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# Classical decomposition

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• The following presentation is meant to familiarize individuals with classical decomposition. It does not contain an entirely comprehensive study of this statistical tool; however, it should make individuals aware of the benefits that classical decomposition can provide.   Individuals who will benefit the most from this learning tool will have a basic background in introductory business statistics and knowledge of simple linear regression.   Sources that are used throughout this presentation are cited in the notes below each slide and in the bibliography and readings list.   Please contact the creator of this presentation with any questions or comments: Kurt Folke E-mail: kfolke75@hotmail.com Boise State University College of Business &amp; Economics
• Also included in this presentation are solutions to the exercise. These solutions are available in hidden slide form in Appendix B.
• Definitions taken from … StatSoft Inc. (2003). Time Series Analysis. Retrieved April 21, 2003, from http://www.statsoft.com/textbook/sttimser.html Time series models are functions that relate time to previous values of the model. Accordingly, these models presume what has happened in the past will reoccur in the future.   Further examples of possible time series data include earnings, market share, and cash flows.
• The multiplicative model Y = TCSe is the product of the trend, cyclical, seasonal, and error (or random) components at some time t . Likewise, the additive model Y = T+C+S+e is the sum of trend, cyclical, seasonal, and error (or random) components at some time t. The trend component is the gradual upward or downward movement found in a time series as a result of many possible factors (such as demand). Cyclical influences are recurrent up or down movements that last for long periods of time (longer than a year). Examples of events that might trigger a cyclical influence include recessions or booms. The seasonal component is an upward and downward movement that is repeated periodically as a result of holidays, seasons, etc. These seasonal influences may be observed as being weekly, monthly, quarterly, yearly, or some other periodic term. The error or random component of a time series is the usually small, erratic movement that does not follow a pattern and can be the result of the weather, strikes, and other unpredictable events.
• Definitions taken from … Shim, Jae K. Strategic Business Forecasting. New York: St Lucie, 2000. 269. Classical decomposition is a powerful tool for decomposing the elements of a time series model and studying each component’s sub-patterns. By analyzing each component, management can make educated decisions concerning trend and demand for future periods.
• A possible example is … classical decomposition can help a company learn when it is experiencing an abnormally high/low demand for its products.
• Steps summarized from … DeLurgio, Stephen, and Bhame, Carl. Forecasting Systems for Operations Management . Homewood: Business One Irwin, 1991. 297-298.   The basic steps outlined above encompass the major tasks in classical decomposition. Many sub-steps of these general tasks are shown in the explanation and illustration that follows.
• Although both multiplicative and additive time series models can be used in classical decomposition, this presentation will only include the multiplicative model as it is most commonly used. Methods for choosing between using the additive or multiplicative model can be found in … DeLurgio, Stephen, and Bhame, Carl. Forecasting Systems for Operations Management . Homewood: Business One Irwin, 1991. 289-290.
• The classical decomposition demonstrated here models trend and cyclical effects together for simplicity and lack of a simple modeling technique for cyclical influences. Note: This assumption is appropriate for short-term forecasts, but forecasts for periods longer than one year should include an adjustment for cyclical influences.   Just as the four-quarter moving average is used when dealing with quarterly data, likewise the 12-month moving average should be used when working with monthly data. Regardless of the periodic term used in the moving average, the outcome is the same: the seasonal influences are averaged, and therefore are neither seasonally high, nor seasonally low. Note: The hyperlinks provided in these slides will navigate the operator between the classical decomposition explanation and the classical decomposition illustration . This provides the learner with a conceptual explanation followed by actual application of the process.
• Since the simple moving average is centered at the end of one period and the beginning of the next, computing the centered moving average is necessary to ensure that the average is centered at the middle of the period. Through this process, the centered moving average is created to contain no seasonality, and therefore is the trend-cyclical component of the model.
• Using the identity Se = (Y/TC) , the seasonal-error component is derived by dividing the original data ( Y ) by the trend-cyclical data ( TC ). The trend-cyclical data is the centered moving average that was developed earlier.
• By taking the seasonal-error components and averaging them across the available periods, the unadjusted seasonal index is computed. This computation is demonstrated on slide 22. The adjusting factor is created by dividing the number of periods per year (four since the data is quarterly) by the sum of the unadjusted seasonal indexes. This ensures that the average seasonal index is one since all of the seasonal indexes must equal the number of periods in the year. If this were not done, error would be introduced into the final forecast.
