it is aligned with 99.! Also note that there are now three lost values (the seven original records reduced to four). In other words, the moving average is not centred properly. To align the data correctly, an additional step needs to be performed; this is called centring.

27 Chapter 4 Time series 7 Use the following procedure to centre the data: Step. Find the average of the first two smoothed points and align it with the rd time point. Step 2. Find the average of the next two smoothed points and align it with the 4th time point. Step. Repeat, leaving two blank entries at both top and bottom of the table. This is demonstrated in the following table, using the data from the previous table. Time y-value 4-point average (smoothed value) 4-point average after centring Calculation Result Calculation Result ( ) (0. +.7) 2.2 ( ) (.7 + ) ( ) ( +.) ( ) The first average (.2) is now aligned with 994, the second (2.7) aligned with 99 and so on. This process not only introduces an extra step, but an extra averaging (or smoothing) as well. It is usually preferable to stick with an odd-point smoothing to reduce these difficulties. However, if the original time series has seasonal variation, by taking a 4-point moving average we get a point from each season (spring, summer, autumn, winter) in our moving average. Thus, we may remove much of the seasonal fluctuation in the data.

28 72 Further Mathematics WORKED Example 7 The quarterly sales figure for a dress shop (in thousands of dollars) were recorded over a 2-year period. Perform a centred 4-point moving average on the data and plot the result. Comment on any trends you find. Time Summer Autumn Winter Spring Summer Autumn Winter Spring Sales ( 000) THINK WRITE 2 Put the data in a table. Note: Code the time column. Calculate a 4-point moving average in column. Average the pairs of averages to find the 4-point centred data. This is done in column 4. Time Sales 4-point moving average 4-point centred m.a ( ) 4 = ( ) 2 = 2.7 ( ) 4 = ( ) 2 = 24.6 ( ) 4 = 2.00 ( ) 2 = 2.8 ( ) 4 = ( ) 2 = 26.2 ( ) 4 = Plot the data. The smoothed line is the thicker, red one. Note: The smoothed data start at the rd time point and finish at the 6th point. Interpret the results Sales ( $000) Time Observe the steadily increasing trend (even with only four smoothed points) that was not obvious from original data. 8

29 Chapter 4 Time series 7 Even point smoothing with spreadsheets The spreadsheet for the 4-point moving average of worked example 7 is shown below. t y 4-point 4-point-C* The formulas are shown below. Note the cell row and column labels. C D E F 4 t y 4-point 4-point-C* =SUM(D:D8)/4 7 9 =SUM(D6:D8)/4 =SUM(E6:E7)/ =SUM(D7:D8)/4 =SUM(E7:E8)/2 9 =SUM(D8:D)/4 =SUM(E8:E9)/ =SUM(D9:D2)/4 =SUM(E9:E0)/ There is little difference between this and a -point moving-average spreadsheet, except that the SUMs are located (columns E and F) to correspond to the appropriate term in the time series (columns C and D). remember remember. Smoothing involves replacing the original time series with another one where most of the variation has been removed, to see if there is a secular trend. 2. Moving averages work best with an odd number of points. For a -point moving average, one point is lost at either end of the time series.. Moving average smoothing with an even number of points is a 2-step process. First, we perform a 4-point moving average, then centre by averaging pairs of the 4-point averages. For 4-point averages, two points are lost at each end of the time series.

30 74 Further Mathematics 4E Smoothing with an even number of points WORKED Example 7 Perform a 4-point centred moving average to smooth the following data and plot the result. Comment on any trends that you find. t y The price of oranges fluctuates from season to season. Data have been recorded for years. Perform a 4-point centred moving average, plot the data and comment on any trends. t Autumn 998 Winter 998 Spring 998 Summer 998 Autumn 999 Winter 999 Spring 999 Summer 999 Autumn 2000 Winter 2000 Spring 2000 Summer 2000 Price Use a spreadsheet to complete the following table. The time series represents the temperature of a hospital patient over days Day Temperature 4-point moving average 4-point centred moving average

