1. Classical Decomposition Boise State University By: Kurt Folke Spring 2003

2. Overview: Time series models & classical decompositionTime series models & classical decomposition Brainstorming exerciseBrainstorming exercise Classical decomposition explainedClassical decomposition explained Classical decomposition illustrationClassical decomposition illustration ExerciseExercise SummarySummary Bibliography & readings listBibliography & readings list Appendix A: exercise templatesAppendix A: exercise templates

3. Time Series Models & Classical Decomposition Time series models are sequences of data that follow non-random ordersTime series models are sequences of data that follow non-random orders Examples of time series data:Examples of time series data: Sales Sales Costs Costs Time series models are composed of trend, seasonal, cyclical, and random influencesTime series models are composed of trend, seasonal, cyclical, and random influences

4. Time Series Models & Classical Decomposition Decomposition time series models:Decomposition time series models: Multiplicative:Y = T x C x S x eMultiplicative:Y = T x C x S x e Additive:Y = T + C + S + eAdditive:Y = T + C + S + e T = Trend componentT = Trend component C = Cyclical componentC = Cyclical component S = Seasonal componentS = Seasonal component e = Error or random componente = Error or random component

5. Time Series Models & Classical Decomposition Classical decomposition is used to isolate trend, seasonal, and other variability components from a time series modelClassical decomposition is used to isolate trend, seasonal, and other variability components from a time series model Benefits:Benefits: Shows fluctuations in trend Shows fluctuations in trend Provides insight to underlying factors affecting the time series Provides insight to underlying factors affecting the time series

6. Brainstorming Exercise Identify how this tool can be used in your organization…Identify how this tool can be used in your organization…

7. Classical Decomposition Explained Basic Steps: 1.Determine seasonal indexes using the ratio to moving average method 2.Deseasonalize the data 3.Develop the trend-cyclical regression equation using deseasonalized data 4.Multiply the forecasted trend values by their seasonal indexes to create a more accurate forecast

8. Classical Decomposition Explained: Step 1 Determine seasonal indexesDetermine seasonal indexes Start with multiplicative model…Start with multiplicative model… Y = TCSe Equate…Equate… Se = (Y/TC)

9. Classical Decomposition Explained: Step 1 To find seasonal indexes, first estimate trend-cyclical componentsTo find seasonal indexes, first estimate trend-cyclical components Se = (Y/TC) Use centered moving averageUse centered moving average Called ratio to moving average method Called ratio to moving average method For quarterly data, use four-quarter moving averageFor quarterly data, use four-quarter moving average Averages seasonal influences Averages seasonal influences Example

10. Classical Decomposition Explained: Step 1 Four-quarter moving average will position average at…Four-quarter moving average will position average at… end of second period and end of second period and beginning of third period beginning of third period Use centered moving average to position data in middle of the periodUse centered moving average to position data in middle of the period Example

11. Classical Decomposition Explained: Step 1 Find seasonal-error components by dividing original data by trend- cyclical componentsFind seasonal-error components by dividing original data by trend- cyclical components Se = (Y/TC) Se = Seasonal-error componentsSe = Seasonal-error components Y = Original data valueY = Original data value TC = Trend-cyclical componentsTC = Trend-cyclical components (centered moving average value) Example

12. Classical Decomposition Explained: Step 1 Unadjusted seasonal indexes (USI) are found by averaging seasonal- error components by periodUnadjusted seasonal indexes (USI) are found by averaging seasonal- error components by period Develop adjusting factor (AF) so USIs are adjusted so their sum equals the number of quarters (4)Develop adjusting factor (AF) so USIs are adjusted so their sum equals the number of quarters (4) Reduces error Reduces error Example

13. Classical Decomposition Explained: Step 1 Adjusted seasonal indexes (ASI) are derived by multiplying the unadjusted seasonal index by the adjusting factorAdjusted seasonal indexes (ASI) are derived by multiplying the unadjusted seasonal index by the adjusting factor ASI = USI x AF ASI = Adjusted seasonal indexASI = Adjusted seasonal index USI = Unadjusted seasonal indexUSI = Unadjusted seasonal index AF = Adjusting factorAF = Adjusting factor Example

14. Classical Decomposition Explained: Step 2 Deseasonalized data is produced by dividing the original data values by their seasonal indexesDeseasonalized data is produced by dividing the original data values by their seasonal indexes (Y/S) = TCe Y/S = Deseasonalized dataY/S = Deseasonalized data TCe = Trend-cyclical-error componentTCe = Trend-cyclical-error component Example

15. Classical Decomposition Explained: Step 3 Develop the trend-cyclical regression equation using deseasonalized dataDevelop the trend-cyclical regression equation using deseasonalized data T t = a + bt T t = a + bt T t = Trend value at period tT t = Trend value at period t a = Intercept valuea = Intercept value b = Slope of trend lineb = Slope of trend line Example

16. Classical Decomposition Explained: Step 4 Use trend-cyclical regression equation to develop trend dataUse trend-cyclical regression equation to develop trend data Create forecasted data by multiplying the trend data values by their seasonal indexesCreate forecasted data by multiplying the trend data values by their seasonal indexes More accurate forecast More accurate forecast Example

