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# Classical Decomposition - PowerPoint PPT Presentation

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

Boise State University

By: Kurt Folke

Spring 2003

• Time series models & classical decomposition

• Brainstorming exercise

• Classical decomposition explained

• Classical decomposition illustration

• Exercise

• Summary

• Appendix A: exercise templates

• 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

• 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

• 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

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

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

• Determine seasonal indexes

Y = TCSe

• Equate…

Se = (Y/TC)

• 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

• 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

• 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

• 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

• 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

Example

• 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

• 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

• 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

Summarized Steps:

• Determine seasonal indexes

• Deseasonalize the data

• Develop the trend-cyclical regression equation

• Create forecast using trend data and seasonal indexes

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

• (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

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 1

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

Ex: percent MA at Qtr 3

(65/60.500)

= 1.074

Explain

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 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 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 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 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 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 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 DecompositionIllustration: Graphical Look

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

• 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

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

Appendix A:Exercise Templates

Appendix A:Exercise Templates

Appendix A:Exercise Templates

Appendix A:Exercise Templates

Appendix A:Exercise Templates