31 Chapter 4 Time series 7 4 The sales of summer clothing vary according to the season. The following table gives seasonal sales data (in thousands of dollars) for years at a Sydney Jones department store. Season Q-96 Q4-96 Q-97 Q2-97 Q-97 Q4-97 Q-98 Q2-98 Q-98 Q4-98 Q-99 Q2-99 Sales a b c Calculate a 4-point centred moving average. Plot the original and smoothed data. Determine if there is an underlying trend upwards or downwards. Calculate a 6-point centred moving average on the data from question. 6 An athlete wishes to measure her performance in running a km race. She records her times over the last 0 days. Day Time (s) a Calculate a 4-point centred moving average. b Plot the original and smoothed data. c Determine if there is a significant improvement in her times. 7 The following table shows the share price index of Industrial Companies during an unstable fortnight s trading. By calculating a 4-point centred moving average, determine if there seems to be an upward or downward trend. Day Index

32 76 Further Mathematics Median smoothing An alternative to moving average smoothing is to replace the averaging of a group of points with the median of each group. Although no particular mathematical advantage is gained, it is a faster technique requiring no calculations (provided you use odd-point median smoothing). Often it can be done directly on a graph of a time series. Median smoothing from a table By placing the data in a table, median smoothing can be performed simply and quickly. Look at each group of three points (for smoothing with -point medians) and choose the middle value. Progress through the table one point at a time. As with other methods, points will be lost at the beginning and end of the table. WORKED Example 8 Perform a -point median smoothing on the data in the table below which represents the cost of an airline ticket between Sydney and Melbourne over an 8-month period. THINK WRITE 2 Put data in the table. Find the median of each group of data points. Again, note the lost values at t = and t = 8. Time Cost -point median smoothed cost median of (40, 0, 20) = median of (0, 20, 40) = median of (20, 40, 00) = median of (40, 00, 0) = median of (00, 0, 0) = median of (0, 0, 0) = Generally, the effect of median smoothing is to remove some random fluctuations. It probably performs poorly on cyclical or seasonal fluctuations unless the size of the range being used (,, 7,... points) is chosen carefully. Median smoothing from a graph Provided the graph has clearly marked data points, it is possible to find a median smooth directly from it.

33 Chapter 4 Time series 77 WORKED Example Perform a -point median smoothing on the following graph of a time series. THINK 9 WRITE y x 2 Read the data values and compute the median. Plot the medians on the graph. Note: Median smoothing has indicated a downward trend that is probably not in the real time series. This indicates that moving average smoothing would be the preferred option. The st data points are 2, 8, 6 so median = 6. The 2nd data points are 8, 6, 8 so median = 6. The rd data points are 6, 8, 2 so median = 2. The 4th data points are 8, 2, 6 so median = 2. The th data points are 2, 6, 2 so median = 2. The 6th data points are 6, 2, 8 so median = 2. The 7th data points are 2, 8, 0 so median = 0. The 8th data points are 8, 0, 4 so median = 0. y x remember remember. Smoothing involves replacing the original time series with another one where most of the variation has been removed, in order to see if there is a secular trend. There are three basic smoothing techniques. (a) Moving average smoothing works best with an odd number of points. For a -point smooth, one point is lost at either end of the time series. (b) Moving average smoothing with an even number of points is a 2-step process. First you perform a 4-point moving average, then centre by averaging pairs of the 4-point smooth. For a 4-point centred smooth, two points are lost at each end of the time series. (c) Median smoothing is usually done with an odd number of points. The number of points lost is the same as for moving average smoothing. 2. In all cases, points are lost at the start and end of the time series. Refer to the text for detailed descriptions of the techniques involved.

