double exponential smoothing using holt’s method

7/17/2019 Double Exponential Smoothing Using Holt’s Method

http://slidepdf.com/reader/full/double-exponential-smoothing-using-holts-method 1/23

2-1

Double Exponential Smoothing Using Holt’s

Method• Holt’s Method is a type of double exponential

smoothing designed to track time series with

linear trend.

• t has two smoothing constants! α and β! and

uses two smoothing e"uations# one for the $alue

of the series %the intercept& and one for the trend%the slope&.



2-2

The equations are:

't ( α)t * %1- α&%'t-1 * +t-1&+t ( β%'t , 't-1& * %1- β&+t-1

• 't is the $alue of intercept %series& at time t.

• +t is the $alue of slope %trend& at time t.

• he τ-step forecast made in period t is#

• t! t* τ ( 't * τ +t

• he smoothing constants may ha$e the same

$alue but for most applications more stability is

gi$en to the slope estimate β ≤ α



2-/

Example 2.5

• 0et’s apply the Holt’s method to the enginefailure data# 2! 2! 13! 145! 22! 24! /!

16. 7ssume that both α and β are e"ual to .1.

'uppose the ' ( 2 and + ( 1.'1 ( .1%2& * %1- .1&%2 * 1& ( 26

+1 ( .1%26 , 2& * %1- .1& 1 ( 6.6

'2 ( .1%2& * %1- .1&%26 * 6.6& ( 222

+2 ( .1%222 , 26& * %1- .1& 6.6 ( 1.2



2-8

ontinue:

'/ ( .1%13& * %1- .1&%222 * 1.2& ( 225.+/ ( .1%225. , 222& * %1- .1&1.2 ( 6.5

as t! t* τ ( 't * τ +t then forecast of period 8 at

period three is /! 8 ( 225. * 6.5 ( 2/5.1 ( 2/5Period Failures Forecast ABS(Error)

! "#$ 2%$ 5&

5 225 2!& "5

$ 2#5 2!# %'

' %&5 2$" !!

# "(& 2'5 #5

M)D * !$.2&



2-

Multiple Step )head +ore,asts

• 9x# :hat is the forecast for period if we are at

period 2;

as t! t*τ

( 't * τ +t then forecast of period at period 2 is 2! ( 222 */ %1.2& ( 22.5 ( 2/

• :hat is the forecast for period 5 if we are at

period 2; 2! 5 ( 222 *8 %1.2& ( 252.5 ( 25/



2-5

Ho- to initialie Holt’s method/

How to find the initial $alues of intercept %'t&

and the slope %+t&;

he best approach is to establish some set of

initial periods as a baseline and use regression

analysis to determine estimates of the slope andintercept using the baseline data.


http://slidepdf.com/reader/full/double-exponential-smoothing-using-holts-method 7/232-3

Example: 0ne 1ear sales re,ords or super3gadget ,ompan14 use $

periods as a base line or itting regression line and ore,ast or

September and 0,tober b1 Holt’s method4 α and β are equal to &."

Month Demandan. 5&&

+eb. 5"&

Mar. !#&

)pr. $&&

Ma1 $&&une $$&

ul1 5(&

)ug. '&&

<alue of regression in =une ( 8/6./ * /8%5&( 58/./ then '5 ( 58/./ and +5 ( /8

't ( α)t * %1- α&%'t-1 * +t-1&

'3 ( .1%6& * %1- .1&%58/./ * /8& ( 554.5

+t ( β%'t , 't-1& * %1- β&+t-1

+3 ( .1%554.5 , 58/./& * %1- .1& /8 ( //.1

'4 ( .1%3& * %1- .1&%554.5 * //.1& ( 31.

+4 ( .1%31. , 554.5& * %1- .1& //.1 ( //.1

4! 6 ( '4 * +4

4! 6 ( 31. * //.1 ( 3/8.5 ( 735

4! 1 ( '4 * 2+4 ( ( (

>sing the ?egression

7nalysis to the first

six months# =an. to

=une we find a

( intercept& ( 8/6./

and b %slope& ( /8.



Methods or Seasonal Series

• 7 seasonal series is the one that has a pattern that repeats

e$ery @ %which is at least /& periods.• @ is the length of the season.