17. Classical Decomposition Explained: Step Summary Summarized Steps: 1.Determine seasonal indexes 2.Deseasonalize the data 3.Develop the trend-cyclical regression equation 4.Create forecast using trend data and seasonal indexes

18. Classical Decomposition: Illustration Gem Company’s operations department has been asked to deseasonalize and forecast sales for the next four quarters of the coming yearGem Company’s operations department has been asked to deseasonalize and forecast sales for the next four quarters of the coming year The Company has compiled its past sales data in Table 1The Company has compiled its past sales data in Table 1 An illustration using classical decomposition will followAn illustration using classical decomposition will follow

19. Classical Decomposition Illustration: Step 1 (a) Compute the four-quarter simple moving average(a) Compute the four-quarter simple moving average Ex: simple MA at end of Qtr 2 and beginning of Qtr 3 (55+47+65+70)/4 = 59.25 Explain

20. Classical Decomposition Illustration: Step 1 (b) Compute the two-quarter centered moving average(b) Compute the two-quarter centered moving average Ex: centered MA at middle of Qtr 3 (59.25+61.25)/2 = 60.500 Explain

21. Classical Decomposition Illustration: Step 1 (c) Compute the seasonal-error component (percent MA)(c) Compute the seasonal-error component (percent MA) Ex: percent MA at Qtr 3 (65/60.500) = 1.074 Explain

22. Classical Decomposition Illustration: Step 1 (d) Compute the unadjusted seasonal index using the seasonal-error components from Table 2(d) Compute the unadjusted seasonal index using the seasonal-error components from Table 2 Ex (Qtr 1): [(Yr 2, Qtr 1) + (Yr 3, Qtr 1) + (Yr 4, Qtr 1)]/3 = [0.989+0.914+0.926]/3 = 0.943 Explain

23. Classical Decomposition Illustration: Step 1 (e) Compute the adjusting factor by dividing the number of quarters (4) by the sum of all calculated unadjusted seasonal indexes(e) Compute the adjusting factor by dividing the number of quarters (4) by the sum of all calculated unadjusted seasonal indexes = 4.000/(0.943+0.851+1.080+1.130) = (4.000/4.004) Explain

24. Classical Decomposition Illustration: Step 1 (f) Compute the adjusted seasonal index by multiplying the unadjusted seasonal index by the adjusting factor(f) Compute the adjusted seasonal index by multiplying the unadjusted seasonal index by the adjusting factor Ex (Qtr 1): 0.943 x (4.000/4.004) = 0.942 Explain

25. Classical Decomposition Illustration: Step 2 Compute the deseasonalized sales by dividing original sales by the adjusted seasonal indexCompute the deseasonalized sales by dividing original sales by the adjusted seasonal index Ex (Yr 1, Qtr 1): (55 / 0.942) = 58.386 Explain

26. Classical Decomposition Illustration: Step 3 Compute the trend- cyclical regression equation using simple linear regressionCompute the trend- cyclical regression equation using simple linear regression T t = a + bt t-bar = 8.5 T-bar = 69.6 b = 1.465 a = 57.180 T t = 57.180 + 1.465t Explain

27. Classical Decomposition Illustration: Step 4 (a) Develop trend sales(a) Develop trend sales T t = 57.180 + 1.465t Ex (Yr 1, Qtr 1): T 1 = 57.180 + 1.465(1) = 58.645 Explain

28. Classical Decomposition Illustration: Step 4 (b) Forecast sales for each of the four quarters of the coming year(b) Forecast sales for each of the four quarters of the coming year Ex (Yr 5, Qtr 1): 0.942 x 82.085 = 77.324 Explain

29. Classical Decomposition Illustration: Graphical Look

30. Classical Decomposition: Exercise 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 salesAssume 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 Use the steps and examples shown in the explanation and illustration as a referenceUse the steps and examples shown in the explanation and illustration as a reference Basic Steps Basic Steps Explanation Illustration Templates

31. Summary Time series models are sequences of data that follow non-arbitrary ordersTime series models are sequences of data that follow non-arbitrary orders Classical decomposition isolates the components of a time series modelClassical decomposition isolates the components of a time series model Benefits:Benefits: Insight to fluctuations in trend Insight to fluctuations in trend Decomposes the underlying factors affecting the time series Decomposes the underlying factors affecting the time series

32. Bibliography & Readings List DeLurgio, Stephen, and Bhame, Carl. Forecasting Systems for Operations Management. Homewood: Business One Irwin, 1991. Shim, Jae K. Strategic Business Forecasting. New York: St Lucie, 2000. StatSoft Inc. (2003). Time Series Analysis. Retrieved April 21, 2003, from http://www.statsoft.com/textbook/sttimser.html

33. Appendix A: Exercise Templates

38. Appendix B: Exercise Solutions

41. Trend-cyclical Regression Equation T t = 5.402 + 0.514t

42. Appendix B: Exercise Solutions