34 78 Further Mathematics 4F Median smoothing WORKED Example 8 Perform a -point median smooth on the following data and plot the result. Comment on any trends that you find. These are the same data as in question, Exercise 4E, so compare the graphs of the median smooth with the moving average smooth. t y WORKED Example 9 2 The maximum daily temperatures for a year were recorded as a monthly average. Perform a -point median smooth on the data. Comment on your result. Month Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec. Temp. ( C) Perform a -point median smooth on the graphical time series shown at right. Comment on the effectiveness of the result. 4 Perform a -point median smooth on the graphical time series shown at right. Comment on the effectiveness of the result. Perform a -point median smooth on the data in the following table, which represents the share price of the HAL computer company over the last days. y x y x Day Price WorkSHEET Perform a -point median smooth on the data in the following table, which represents the share price of the Pear-Shaped Computer Company over an 8-week trading period. Day Price Day Price Day Price Day Price

35 Chapter 4 Time series 79 Seasonal adjustment As we have seen in the sections on fitting a straight line to a time series, it is difficult to find an effective linear equation for such data. As well, the sections on smoothing indicated that seasonal data may not lend themselves to the techniques of moving-average or median smoothing. We may just have to accept that the data vary from season to season and treat each record individually. For example, the unemployment rate in Australia is often quoted as 6.8% seasonally adjusted. The Government has accepted that each season has its own time series, more or less independent of the other seasons. How can we remove the effect of the season on our time series? The technique of seasonally adjusting, or de-seasonalising, will modify the original time series, hopefully removing the seasonal variation, and exposing any other trends (secular, cyclic, random) which may be hidden by seasonal variation. De-seasonalising time series The method of de-seasonalising time series is best demonstrated with an example. Observe carefully the various steps, which must be performed in the order shown. Season Unemployment figures have been collected Summer over a -year period and presented in this Autumn table. It is difficult to see any trends, other than seasonal ones. Winter a Compute the seasonal indices. Spring b De-seasonalise the data using the seasonal indices. c Plot the original and deseasonalised data. d Comment on your results, supporting your statements with mathematical evidence. THINK WRITE a WORKED Example 2 Find the yearly averages over the four seasons for each year and put them in a table. Divide each term in the original time series by its yearly average. 0 a 994: ( ) 4 = : ( ) 4 = : ( ) 4 = : ( ) 4 = : ( ) 4 = Year Average Summer 994: = Autumn 994: =.098 Winter 994: =.0847 Spring 994: = Summer 99: = Spring 998: = Summer Autumn Winter Spring Continued over page

36 80 Further Mathematics THINK Find the seasonal averages from this second table. These are called seasonal indices. b Divide each term in the original series by its seasonal index. This is the seasonally adjusted time series. Note: Your answers may vary a little, depending upon how and when you rounded your calculations. c Graph the original and the seasonally adjusted time series. d Note that most, but not all, of the seasonal variation has been removed. However, by using least squares we could more confidently fit a straight line to the de-seasonalised data. WRITE Summer: ( ) = Autumn: ( ) =.0772 Winter: ( ) =.0692 Spring: ( ) = b Summer 94: = Autumn 94: = c Season Seasonal index Summer Autumn.0772 Winter.0692 Spring Spring 98: = Summer Autumn Winter Spring Unemployment figures Time period d There appears to be a slight upward trend in unemployment.

37 Chapter 4 Time series 8 Spreadsheet solution A spreadsheet using the data in worked example 0 has been constructed. Note: The input data are in the table at the top of the spreadsheet. Step. Yearly averages are calculated just below the data table. Step 2. Each term is divided by the appropriate yearly average. Step. Seasonal indices are calculated (to the right of step 2). Step 4. De-seasonalised data are calculated (below step 2). Season Summer Autumn Winter Spring Step Yearly ave Step Step 2 Season Seasonal indices Summer Autumn Winter Spring Step 4 Season Summer Autumn Winter Spring Original data Seasonalised

