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Classical Decomposition. Boise State University By: Kurt Folke Spring 2003. Overview:. Time series models & classical decomposition Brainstorming exercise Classical decomposition explained Classical decomposition illustration Exercise Summary Bibliography & readings list

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Classical decomposition l.jpg

Classical Decomposition

Boise State University

By: Kurt Folke

Spring 2003


Overview l.jpg
Overview:

  • Time series models & classical decomposition

  • Brainstorming exercise

  • Classical decomposition explained

  • Classical decomposition illustration

  • Exercise

  • Summary

  • Bibliography & readings list

  • Appendix A: exercise templates


Time series models classical decomposition l.jpg
Time Series Models & Classical Decomposition

  • Time series models are sequences of data that follow non-random orders

  • Examples of time series data:

    • Sales

    • Costs

  • Time series models are composed of trend, seasonal, cyclical, and random influences


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Time Series Models & Classical Decomposition

  • Decomposition time series models:

  • Multiplicative: Y = T x C x S x e

  • Additive: Y = T + C + S + e

  • T = Trend component

  • C = Cyclical component

  • S = Seasonal component

  • e = Error or random component


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Time Series Models & Classical Decomposition

  • Classical decomposition is used to isolate trend, seasonal, and other variability components from a time series model

  • Benefits:

    • Shows fluctuations in trend

    • Provides insight to underlying factors affecting the time series


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Brainstorming Exercise

  • Identify how this tool can be used in your organization…


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Classical Decomposition Explained

Basic Steps:

  • Determine seasonal indexes using the ratio to moving average method

  • Deseasonalize the data

  • Develop the trend-cyclical regression equation using deseasonalized data

  • Multiply the forecasted trend values by their seasonal indexes to create a more accurate forecast


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Classical Decomposition Explained: Step 1

  • Determine seasonal indexes

  • Start with multiplicative model…

    Y = TCSe

  • Equate…

    Se = (Y/TC)


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Classical Decomposition Explained: Step 1

  • To find seasonal indexes, first estimate trend-cyclical components

    Se = (Y/TC)

  • Use centered moving average

    • Called ratio to moving average method

  • For quarterly data, use four-quarter moving average

    • Averages seasonal influences

Example


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Classical Decomposition Explained: Step 1

  • Four-quarter moving average will position average at…

    • end of second period and

    • beginning of third period

  • Use centered moving average to position data in middle of the period

  • Example


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    Classical Decomposition Explained: Step 1

    • Find seasonal-error components by dividing original data by trend-cyclical components

      Se = (Y/TC)

    • Se = Seasonal-error components

    • Y = Original data value

    • TC = Trend-cyclical components

      (centered moving average value)

    Example


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    Classical Decomposition Explained: Step 1

    • Unadjusted seasonal indexes (USI) are found by averagingseasonal-error components by period

    • Develop adjusting factor (AF) so USIs are adjusted so their sum equals the number of quarters (4)

      • Reduces error

    Example

    Example


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    Classical Decomposition Explained: Step 1

    • Adjusted seasonal indexes (ASI) are derived by multiplying the unadjusted seasonal index by the adjusting factor

      ASI = USI x AF

    • ASI = Adjusted seasonal index

    • USI = Unadjusted seasonal index

    • AF = Adjusting factor

    Example


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    Classical Decomposition Explained: Step 2

    • Deseasonalized data is produced by dividing the original data values by their seasonal indexes

      (Y/S) = TCe

    • Y/S = Deseasonalized data

    • TCe = Trend-cyclical-error component

    Example


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    Classical Decomposition Explained: Step 3

    • Develop the trend-cyclical regression equation using deseasonalized data

      Tt = a + bt

    • Tt = Trend value at period t

    • a = Intercept value

    • b = Slope of trend line

    Example


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    Classical Decomposition Explained: Step 4

    • Use trend-cyclical regression equation to develop trend data

    • Create forecasted data by multiplying the trend data values by their seasonal indexes

      • More accurate forecast

    Example

    Example


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    Classical Decomposition Explained: Step Summary

    Summarized Steps:

    • Determine seasonal indexes

    • Deseasonalize the data

    • Develop the trend-cyclical regression equation

    • Create forecast using trend data and seasonal indexes


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    Classical Decomposition:Illustration

    • Gem 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 1

    • An illustration using classical decomposition will follow


    Classical decomposition illustration step 1 l.jpg
    Classical Decomposition Illustration: Step 1

    • (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


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    Classical DecompositionIllustration: Step 1

    • (b) Compute the two-quarter centered moving average

      Ex: centered MA at middle of Qtr 3

      (59.25+61.25)/2

      = 60.500

    Explain


    Classical decomposition illustration step 121 l.jpg
    Classical Decomposition Illustration: Step 1

    • (c) Compute the seasonal-error component (percent MA)

      Ex: percent MA at Qtr 3

      (65/60.500)

      = 1.074

    Explain


    Classical decomposition illustration step 122 l.jpg
    Classical DecompositionIllustration: Step 1

    • (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


    Classical decomposition illustration step 123 l.jpg
    Classical DecompositionIllustration: Step 1

    • (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


    Classical decomposition illustration step 124 l.jpg
    Classical DecompositionIllustration: Step 1

    • (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


    Classical decomposition illustration step 2 l.jpg
    Classical DecompositionIllustration: Step 2

    • Compute the deseasonalized sales by dividing original sales by the adjusted seasonal index

      Ex (Yr 1, Qtr 1):

      (55 / 0.942)

      = 58.386

    Explain


    Classical decomposition illustration step 3 l.jpg
    Classical DecompositionIllustration: Step 3

    • Compute the trend-cyclical regression equation using simple linear regression

      Tt = a + bt

      t-bar = 8.5

      T-bar = 69.6

      b = 1.465

      a = 57.180

      Tt = 57.180 + 1.465t

    Explain


    Classical decomposition illustration step 4 l.jpg
    Classical DecompositionIllustration: Step 4

    • (a) Develop trend sales

      Tt = 57.180 + 1.465t

      Ex (Yr 1, Qtr 1):

      T1 = 57.180 + 1.465(1) = 58.645

    Explain


    Classical decomposition illustration step 428 l.jpg
    Classical DecompositionIllustration: Step 4

    • (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


    Classical decomposition illustration graphical look l.jpg
    Classical DecompositionIllustration: Graphical Look


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    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 sales

    • Use the steps and examples shown in the explanation and illustration as a reference

      Basic Steps

      Explanation

      Illustration

      Templates


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    Summary

    • Time series models are sequences of data that follow non-arbitrary orders

    • Classical decomposition isolates the components of a time series model

    • Benefits:

      • Insight to fluctuations in trend

      • Decomposes the underlying factors affecting the time series


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    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


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    Appendix A:Exercise Templates


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    Appendix A:Exercise Templates


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    Appendix A:Exercise Templates


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    Appendix A:Exercise Templates


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    Appendix A:Exercise Templates