Determine the trend for time series data

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1 Extra Online Questions Determine the trend for time series data Covers AS (Statistics and Modelling 3.1) Scholarship Statistics and Modelling Chapter 1 Essent ial exam notes Time series 1. The value of the retail sales at NAILS (a large hardware store) was recorded each day between Monday 1 st January 2001 and Friday 30 th April You have been provided with some of the resulting time series data, which are shown in the tables and graphs that follow. Table 1 shows the monthly retail sales from July 2001 to October 2003 inclusive. Centred moving means have been calculated and are shown in the table. For these data, the least squares regression line fitting the centred moving means was obtained. The equation of the regression line is y = 0.91x , where x represents the number of months since June 2001 (so x = 1 corresponds with July 2001, etc) and y represents the values of sales (in thousands of dollars, $000). Table 2 shows the daily retail sales for the five weeks from Sunday 28 th September 2003 until Saturday 1 st November 2003 inclusive; moving means of order seven have been calculated. Graph 1 shows the value of the monthly retail sales from 1 st July 2001 to 31 st October Centred moving means of order 12 have also been plotted on the graph. Graph 2 shows the value of the daily retail sales for the five weeks from Sunday 28 th September 2003 until Saturday 1 st November 2003 inclusive. Moving means of order seven have also been plotted on the graph. Year 2004 Ans. p. 15 TABLE 1 Month Sales ($000) Centred Moving Mean Month Sales ($000) Centred Moving Mean Jul Sep Aug Oct Sep Nov Oct Dec Nov Jan Dec Feb Jan Mar Feb Apr Mar May Apr Jun May Jul Jun Aug Jul Sep Aug Oct

2 2 Scholarship Statistics and Modelling (Chapter 1) TABLE 2 Day Date Sales ($000) Moving Mean Day Date Sales ($000) Moving Mean Sun 28-Sep Thu 16-Oct Mon 29-Sep Fri 17-Oct Tue 30-Sep Sat 18-Oct Wed 1-Oct Sun 19-Oct Thu 2-Oct Mon 20-Oct Fri 3-Oct Tue 21-Oct Sat 4-Oct Wed 22-Oct Sun 5-Oct Thu 23-Oct Mon 6-Oct Fri 24-Oct Tue 7-Oct Sat 25-Oct Wed 8-Oct Sun 26-Oct Thu 9-Oct Mon 27-Oct Fri 10-Oct Tue 28-Oct Sat 11-Oct Wed 29-Oct Sun 12-Oct Thu 30-Oct Mon 13-Oct Fri 31-Oct Tue 14-Oct Sat 1-Nov Wed 15-Oct GRAPH NAILS Retail Sales Jul-01 Oct-01 Jan-02 Apr-02 Jul-02 Oct-02 Jan-03 Apr-03 Jul-03 Oct-03 Sales ($000) Centred Moving Mean

3 Determine the trend for time series data 3 GRAPH 2 25 NAILS Retail Sales Sep 1-Oct 4-Oct 7-Oct 10-Oct 13-Oct 16-Oct 19-Oct 22-Oct 25-Oct 28-Oct 31-Oct Sales ($000) Moving Mean a. To improve sales, the duty manager for each day is offered a bonus payment if the sales for that day exceed the expected value by at least 5%. i. What features of the time series should be considered in setting up this bonus payment scheme for managers? ii. Explain how the daily sales targets could be calculated.

4 4 Scholarship Statistics and Modelling (Chapter 1) b. Labour Day, the last Monday of October, results in a three-day weekend, which is traditionally used for home renovation and gardening. This produces high sales for hardware stores throughout New Zealand at this time. i. What effect does the high sales on Labour Day 2003 have on the (centred) moving mean for the daily retail sales? ii. How would you allow for the high sales figures for Labour Days in the calculation of a sales forecast for Mondays? c. Using the given information, forecast the sales for Tuesday 7 th December You must make clear the method you are using to make your forecast and justify your reasoning.