38 82 Further Mathematics The formulas corresponding to the spreadsheet above follow. Note carefully the row and column addresses. B C D E F G H I 4 Season Summer Autumn Winter Spring Step Yearly ave. =SUM(D:D8)/4 =SUM(E:E8)/4 =SUM(F:F8)/4 =SUM(G:G8)/4 =SUM(H:H8)/4 Step 2 Step 2 Season Seasonal indices 4 6 Summer Autumn Winter Spring =D/D$0 =D6/D$0 =D7/D$0 =D8/D$0 =E/E$0 =E6/E$0 =E7/E$0 =E8/E$0 =F/F$0 =F6/F$0 =F7/F$0 =F8/F$0 =G/G$0 =G6/G$0 =G7/G$0 =G8/G$0 =H/H$0 =H6/H$0 =H7/H$0 =H8/H$0 =SUM(D:H)/ =SUM(D4:H4)/ =SUM(D:H)/ =SUM(D6:H6)/ 7 8 Step 4 Season Summer Autumn Winter Spring =D/$I$ =D6/$I$4 =D7/$I$ =D8/$I$6 =E/$I$ =E6/$I$4 =E7/$I$ =E8/$I$6 =F/$I$ =F6/$I$4 =F7/$I$ =F8/$I$6 =G/$I$ =G6/$I$4 =G7/$I$ =G8/$I$6 =H/$I$ =H6/$I$4 =H7/$I$ =H8/$I$6 Note the following:. By adding/deleting columns between columns D and H, you could increase/decrease the number of years. Remember to change the denominator in the seasonal indices (I... I6) 2. By adding/deleting more rows between Rows and 8, you could increase/decrease the number of seasons (see Exercise 4G, question 4). Do not forget to change the denominator in row 0.. The sum of the seasonal indices always equals the number of seasons (4 in this example). remember remember De-seasonalising a time series involves replacing the original time series with another one where most or all of the seasonal variation is removed. To deseasonalise data: (a) Average over all seasons for each year these are the yearly averages. (b) Divide each point in the original time series by its corresponding yearly average. (c) Using this new series, average over all years for each season these are the seasonal indices. (d) Divide each point in the original time series by its corresponding seasonal index.

39 Chapter 4 Time series 8 4G Seasonal adjustment WORKED Example 0 Note: Your answers may vary slightly, depending upon rounding. Try to round to 4 decimal places for all intermediate calculations. The price of sugar ($/kilo) has been recorded over years on a seasonal Season basis. Summer a Compute the seasonal indices. b De-seasonalise the data using the Autumn c seasonal indices. Winter Plot the original and de-seasonalised data. Spring d Comment on your results, supporting your statements with mathematical evidence. 2 Data on the total seasonal rainfall (in mm) have been accumulated over a 6-year period. Season a b c d Summer Autumn Winter Spring Compute the seasonal indices. De-seasonalise the original time series. Plot the original and de-seasonalised time series. Comment on your result, supporting your statements with mathematical evidence. It is known that young people (8 2) have problems in Season finding work that are different from those facing older people. The youth unemployment statistics are recorded separately from the overall data. Using the youth unemployment figures for five years shown at right: a Compute the seasonal indices. Summer Autumn Winter Spring b De-seasonalise the time series. c Plot the original and de-seasonalised time series. d Comment on your result, supporting your statements with mathematical evidence. 4 It is possible to seasonally adjust time series for other than the usual 4 seasons. Consider an expensive restaurant that wishes to study its customer patterns on a daily basis. In this case a season is a single day and there are 7 seasons in a weekly cycle. Data are total revenue each day shown in the table which follows. Modify the spreadsheet solution to allow for these 7 seasons and de-seasonalise the following data over a -week period. Comment on your result, supporting your statements with mathematical evidence. EXCEL Seasonal adjustment Spreadsheet

40 84 Further Mathematics The unemployment rate in a successful European economy is given in Quarter 2 4 the table at right a Compute the seasonal indices. b De-seasonalise the time series c Plot the original and deseasonalised time series d Comment on your result, supporting your statements with mathematical evidence. 6 The price of wheat over 4 years and 4 seasons is given in the table Year at right. Spring a Compute the seasonal indices. b De-seasonalise the time series. Summer c Plot the original and deseasonalised time series. Autumn d Comment on your result, supporting your statements with mathematical Winter evidence. 7 Sales (in thousands) of a new computer chip have been recorded for each quarter over years. a Compute the seasonal indices. b De-seasonalise the time series. c d Season Week Week 2 Week Week 4 Week Monday Tuesday Wednesday Thursday Friday Saturday Sunday Plot the original and de-seasonalised time series. Comment on your result, supporting your statements with mathematical evidence. Year The Gross National Product (GNP) is a measure of a nation s economy. A measure of a government s efficiency is the ratio of government spending to GNP. The seasonal data for years have Year Spring Summer been recorded for the island state Autumn of Maximinia. a Compute the seasonal indices. Winter b De-seasonalise the time series. c Plot the original and de-seasonalised time series. d Comment on your result, supporting your statements with mathematical evidence.