• The adjusted seasonal index is the product of the unadjusted seasonal index and the adjusting factor. A quarterly adjusted seasonal index of 0.942 suggests the data was 94.2 percent of the typical trend-cyclical value during the quarter. Accordingly, adjusted seasonal indexes greater than one indicate that the data was higher than the typical trend-cyclical values for that quarter.
• To create the deseasonalized data, the original data values ( Y ) must be divided by their appropriate seasonal indexes ( S ). Once the deseasonalized data is computed, it can be analyzed to identify true fluctuations in the time series. These fluctuations can help management in strategic planning.
• By using simple linear regression, the trend of the time series can be estimated. This process is done by using the deseasonalized values to create a trend-cyclical regression equation of the form Tt = a + bt…where t is equal to the period. Hence, t = 1 refers to year 1, quarter 1. Although actual simple linear regression computations are not shown, recommendations are to use Excel or Minitab when creating the trend-cyclical regression equations from the deseasonalized data.
• Using the trend-cyclical regression equation, the trend data can be created by imputing each period’s assigned number into the equation. Note: At this point, it would be suitable to check the model to see how closely it fits the data. This process is omitted from this demonstration; however, in practice it should be performed before continuing. A demonstration of this is presented in … DeLurgio, Stephen, and Bhame, Carl. Forecasting Systems for Operations Management . Homewood: Business One Irwin, 1991. 296-297. The final forecast is developed by multiplying the trend values by their appropriate seasonal indexes. This produces a more accurate forecast for management. Note: As previously mentioned, this method is appropriate for short-term forecasts, but forecasts for periods longer than one year should include an adjustment for cyclical influences.
• Steps summarized from … DeLurgio, Stephen, and Bhame, Carl. Forecasting Systems for Operations Management . Homewood: Business One Irwin, 1991. 297-298.
• Use the hyperlink to navigate to the explanation slide that includes conceptual details in the notes.
• Use the hyperlink to navigate to the explanation slide that includes conceptual details in the notes.
• Use the hyperlink to navigate to the explanation slide that includes conceptual details in the notes.
• Use the hyperlink to navigate to the explanation slide that includes conceptual details in the notes.
• Use the hyperlink to navigate to the explanation slide that includes conceptual details in the notes.
• Use the hyperlink to navigate to the explanation slide that includes conceptual details in the notes.
• Use the hyperlink to navigate to the explanation slide that includes conceptual details in the notes.
• Use the hyperlink to navigate to the explanation slide that includes conceptual details in the notes.
• Use the hyperlink to navigate to the explanation slide that includes conceptual details in the notes.
• Use the hyperlink to navigate to the explanation slide that includes conceptual details in the notes.
• Graphing the trend, original, and deseasonalized data can be very helpful for identifying fluctuations in trend. Deviations from the norm can be invaluable knowledge for management to analyze and use for planning future capacity, production, and allocations of resources.
• This example provides individuals with the opportunity to apply the new skills they have learned through this presentation. It is highly recommended that a spreadsheet program such as Excel or Minitab be used for computations and for building the trend-cyclical regression equation. In Excel, simple linear regression can be performed by going to Tools , Data Analysis , and using the Regression tool. Preformatted Excel templates have been created for this exercise and are available in Appendix A. Solutions for all steps are presented in hidden slides in Appendix B.
• Definitions taken from … StatSoft Inc. (2003). Time Series Analysis. Retrieved April 21, 2003, from http://www.statsoft.com/textbook/sttimser.html Time series models are based on the assumption that what has happened in the past will reoccur in the future. Classical decomposition can be used to segregate the elements of a time series model; after studying each component’s sub-patterns, management can apply the new learned knowledge when making decisions regarding strategic planning.
• The sources provided in the bibliography and readings list are highly recommended to individuals wishing to expand their knowledge in classical decomposition and similar statistical tools.
