Double Exponential Smoothing Using Holt’s Method
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7/17/2019 Double Exponential Smoothing Using Holt’s Method
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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&.
7/17/2019 Double Exponential Smoothing Using Holt’s Method
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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
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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
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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&
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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/
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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.
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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.
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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.
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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.&
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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
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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.
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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.
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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
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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,!"
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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,!"
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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
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!. +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$
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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.
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2-16
Seasonal Series
-ith =n,reasing Trend
Fig. 2-10
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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 - @
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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
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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
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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