41 Chapter 4 Time series 8 summary Time series A time series is a set of measurements taken over (usually) equally spaced time intervals, such as hourly, daily, weekly, monthly or annually. Trend lines There are 4 basic types of trend:. secular: increasing or decreasing steadily 2. seasonal: varying from season to season. cyclic: similar to seasonal but not tied to a calendar cycle 4. random: variations caused by external triggers happening at random. Fitting trend lines The trend line is a straight line that can be used to represent the entire time series. Trend lines can be used for predicting the future values of the time series. The line can be found in several ways:. By eye: Try to fit a straight line using methods of Chapter. 2. The 2-mean method: (a) Divide the time series into lower and upper halves. (b) Compute the means of the t and y values for each half. These are called t L, y L, for the lower half, and t U, y U, for the upper half. y U y L (c) Compute the gradient using the formula: m = t U t L (d) Compute the y-intercept using the formula: b = y L m t L. (e) Write the equation of the straight line: y = mt + b.. The -median method: (a) Divide the data into three groups: lower, middle and upper. (b) Find the median of each of the three groups. y U y L (c) Calculate the gradient using the formula: m = t U t L (d) Calculate the y-intercept using the formula: b = -- [(y L + y M + y U ) m(t L + t M + t U )]. (e) Write the equation of the straight line: y = mt + b. Least-squares regression Least-squares regression techniques can be used to find a trend line. Centre the data so that the sum of all t values = 0. There are two methods, depending on whether there are even or odd numbers of points in the time series. Σty 2. The gradient can be calculated simply from the formula: m = Σt 2. The y-intercept can be calculated from: b = y.

42 86 Further Mathematics Smoothing time series Smoothing involves replacing the original time series with another one where most of the variation has been removed, in order to see if there is a secular trend. There are three basic smoothing techniques. In all cases, points are lost at the start and end of the time series. Refer to the text for detailed descriptions of the techniques involved. Moving average smoothing with an odd number of points Moving average smoothing works best with an odd number of points. For a -moving average, two points are lost; one point each at end of the time series. Moving average smoothing with an even number of points Moving average smoothing with an even number of points is a 2-step process. First you perform a 4-point moving average, then centre by averaging pairs of the 4-point averages. For a 4-point centred smooth, four points are lost; two points at each end of the time series. Median smoothing Median smoothing is usually done with an odd number of points. The number of points lost is the same as for moving average smoothing. De-seasonalisation De-seasonalising a time series involves replacing the original time series with another one where most or all of the seasonal variation is removed:. Average over all seasons for each year these are the yearly averages. 2. Divide each point in the original time series by its corresponding yearly average.. Using this new series, average over all years for each season these are the seasonal indices. 4. Divide each point in the original time series by its corresponding seasonal index.

43 Chapter 4 Time series 87 CHAPTER review Multiple choice The price of oranges over a 6-month period is recorded in the figure. The trend can be described as: A Cyclic B Seasonal C Random D Secular E There is no trend. 2 A 2-mean trend line was fitted to the data from question using the values below. The gradient of this line is: t Price of oranges Months 6 t 4A 4B Price A 4.9 B 0.20 C 0. D.0 E 0.0 From another 6-month time series for the price of apples, it was found that the 2-mean trend line was: price = 0.4 month A prediction for the price of apples in month 8 is: A 8.4 B 0.42 C 6.64 D.09 E Cannot be determined with the above information. 4 A least squares trend line has been fitted to the time series in the figure. Its equation is most likely to be: A y = 0t B y = 8t + 0 C y = 8t D y = 8t 0 E y = 8t + 0 y t The following data represent the number of employees in a car manufacturing plant. The data is smoothed using a -point moving average. Year B 4C 4D Number The first two points in the smoothed trend line are: A 20 and 00 B 20 and 0 C 2 and 0 D and 0 E 2 and 27 6 How many points would be obtained from the smoothed trend in question? A 8 B 7 C 6 D E 4 4D