• Directions for use: Double-click on the desired table Highlight the cells of the table Select “copy” from the right-click pop-up menu or the Edit pull-down menu Open a spreadsheet program Paste the table into the spreadsheet program
• Directions for use: Double-click on the desired table Highlight the cells of the table Select “copy” from the right-click pop-up menu or the Edit pull-down menu Open a spreadsheet program Paste the table into the spreadsheet program
• Directions for use: Double-click on the desired table Highlight the cells of the table Select “copy” from the right-click pop-up menu or the Edit pull-down menu Open a spreadsheet program Paste the table into the spreadsheet program
• Directions for use: Double-click on the desired table Highlight the cells of the table Select “copy” from the right-click pop-up menu or the Edit pull-down menu Open a spreadsheet program Paste the table into the spreadsheet program
• Directions for use: Double-click on the desired table Highlight the cells of the table Select “copy” from the right-click pop-up menu or the Edit pull-down menu Open a spreadsheet program Paste the table into the spreadsheet program
• To show these slides in the presentation: Select the Normal View tab In the left-hand screen, select slide From the Slide Show pull down menu, press Hide Slide
• To show these slides in the presentation: Select the Normal View tab In the left-hand screen, select slide From the Slide Show pull down menu, press Hide Slide
• To show these slides in the presentation: Select the Normal View tab In the left-hand screen, select slide From the Slide Show pull down menu, press Hide Slide
• To show these slides in the presentation: Select the Normal View tab In the left-hand screen, select slide From the Slide Show pull down menu, press Hide Slide
• To show these slides in the presentation: Select the Normal View tab In the left-hand screen, select slide From the Slide Show pull down menu, press Hide Slide
• ### Classical decomposition

1. 1. Classical Decomposition Boise State University By: Kurt Folke Spring 2003
2. 2. Overview: <ul><li>Time series models & classical decomposition </li></ul><ul><li>Brainstorming exercise </li></ul><ul><li>Classical decomposition explained </li></ul><ul><li>Classical decomposition illustration </li></ul><ul><li>Exercise </li></ul><ul><li>Summary </li></ul><ul><li>Bibliography & readings list </li></ul><ul><li>Appendix A: exercise templates </li></ul>
3. 3. Time Series Models & Classical Decomposition <ul><li>Time series models are sequences of data that follow non-random orders </li></ul><ul><li>Examples of time series data: </li></ul><ul><ul><li>Sales </li></ul></ul><ul><ul><li>Costs </li></ul></ul><ul><li>Time series models are composed of trend, seasonal, cyclical, and random influences </li></ul>
4. 4. Time Series Models & Classical Decomposition <ul><li>Decomposition time series models: </li></ul><ul><li>Multiplicative: Y = T x C x S x e </li></ul><ul><li>Additive: Y = T + C + S + e </li></ul><ul><li>T = Trend component </li></ul><ul><li>C = Cyclical component </li></ul><ul><li>S = Seasonal component </li></ul><ul><li>e = Error or random component </li></ul>
5. 5. Time Series Models & Classical Decomposition <ul><li>Classical decomposition is used to isolate trend, seasonal, and other variability components from a time series model </li></ul><ul><li>Benefits: </li></ul><ul><ul><li>Shows fluctuations in trend </li></ul></ul><ul><ul><li>Provides insight to underlying factors affecting the time series </li></ul></ul>
6. 6. Brainstorming Exercise <ul><li>Identify how this tool can be used in your organization… </li></ul>
7. 7. Classical Decomposition Explained <ul><li>Basic Steps: </li></ul><ul><li>Determine seasonal indexes using the ratio to moving average method </li></ul><ul><li>Deseasonalize the data </li></ul><ul><li>Develop the trend-cyclical regression equation using deseasonalized data </li></ul><ul><li>Multiply the forecasted trend values by their seasonal indexes to create a more accurate forecast </li></ul>