More about seasonalit1:

• 'ome examples of seasonal series are#

, :eather $ariations

, 'ales of winter and summer sports e"uipment

, 7irline ticket sales , +reeting card sales

• 'ometimes series might ha$e more than one seasonal

$ariation. 9x# Aanks may experience daily seasonal

seasonal $ariation %hea$ier traffic hours after 8# pm&!weekly $ariation %hea$ier toward the end of the week&

and monthly $ariation %hea$iest around the end of

month! checks being cashed! deposit etc.&



omponents o an 0bser6ation

Bbser$ed demand %B& (

'ystematic component %'& * ?andom component %?& Level %current deseasonaliCed demand&

Trend %growth or decline in demand&

Seasonality %predictable seasonal fluctuation&

• 'ystematic component# 9xpected $alue of demand• ?andom component# he part of the forecast that de$iates

from the systematic component• orecast error# difference between forecast and actual demand



2-11

Seasonal a,tors method or stationar1

series

• his method can be applied to series with

seasonal $ariation and no trend.• t re"uires a minimum of two seasons of

data.



2-12

Steps o the method:

1. Dompute the sample mean of all the data.

2. )i$ide each obser$ation by the sample mean. hisgi$es seasonal factors for each period of obser$eddata.

/. 7$erage the factors for like periods within eachseason. hat is a$erage all the factors correspondingto the first period of a season! all the factorscorresponding to the second period of a season! andso on. he resulting a$erages are the @ seasonalfactors.

8. orecasts can be obtained by multiplying the samplemean with appropriate seasonal factors.



2-1/

7umber o ,ars 8in thousands o ,ars9 using the toll bridge at

-oring da1s are gi6en belo-:

;ee " Mon. "$42

Tues. "242;ed. "!42

Thur. "'4%

+ri. 2245

;ee 2 Mon. "'4%

Tues. ""45

;ed. "5

Thur. "'4$

+ri. 2%45

;ee % Mon. "!4$Tues. "%4"

;ed. "%

Thur. "$4(

+ri. 2"4(

&

5

"&

"5

2&

25

" 2 % ! 5 $ ' # ( "& "" "2 "% "! "5



2-18

". ompute the sample mean o all the data.

;ee " Mon. "$42

Tues. "242

;ed. "!42Thur. "'4%

+ri. 2245

;ee 2 Mon. "'4%

Tues. ""45

;ed. "5Thur. "'4$

+ri. 2%45

;ee % Mon. "!4$

Tues. "%4"

;ed. "%Thur. "$4(

+ri. 2"4(

Average 16,!"



2-1

2. Di6ide ea,h obser6ation b1 the sample mean. This gi6es

seasonal a,tors or ea,h period o obser6ed data.

;ee " Mon. "$42 &4(#!$&%

Tues. "242 &4'!"!(";ed. "!42 &4#$%&!'

Thur. "'4% "4&5"!5(

+ri. 2245 "4%$'5&!

;ee 2 Mon. "'4% "4&5"!5(

Tues. ""45 &4$(#(!';ed. "5 &4(""$$(

Thur. "'4$ "4&$($(2

+ri. 2%45 "4!2#2#2

;ee % Mon. "!4$ &4##'%5#

Tues. "%4" &4'($"(";ed. "% &4'(&""%

Thur. "$4( "4&2'"!'

+ri. 2"4( "4%%"&%'

Average 16,!"



2-15

%. )6erage the a,tors or lie periods -ithin ea,h season. That is

a6erage all the a,tors ,orresponding to the irst period o a

season4 all the a,tors ,orresponding to the se,ond period oa season4 and so on. The resulting a6erages are the 7

seasonal a,tors.;ee " Mon. "$42 &4(#!$&%

Tues. "242 &4'!"!("

;ed. "!42 &4#$%&!'

Thur. "'4% "4&5"!5(+ri. 2245 "4%$'5&!

;ee 2 Mon. "'4% "4&5"!5(

Tues. ""45 &4$(#(!'

;ed. "5 &4(""$$(

Thur. "'4$ "4&$($(2

+ri. 2%45 "4!2#2#2

;ee % Mon. "!4$ &4##'%5#

Tues. "%4" &4'($"("

;ed. "% &4'(&""%

Thur. "$4( "4&2'"!'

+ri. 2"4( "4%%"&%'

Mon. &4('!5

Tues. &4'!55

;ed. &4#5!(

Thur. "4&!(!