5 Determine the trend for time series data 5 d. Describe two limitations of the forecast you made in part c.

6 Year 2005 Ans. p Scholarship Statistics and Modelling (Chapter 1) 2. You are contracted as a statistical analyst to investigate sales patterns for an internet café over the previous three years, and to make a sales forecast for February You have been provided with the following data: The value of sales ($000) from the café for each month from November 2002 (t = 1) to October 2005 (t = 36) inclusive. The 12-point centred moving average (CMA) sales values ($000) for the months May 2003 (x = 1) to April 2005 (x = 24) inclusive. In addition you are provided with the following statistical output or information: A graph of the value of the monthly sales on which the CMA values have also been plotted. A table showing some summary statistics for the sales over each six-monthly period. A linear regression line fitted to the plotted CMA points has equation: y = x and R 2 = A quadratic regression curve fitted to the plotted CMA points has the equation: y = x x and R 2 = Write a report, no more than a page long (excluding calculations), to the owner of the internet café that summarises the output. Include in your report two calculations, one using the line and the other using the curve, to forecast sales for February Comment on the usefulness and limitations of your forecasts. 40 Internet Café Sales ($000) Months since Oct 2002 Sales ($000) CMA($000) Summary of Monthly Sales ($000) in each Six-monthly Period from Nov 2002 Summary Statistics Nov 02 to Apr 03 May 03 to Oct 03 Nov 03 to Apr 04 May 04 to Oct 04 Nov 04 to Apr 05 May 05 to Oct 05 Mean Median Standard Deviation

7 Determine the trend for time series data 7 t-value Month Year Sales ($000) x-value CMA($000) 1 Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct

8 8 Scholarship Statistics and Modelling (Chapter 1)

9 Determine the trend for time series data 9 3. A large swimming pool complex has both indoor and outdoor facilities. At some times of the year (for example during the April, July and October school holidays) there is an increase in the number of admissions. Ans. p Swimming Pool Admissions Apr-95 Jul-95 Oct-95 Jan-96 Apr-96 Jul-96 Oct-96 Jan-97 Apr-97 Jul-97 Oct-97 Jan-98 Apr-98 Jul-98 Oct-98 Jan-99 Apr-99 Jul-99 Number of admissions The number of admissions each month has been recorded over a four-year period. The data are shown in the spreadsheet at the end of this question. a. What is/are the order(s) of the seasonal effect(s) shown in those data? Explain. b. Which of the following moving averages of order 3 would show the seasonal effects more obviously a moving median or a moving mean? Support your answer with an explanation.

10 10 Scholarship Statistics and Modelling (Chapter 1) c. Use the data for December 1995, December 1996 and December 1997 to seasonally adjust the data for December d. Explain whether the number of admissions in December 1998 was higher or lower than expected. Month/Year Number of Admissions Jul Aug Sep Oct Nov Moving Mean (order 12) Seasonal Effect Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun

11 Determine the trend for time series data 11 Month/Year Number of Admissions Moving Mean (order 12) Seasonal Effect Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun

12 Ans. p Scholarship Statistics and Modelling (Chapter 1) 4. This graph shows total Australian electricity production in millions of kw-hours measured at quarterly intervals from March 1960 to December Mar Dec Sep Jun Mar Dec Sep Jun Mar 1982 a. Give a full description of the features of this time series. The first 24 items of data are shown in the spreadsheet below. The spreadsheet shows some of the data values and calculations that can be made from these. A B C D 1 Quarter Electricity use Moving mean 2 3 Mar Jun Sep Dec Mar Jun Sep Dec Mar Jun Sep Dec Mar Jun Sep Dec Mar Jun Sep Dec Mar Jun Sep Dec

13 Determine the trend for time series data 13 b. What is the order of the moving means shown in column C? c. Show that the best estimate, from the available data, of the seasonal adjustment for the June quarter is million kw-hours. d. The moving mean for the September 1980 quarter is million kw-hours. Use this information and the first smoothed value, to predict the electricity use for the June quarter for Include an explanation, with working, of the method you used.

14 14 Scholarship Statistics and Modelling (Chapter 1)

15 Answers 3.1 Time series (page 1) 1. a. i. The features of the time series include: The long-term trend of a gradual increase in sales over time. The seasonal effects at a monthly level, peak sales in December and low sales in July and August. The seasonal effects at a daily level, peak sales during the weekends and low sales mid-week. ii. Obtain a moving mean forecast for the month in question. Obtain the average seasonal effect for that month and adjust the forecast appropriately. Divide the month s forecast to get the daily forecast for each day in that month. Find the average seasonal effects for the different days of the week. Adjust the daily forecast by this average seasonal effect for the day under consideration. To calculate a sales target of 5% above the expected sales, multiply by b. i. The high sales on Labour Day 2003 has the effect of causing seven values of the centred moving mean to increase. This means that the centred moving mean values from the Friday before to the Thursday after Labour Day are larger than the rest of the data would suggest they should be. This would cause any analysis of this area of the time series to have an inflated daily value trend. ii. Labour Day could be treated as an outlier and as such ignored during the calculation of Mondays average seasonal effect. A particular Monday could have a moving mean value forecasted and then seasonally adjusted. If this particular Monday happened to be a Labour day, it could be adjusted further by using any historical data for the seasonal effects of a Labour day. c. December 2004 corresponds to 42 months after June 2001 hence, the raw forecast value for December 2004 is: y = = The average seasonal effect for December is needed: = Forecasted daily sales for December 2004: = [divide by 30 since no sales on Christmas day] The average seasonal effect for Tuesdays is needed: = Forecasted sales for Tuesday 7th December 2004 is: ( ) = $ [important to include 1 000] d. Any two of the following would be appropriate. Any statement made must be consistent with anything that has been said so far. The trend within the month of December was to gradually increase, no account has been taken into consideration for the predicted value being at the start of the month and hence possibly lower than if it were later in the month. The trendline was calculated on data ending in October It is therefore assumed that the slow increasing trend continues up to and including the month of December This is over 12 months after advertising and hence might not be the case.