8. 8. Classical Decomposition Explained: Step 1 <ul><li>Determine seasonal indexes </li></ul><ul><li>Start with multiplicative model… </li></ul><ul><li>Y = TCSe </li></ul><ul><li>Equate… </li></ul><ul><li>Se = (Y/TC) </li></ul>
9. 9. Classical Decomposition Explained: Step 1 <ul><li>To find seasonal indexes, first estimate trend-cyclical components </li></ul><ul><li>Se = (Y/ TC ) </li></ul><ul><li>Use centered moving average </li></ul><ul><ul><li>Called ratio to moving average method </li></ul></ul><ul><li>For quarterly data, use four-quarter moving average </li></ul><ul><ul><li>Averages seasonal influences </li></ul></ul>Example
10. 10. Classical Decomposition Explained: Step 1 <ul><li>Four-quarter moving average will position average at… </li></ul><ul><ul><ul><li>end of second period and </li></ul></ul></ul><ul><ul><ul><li>beginning of third period </li></ul></ul></ul><ul><li>Use centered moving average to position data in middle of the period </li></ul>Example
11. 11. Classical Decomposition Explained: Step 1 <ul><li>Find seasonal-error components by dividing original data by trend-cyclical components </li></ul><ul><li>Se = ( Y/TC ) </li></ul><ul><li>Se = Seasonal-error components </li></ul><ul><li>Y = Original data value </li></ul><ul><li>TC = Trend-cyclical components </li></ul><ul><li>(centered moving average value) </li></ul>Example
12. 12. Classical Decomposition Explained: Step 1 <ul><li>Unadjusted seasonal indexes (USI) are found by averaging seasonal-error components by period </li></ul><ul><li>Develop adjusting factor (AF) so USIs are adjusted so their sum equals the number of quarters (4) </li></ul><ul><ul><li>Reduces error </li></ul></ul>Example Example
13. 13. Classical Decomposition Explained: Step 1 <ul><li>Adjusted seasonal indexes (ASI) are derived by multiplying the unadjusted seasonal index by the adjusting factor </li></ul><ul><li>ASI = USI x AF </li></ul><ul><li>ASI = Adjusted seasonal index </li></ul><ul><li>USI = Unadjusted seasonal index </li></ul><ul><li>AF = Adjusting factor </li></ul>Example
14. 14. Classical Decomposition Explained: Step 2 <ul><li>Deseasonalized data is produced by dividing the original data values by their seasonal indexes </li></ul><ul><li>( Y/S ) = TCe </li></ul><ul><li>Y/S = Deseasonalized data </li></ul><ul><li>TCe = Trend-cyclical-error component </li></ul>Example
15. 15. Classical Decomposition Explained: Step 3 <ul><li>Develop the trend-cyclical regression equation using deseasonalized data </li></ul><ul><li> T t = a + bt </li></ul><ul><li>T t = Trend value at period t </li></ul><ul><li>a = Intercept value </li></ul><ul><li>b = Slope of trend line </li></ul>Example
16. 16. Classical Decomposition Explained: Step 4 <ul><li>Use trend-cyclical regression equation to develop trend data </li></ul><ul><li>Create forecasted data by multiplying the trend data values by their seasonal indexes </li></ul><ul><ul><li>More accurate forecast </li></ul></ul>Example Example
17. 17. Classical Decomposition Explained: Step Summary <ul><li>Summarized Steps: </li></ul><ul><li>Determine seasonal indexes </li></ul><ul><li>Deseasonalize the data </li></ul><ul><li>Develop the trend-cyclical regression equation </li></ul><ul><li>Create forecast using trend data and seasonal indexes </li></ul>
18. 18. Classical Decomposition: Illustration <ul><li>Gem Company’s operations department has been asked to deseasonalize and forecast sales for the next four quarters of the coming year </li></ul><ul><li>The Company has compiled its past sales data in Table 1 </li></ul><ul><li>An illustration using classical decomposition will follow </li></ul>
19. 19. Classical Decomposition Illustration: Step 1 <ul><li>(a) Compute the four-quarter simple moving average </li></ul><ul><li>Ex: simple MA at end of Qtr 2 and beginning of Qtr 3 </li></ul><ul><li>(55+47+65+70)/4 = 59.25 </li></ul>Explain