+ri. "4%'5$

Sum* 5



2-13

!. +ore,asts ,an be obtained b1 multipl1ing the sample

mean -ith appropriate seasonal a,tors.

Sample Mean +ore,asts

Mon. &4('!5 "$4!5% "$4&

Tues. &4'!55 "$4!5% "24%

;ed. &4#5!( "$4!5% "!4"

Thur. "4&!(! "$4!5% "'4%

+ri. "4%'5$ "$4!5% 224$



2-14

;inter’s Method or Seasonal <roblems

• :inter’s method is a type of triple exponential

smoothing.

• :e assume a model of the form#

)t ( %µ * +t &ct * εt

where µ is the intercept at time t ( excluding seasonality!

+ as the trend or slope component!

ct is the multiplicati$e seasonal component in period t!

and εt is the error.



2-16

Seasonal Series

-ith =n,reasing Trend

Fig. 2-10



2-2

The Exponential Smoothing Equations

1. he series# he current le$el of deseasonaliCedseries! 't is

't ( α%)t E ct-@&* %1- α&%'t-1 * +t-1&

2. he trend# he trend is updated like Holt’s

method.+t ( β%'t , 't-1& * %1- β&+t-1

/. he seasonal factors#

ct ( γ %)tE 't & * %1- γ &ct-@

where @ is the length of seasonal period

• he forecast is# t! t * τ ( %'t * τ +t& ct * τ - @



2-21

hoi,e o the smoothing ,onstants: α4 β4

and γ

• 'ame issues as with exponential smoothing andHolt’s method.

• 0arge $alues of the smoothing constants will result

in more responsi$e but less stable forecasts.• he most conser$ati$e approach is choosing

$alues between .1 and ./ for ha$ing stableforecasts.

• Bne method is experimenting different $alues ofthese parameters and use the ones which gi$es the best fit to the historical data.

le6el Trend Sesonal



2-22

Example:

le6el Trend Sesonal

&4"5 &4"5 &4"5 "%$" 5"%"5#2

&4"5 &4"5 &4%& "%%% 52!5(&"

&4"5 &4"5 &4!5 "%&% 5!"'"'2

&4"5 &4%& &4"5 """( %('5(!2

&4"5 &4%& &4%& ""$' !2'$'(5&4"5 &4%& &4!5 "25$ !$(&(2&

&4"5 &4!5 &4"5 ""2( %$"$&#"

&4"5 &4!5 &4%& "2$5 !2'$'(5

&4"5 &4!5 &4!5 "!2" !$(&(2&

&4%& &4"5 &4"5 '#% 2%!2'&!

&4%& &4"5 &4%& '($ 25&!'(&

&4%& &4"5 &4!5 #2' 2'&"%$!&4%& &4%& &4"5 '"5 "(%%%25

&4%& &4%& &4%& '$5 2"2%&($

&4%& &4%& &4!5 #'& 2%#($5!

&4%& &4!5 &4"5 $'$ "'$%!("

&4%& &4!5 &4%& '(& "(((&#&

&4%& &4!5 &4!5 ((' 2!"#$$2

&4!5 &4"5 &4"5 5(" "$%"#%&&4!5 &4"5 &4%& $&( "'2'!2$

&4!5 &4"5 &4!5 $%# "#!&!%2

&4!5 &4%& &4"5 525 "!!'%!(

&4!5 &4%& &4%& 5#2 "5!2!!#

&4!5 &4%& &4!5 $!" "$$2'#2

&4!5 &4!5 &4"5 5&% "!2&(55

&4!5 &4!5 &4%& 5#! "5252(%

&4!5 &4!5 &4!5 $#% "$'"2#$

M)D MSEα γ β



2-2/

>asi, )pproa,h to

Demand +ore,asting

• Ae sure to distinguish between demand and sales

• >nderstand the obFecti$es of forecasting

• ntegrate demand planning and forecasting

• dentify maFor factors that influence the demandforecast

• >nderstand and identify customer segments

• )etermine the appropriate forecasting techni"ue

• 9stablish performance and error measures for the

forecast

double exponential smoothing using holt’s method

Documents

period t

double exponential smoothing

smoothing constants

smoothing euations

holtsmethod holts methodis

initialie holts methodhow

period atperiod

tober b1 holts method4