Chapter 2: ForecastingChapter 2: Forecasting

Methods for Seasonal SeriesMethods for Seasonal Series

• Methods for Stationary SeriesMethods for Stationary Series–Seasonal factorsSeasonal factors–Seasonal decomposition using MASeasonal decomposition using MA

• Methods for seasonal series with Methods for seasonal series with trendtrend

–Winter’s MethodWinter’s Method

A Seasonal Demand SeriesA Seasonal Demand SeriesFig. 2-8

Seasonal Series with Increasing TrendSeasonal Series with Increasing TrendFig. 2-10

Winter’s MethodWinter’s Method• We assume a model of the formWe assume a model of the form

• μμ : base signal or intercept at time 0 : base signal or intercept at time 0

• G: trend or slope componentG: trend or slope component

• cctt : multiplicative seasonal component: multiplicative seasonal component

• εεtt : error term: error term

This model assumes that the underlying series has This model assumes that the underlying series has a form similar to that in Figure 2-10.a form similar to that in Figure 2-10.

tttt cGD )(

Assumptions:Assumptions:• The season is exactly The season is exactly NN periods periods

• Seasonal factors are the same each period andSeasonal factors are the same each period and

ΣΣ cct t = N= N

Three exponential smoothing equations are used each period Three exponential smoothing equations are used each period to to

update estimates of :update estimates of :

• Deseasonalized seriesDeseasonalized series

• Seasonal factorsSeasonal factors

• TrendTrend

These equations have different smoothing constants, These equations have different smoothing constants, αα, , ββ, and , and γγ

• The series:The series:

• The trendThe trend

• The seasonal factorsThe seasonal factors

))(1( 11

ttNt

tt GS

c

DS

111 )1()( tttt GGSG

Ntt

tt c

S

Dc )1(

Forecast made in period Forecast made in period tt for for any future period any future period t + t + ττ

N if )(, Nttttt cGSF

Initialization ProcedureInitialization ProcedureSuppose that current period is t=0Suppose that current period is t=0

Past observations are labeled DPast observations are labeled D-2N+1-2N+1, D, D-2N+1-2N+1, …, … , D, D0 0

1.1. Calculate the sample means for the 2 seasons data:Calculate the sample means for the 2 seasons data:

2. Define the initial slope estimate2. Define the initial slope estimate

0

1

12

12

11

NjjN

N

NjjN

DV

DV

NmVV

oNVV

o

N

NjjN

mGG

DV

)1(

12

11

112 seasons 2 m caseIn

Initialization for Winters’s MethodInitialization for Winters’s Method

Fig. 2-11

3. Set the estimate of the value of the series at 3. Set the estimate of the value of the series at t=0t=0

4 (a). 4 (a). Initial SF are obtained by dividing each observation Initial SF are obtained by dividing each observation by the corresponding point on the line connecting Vby the corresponding point on the line connecting V11 and Vand V22 using the formula: using the formula:

i=1,2 for the 1st , 2i=1,2 for the 1st , 2ndnd season season

j : period of the seasonj : period of the season

(b). (b). Average the seasonal factors:Average the seasonal factors:

(c ). (c ). Normalize the SF:Normalize the SF:

]2/)1[(2 NGVS oo

oi

tt GjNV

Dc

]2/)1[(

2,,

2 0112

1oNNN

N

ccc

ccc

01for .1

0

jNNc

cc N

ii

jj

PePerioriodd

DDtt VVii

11 1010

18.2518.25

cc-7-710/10/[[18.25-(5/2-1)18.25-(5/2-1)(.875)(.875)]]=.5904=.5904

22 2020 cc-6-620/20/[[18.25-(5/2-2)18.25-(5/2-2)(.875)(.875)]]=1.123=1.123

33 2626 cc-5-526/26/[[18.25-(5/2-3)18.25-(5/2-3)(.875)(.875)]]=1.391=1.391

44 1717 cc-4-417/17/[[18.25-(5/2-4)(.875)18.25-(5/2-4)(.875)]]=.869=.869

55 1212

21.7521.75

cc-3-312/12/[21.75[21.75-(5/2-1)-(5/2-1)(.875)(.875)]]=.5872=.5872

(.5904+.587(.5904+.5872)/2 =.58882)/2 =.5888

66 2323 cc-2-223/23/[21.75[21.75-(5/2-2)-(5/2-2)(.875)(.875)]]=1.079=1.079

1.10101.1010

77 3030 cc-1-130/30/[21.75[21.75-(5/2-3)-(5/2-3)(.875)(.875)]]=1.352=1.352

1.37201.3720

88 2222 cc0022/22/[21.75[21.75-(5/2-4)-(5/2-4)(.875)(.875)]]=.9539=.9539

.9115.9115

Go=(21.75-18.25)/4Go=(21.75-18.25)/4

=.875=.875

So=21.75+(.875)So=21.75+(.875)(1.5)(1.5)

=23.06=23.06