20. 20. Classical Decomposition Illustration: Step 1 <ul><li>(b) Compute the two-quarter centered moving average </li></ul><ul><li>Ex: centered MA at middle of Qtr 3 </li></ul><ul><li>(59.25+61.25)/2 </li></ul><ul><li>= 60.500 </li></ul>Explain
21. 21. Classical Decomposition Illustration: Step 1 <ul><li>(c) Compute the seasonal-error component (percent MA) </li></ul><ul><li>Ex: percent MA at Qtr 3 </li></ul><ul><li>(65/60.500) </li></ul><ul><li>= 1.074 </li></ul>Explain
22. 22. Classical Decomposition Illustration: Step 1 <ul><li>(d) Compute the unadjusted seasonal index using the seasonal-error components from Table 2 </li></ul><ul><li>Ex (Qtr 1): [(Yr 2, Qtr 1) + (Yr 3, Qtr 1) + (Yr 4, Qtr 1)]/3 </li></ul><ul><li>= [0.989+0.914+0.926]/3 = 0.943 </li></ul>Explain
23. 23. Classical Decomposition Illustration: Step 1 <ul><li>(e) Compute the adjusting factor by dividing the number of quarters (4) by the sum of all calculated unadjusted seasonal indexes </li></ul><ul><li>= 4.000/(0.943+0.851+1.080+1.130) = (4.000/4.004) </li></ul>Explain
24. 24. Classical Decomposition Illustration: Step 1 <ul><li>(f) Compute the adjusted seasonal index by multiplying the unadjusted seasonal index by the adjusting factor </li></ul><ul><li>Ex (Qtr 1): 0.943 x (4.000/4.004) = 0.942 </li></ul>Explain
25. 25. Classical Decomposition Illustration: Step 2 <ul><li>Compute the deseasonalized sales by dividing original sales by the adjusted seasonal index </li></ul><ul><li>Ex (Yr 1, Qtr 1): </li></ul><ul><li>(55 / 0.942) </li></ul><ul><li>= 58.386 </li></ul>Explain
26. 26. Classical Decomposition Illustration: Step 3 <ul><li>Compute the trend-cyclical regression equation using simple linear regression </li></ul><ul><li>T t = a + bt </li></ul><ul><li>t-bar = 8.5 </li></ul><ul><li>T-bar = 69.6 </li></ul><ul><li>b = 1.465 </li></ul><ul><li>a = 57.180 </li></ul><ul><li>T t = 57.180 + 1.465t </li></ul>Explain
27. 27. Classical Decomposition Illustration: Step 4 <ul><li>(a) Develop trend sales </li></ul><ul><li>T t = 57.180 + 1.465t </li></ul><ul><li>Ex (Yr 1, Qtr 1): </li></ul><ul><li>T 1 = 57.180 + 1.465(1) = 58.645 </li></ul>Explain
28. 28. Classical Decomposition Illustration: Step 4 <ul><li>(b) Forecast sales for each of the four quarters of the coming year </li></ul><ul><li>Ex (Yr 5, Qtr 1): </li></ul><ul><li>0.942 x 82.085 </li></ul><ul><li>= 77.324 </li></ul>Explain
29. 29. Classical Decomposition Illustration: Graphical Look
30. 30. Classical Decomposition: Exercise <ul><li>Assume you have been asked by your boss to deseasonalize and forecast for the next four quarters of the coming year (Yr 5) this data pertaining to your company’s sales </li></ul><ul><li>Use the steps and examples shown in the explanation and illustration as a reference </li></ul><ul><li>Basic Steps </li></ul><ul><li>Explanation </li></ul><ul><li>Illustration </li></ul><ul><li>Templates </li></ul>
31. 31. Summary <ul><li>Time series models are sequences of data that follow non-arbitrary orders </li></ul><ul><li>Classical decomposition isolates the components of a time series model </li></ul><ul><li>Benefits: </li></ul><ul><ul><li>Insight to fluctuations in trend </li></ul></ul><ul><ul><li>Decomposes the underlying factors affecting the time series </li></ul></ul>
32. 32. Bibliography & Readings List <ul><li>DeLurgio, Stephen, and Bhame, Carl. Forecasting Systems for Operations Management . Homewood: Business One Irwin, 1991. </li></ul><ul><li>Shim, Jae K. Strategic Business Forecasting . New York: St Lucie, 2000. </li></ul><ul><li>StatSoft Inc. (2003). Time Series Analysis. Retrieved April 21, 2003, from http://www.statsoft.com/textbook/sttimser.html </li></ul>
33. 33. Appendix A: Exercise Templates
34. 34. Appendix A: Exercise Templates
35. 35. Appendix A: Exercise Templates
36. 36. Appendix A: Exercise Templates
37. 37. Appendix A: Exercise Templates
38. 38. Appendix B: Exercise Solutions
39. 39. Appendix B: Exercise Solutions
40. 40. Appendix B: Exercise Solutions
41. 41. Appendix B: Exercise Solutions Trend-cyclical Regression Equation T t = 5.402 + 0.514t
42. 42. Appendix B: Exercise Solutions