Classical Decomposition

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Classical Decomposition

• Boise State University
• By Kurt Folke
• Spring 2003

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Overview

• Time series models classical decomposition
• Brainstorming exercise
• Classical decomposition explained
• Classical decomposition illustration
• Exercise
• Summary
• Bibliography readings list
• Appendix A exercise templates

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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
  averaging seasonal-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 DecompositionIllustration

• Gem Companys 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

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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
• (55476570)/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
• (59.2561.25)/2
• 60.500

Explain
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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
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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
• 0.9890.9140.926/3 0.943

Explain
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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/(0.9430.8511.0801.130) (4.000/4.004)
  

Explain
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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
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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
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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
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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
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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
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Classical DecompositionIllustration Graphical
Look
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Classical DecompositionExercise

• 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 companys 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 AExercise Templates
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Appendix AExercise Templates
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Appendix AExercise Templates
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Appendix AExercise Templates
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Appendix AExercise Templates
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Appendix BExercise Solutions
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Appendix BExercise Solutions
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Appendix BExercise Solutions
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Appendix BExercise Solutions
Trend-cyclical Regression Equation Tt 5.402
0.514t
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Appendix BExercise Solutions