Sep 8 ISMT/Stuart Zhu 1

Lecture 4: Trend and Seasonal Series

• Techniques for trend

• Techniques for seasonality

• Summary

• Readings: Page 79-92

Sep 8 ISMT/Stuart Zhu 2

Techniques for Trend

• Linear Trend Equation

• Ft = Forecast for period t• t = Specified number of time periods• a = Value of Ft at t = 0• b = Slope of the line

tt baF

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Sales Data for Five Weeks

145150

155160165170

175180

1 2 3 4 5

Week

Sa

les

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y

t

btaFt

a

0

∆y∆t y

bt

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Calculate a and b

n

tbya

ttn

ytytnb

n

ii

n

ii

n

ii

n

ii

n

ii

n

ii

n

iii

11

2

11

2

111 ,)(

where n = number of periods,

yi = value of the time series at time ti

(1)

(2)

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Linear Trend Equation Example

Sep 8 ISMT/Stuart Zhu 7

Linear Trend Calculation

y = 143.5 + 6.3t

a = 812 - 6.3(15)

5 =

b = 5 (2499) - 15(812)

5(55) - 225 =

12495-12180

275 -225 = 6.3

143.5

Sep 8 ISMT/Stuart Zhu 8

Techniques for Seasonality

• Seasonal variations– Regularly repeating movements in series values

that can be tied to recurring events.

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1 2 3 4

x

x xx

xx

x xx

xx x x x

xxxxxx x x

xx

x x xx

xx

xx

x

xx

xx

xx

xx

xx

xx

x

x

Year

Seasonal variationSeasonal variation

Linear

Trend

Linear

Trend

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Two Models

• Additive model– Demand = Trend + Seasonality

• Multiplicative model– Demand = Trend * Seasonality

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Using Seasonal Relatives

• Seasonal relative– Percentage of average or trend

• Computing Seasonal relatives – Centered moving average: a moving average

positioned at the center of the data

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Example 7 at Page 84

• Compute estimated relatives for each day

• Suppose that the trend equation is given by

120 +5t

– Give Friday represents t=5, what is the forecast on Friday?

Forecast-s3

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Summary

• Compute the trend equation– a, b,

• How to compute and use seasonal relatives

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Review Problems

• Problem 5 at page 113

• Problem 10 at page 115

• Problem 11 at page 115

• Problem 12 at page 115