44 88 Further Mathematics 4E 7 Consider the following data. Time y-value F 4G 4G 4G The value, after a 4-point smoothing after centring, plotted against the year 2002 is: A 6.2 B 4.2 C. D 7 E A -point median smooth is performed on the data in the figure at right. The last smoothed value is: A 2 B 2.7 C 20 D E 9 9 Seasonal indices and adjustment can be used when: A there are random variations in the data B there are seasonal variations along with a secular trend C there are seasonal variations only D there are seasonal or cyclic variations E there are at least 4 seasons worth of data. y The seasonal indices at right were obtained from a time series: The value of the winter s seasonal index is: Season Index Spring.2 Summer 0.78 Autumn 0.92 Winter A.8 B 0.94 C.08 D.06 E Cannot be determined from the given information. Using the data from question 0, a seasonally adjusted value for the summer of 2000, when the original value was 20, is closest to: A 406 B 666 C 464 D 64 E Cannot be determined without additional information. t 4A Short answer The number of uniforms sold in a school uniform shop is reported in the table. Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec Fit a trend line to these data. What type of trend is best reflected by these data? Can you explain these trends? 4B 2 Fit a 2-mean trend line to the following data, which represent the sales at a snack bar during the recent Melbourne show. State the gradient and y-intercept. Day Sales ($)

45 Chapter 4 Time series 89 Fit a -median trend line to the data at right. a State the gradient and y-intercept as exact values. b Use your line to predict a value when t =. 4 A hotel records the number of rooms booked over an day period. Fit a trend line using the least squares 0 method a State the gradient and y-intercept, rounded to 0 2 decimal places. b Predict the number of rooms booked for days 2 and. Day Rooms Perform a -point moving average on the following rainfall data. Plot the original and smoothed data on the same set of axes. Give all answers rounded to decimal place. Day Rain (mm) y t 4B 4C 4D 6 Fit a -point moving average to the following seasonal data of coat sales. Season Winter 997 Spring 997 Summer 997 Autumn 997 Winter 998 Spring 998 Summer 998 Autumn 998 Sales ($) Fit a 4-point centred moving average to the data from question 6. Compare your results. What do you notice about the number of smoothed data points in each case? 4D 4E 8 Perform a -point median smooth on the data shown at right. Plot the smoothed points joined with a straight line. 9 The seasonal indices for the price of shares in CSP fruit canneries are: Season Winter Spring Summer Autumn Index De-seasonalise the following data: Season (2000) Seasonalised Spring 0 Summer 00 Autumn 00 Winter 400 y 4F Share price De-seasonalised t 4G

46 90 Further Mathematics Analysis The next 8 questons relate to the following data which represent seasonal rainfall (mm) in an Australian city. Season Rainfall Plot the data points and try to fit a straight trend line by eye. Comment on the ease of making this plot. 2 Now, try to plot a trend line using the 2-mean method. Compare the result with that of question. Finally, plot a trend line using the least-squares technique. Again, compare your result with the previous ones. 4 To smooth out the seasonal variation, -point and -point moving average smoothings are tried. Compare the results of these two methods with the results from questions to by plotting the smoothed data. Upon observing the results with the -point smoothing, a trend appears. Take the data from the -point moving average and fit a straight line using least-squares. Put the first smoothed point at t = and then centre the time data. State the y-intercept and gradient. Compare this trend line with that from question. 6 Given the seasonal nature of the data, a 4-point moving average is tried. After calculating the 4-point moving average, fit a straight line using least-squares, following the method of question. Compare the results obtained with those from question. 7 Finally, try seasonal adjustment. Take t = to be summer and find the seasonal indices. Then, seasonally adjust the data. 8 Take the seasonally adjusted data from question 7 and fit a trend line using least-squares. Comment on this result. test yourself ourself CHAPTER 4