16 16 Answers and explanations Both of the calculated average seasonal effects, especially December, have been found using relatively small numbers of values. December s average seasonal effect was found using only two values, this leaves a lot of room for error. The average seasonal effect for Tuesdays was calculated using an inflated value on Tuesday 28th October, due to the inclusion of Labour Day. 2. The following bullet points should form the basis of the report: The shape of the raw data should be commented on. Aside from a peak of $ in July 2004, the sales have fluctuated in value from a minimum of about $ to a maximum of about $ during the first two years to October The last year of data from October 2004 has dropped steadily. Comment on the summary statistics. Within each six-monthly interval, there appears to be little variation in the monthly sales as shown by the low standard deviation. The exception to this is seen in the period May 2004 to October 2005 where the standard deviation is almost double the others. The difference between the mean and median of each interval is minimal and it could be said the mean and medians were approximately equal. Comment on seasonal variation Some months were clearly higher than average as seen by their sales being higher than the moving average for that month. These were July, September, October and November. These months were favourable for sales. January, February, March, May and August had lower than average sales, as seen by the moving average values being higher than the raw data values. These months were not favourable to sales. The shape of the moving average data should be commented on. The moving average graph steadily increased to February 2004 (t = 16) where it reached a peak. The graph steadily decreased from February 2004 to the end of the data. Comment on the fit of the line and the fit of the curve to the moving average data. The line does not fit the data very well, but it seems there is a weak relationship as shown by the coefficient of determination R 2 = The curve fits the data well and a strong relationship is indicated by the coefficient of determination R 2 = Calculation of the seasonal effect for February: ( ) +( ) Seasonal effect = 2 = Calculation of the forecast for February 2006 can be done in two different ways. Using the line: x = 34 y = = Add the seasonal effect to get the prediction: = The line gives a forecast sales value of $

17 Answers and explanations 17 Using the curve: x = 34 y = = Add the seasonal effect to get the prediction: = The curve gives a forecast sales value of $ Comment on the usefulness and limitations of the forecasts made. The fact that the curve forecasts a negative sales figure (which is not possible), indicates that the curve no longer fits the data in February As the line is used to predict a value 10 months in the future, its negative gradient most likely no longer holds. This gives limited use to this forecast. The average seasonal effect for the month of February is calculated using data from only two Februaries. It is most likely that this is not a good representation of what the seasonal effect is in actuality. More data is needed to give a better idea of the seasonal effect of February. 3. a. There are two seasonal effects that can be seen in the graph. There is one that occurs every summer which has order 12 and the other one occurs at every school holidays and has order 3. b. The mean of a set of data is more affected by unusually large or small values than the median. Consequently it would be the moving mean that would show the seasonal effects more obviously. c. The average seasonal effect for the three given Decembers is: = Using this, the seasonally adjusted value for December 1998 is: = d. Since the seasonally adjusted value for December 1998 (4 588) is higher than the moving average value (4 493), it can be said that the value for December 1998 was higher than expected. 4. a. There is an almost linear increasing trend. The seasonal variation is at regular intervals with the highest value of each year being during the September quarter and the lowest being during the March quarter. b. The moving means in the table are order 4. c. Find the individual seasonal effects for June and average them. June 61: = June 62: = 682 June 63: = June 64: = June 65: = 523 The average seasonal effect for June is: =

18 18 Answers and explanations d. For this question it is assumed the moving mean data is increasing with a linear pattern. Using the two points and the fact that the first smoothed data value is the 3rd and September 1980 is the 83rd to find the equation of the line: y = 83 3 y = x For the purpose of this investigation and given the accuracy of the given data, the following is more than accurate enough: y = 293x The June quarter in 1995 is the 142nd data point hence its moving mean value is: y = = Then add in the seasonal adjustment to find the predicted usage for the June quarter: = So the predicted usage of power for the June quarter 1995 is million kw-hours.

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