Types of Exponential Smoothing Methods. Simple Exponential Smoothing. Simple Exponential Smoothing
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1 M Business Forecasing Mehods Exponenial moohing Mehods ecurer : Dr Iris Yeung Room No : P79 Tel No : Types of Exponenial moohing Mehods imple Exponenial moohing Double Exponenial moohing Brown s One-parameer inear Mehod Hol s Two-parameer inear Mehod Winers Three Parameer inear and easonal Exponenial moohing Muliplicaive Mehod Addiive Mehod Properies of Exponenial moohing Mehods Exponenial smoohing weighs all observaions unequally wih heavier weighs given o recen observaions and smaller weighs given o old observaions. The weighs are deermined by smoohing consan(s) which have o be chosen according o some crieria. The exponenial smoohing mehods can be used when he parameers describing he ime series are changing slowly wih ime. imple Exponenial moohing uiable for no rend series y = β + ε, ε ~ N(, σ ) β may change slowly wih ime The basic equaions for simple exponenial smoohing is A = αy + ( α) A - (.) F +m = A, m =,,, (.) A is he simple exponenial smoohed saisic a ime, F +m is he forecas value for period +m made a ime, and α is a smoohing consan beween and. imple Exponenial moohing E Weighs Expanding equaion (.) by replacing A - by is componens, A - by is componens, A by is componens and so on, we have A = αy + ( α)[αy +( α) A ] = αy + α( α)y +( α) A. = αy + α( α)y + α( α) Y + α( α) Y + α( α) Y + α( α) Y + + α( α) Y + ( α) A. (.) o F +m = A represens a weighed moving average of all pas observaions. uppose α =.,.,. or.8. Then he weighs assigned o pas observaions would be as follows: Weigh assigned o: α =. α =. α =. α =.8 Y....8 Y.... Y Y Y (.)(.8) (.)(.) (.)(.) (.8)(.) In each case, - The weighs for all pas daa sum approximaely o one. - The weighs decrease exponenially, hence he name exponenial smoohing.
2 E Weighs Time Alpha=. Alpha=. Alpha=. Alpha=.8 Choice of a in E If daa show large randomness, use small α(. < α <.) If daa show paern, use large α (α, sugges rend or seasonaliy) Choose α which minimize ME, MAPE, over a es se. 8 Iniializaion in E everal alernaives for A A = y, or = mean of all observaions, or = mean of he firs, or observaions, or = mean of half of he daa As he las erm of equaions (.) is ( α) A, so he iniial smoohed value A (or iniial forecas F ) plays a role in all subsequen forecass. Bu he weigh aached o A is ( α), which is usually small. o he choice of A becomes no imporan afer processing many observaions or large α is used. 9 E Example :hipmens of elecric can openers Period Acual α =. Forecas α =. α = Analysis of errors (Tes period -) Mean Error..8.9 Mean Absolue Error Mean Absolue Percenage Error (MAPE) andard Deviaion of Error (Unbiased) Mean quare Error (ME) Theil s U aisics E Example Brown s Mehod Monhs Y alpha=. alpha=. alpha=.9 uiable for linear rend series Y = β + β + ε β and β may change slowly wih ime.
3 Basic Equaions for Brown s Mehod A = αy + ( α) A (.) A = αa + ( α) A (.) a = A A (.) α b = ( A A ) α (.7) F+ m = a + bm (.8) A : simple exponenial smoohed saisic, A : double exponenial smoohed saisic a : esimae of β a ime, and Brown s Mehod I can be shown ha α E( A) = E( y ) β α Tha is he expeced value of he simple exponenial smoohed saisic A will end o lag behind he level of he series a ime, E(y), by an amoun equal o α β. α b : esimae of β a ime. Brown s Mehod imilarly i can be shown ha α E( A ) = E( A ) β α Tha is, he expeced value of he double smoohed saisics A lags behind he expeced value of A by he α β same amoun equal o α. Brown s Mehod The curren level of he daa, β, can be esimaed by a = A + ( A A ) = A A The slope of he series, β, can be esimaed by b α = α ( A A ) The forecas for period + m is obained by exrapolaing he rend m periods ino he fuure F + m = a + b m. Choice of a and Iniializaion in Brown s Mehod Choice of a Choose one ha minimize ME or MAPE. Iniializaion Eiher A = A = y Or Bowerman (P9) A a α = b α A a α = b a α and b are leas squares esimaes of β and β by fiing a sraigh line o, for example, one half of he daa. Brown s Example:Invenory Demand Daa (a =.) Period Acual ingle Exponenial moohing Double Exponenial moohing Forecas ( m = ) ( m = ) ( m = ) ( m = ) 7
4 Hol s Mehod uiable for linear rend series y = β + β + ε β and β may change slowly wih ime. The basic equaions for Hol s Mehod are: A = αy + ( α) (A + T ) (.9) T = β(a A ) + ( β)t (.) F + m = A + mt (.) A esimae he level of he series, β, a ime, and Hol's Mehod Equaion (.9) is similar o equaion (.) excep ha a erm for he rend (T ) is added o adjus for he rend in he daa. The value of his erm is calculaed using equaion(.). The difference beween successive exponenial smoohing values (A A ) is used as an esimae of he rend. The esimae of he rend is smoohed by muliplying i by β and hen muliplying he old esimae of he rend by ( β). T esimae he slope of he series, β, a ime. 9 Hol's Mehod Equaion (.) is similar o equaion (.9) or equaion (.) excep ha he smoohing is no done for he acual daa bu raher for he rend. The final resul of equaion (.) is a smoohed rend ha does no include much randomness. To forecas, he rend is muliplied by he number of periods ahead ha one desires o forecas and hen he produc is added o A (he curren level of he daa ha have been smoohed o eliminae randomness). Choice of a, b andiniializaion in Hol's Mehod Choice of a and b Choose one ha minimize ME or MAPE. Iniializaion Makridakis (P.9) evel : A = y Trend : T = y y or T = y y or T = ( y y ) + ( y y ) Makridakis(P.), Bowerman (P.) Fi a rend line o firs few or one half of he hisorical daa o find A and T. Hol's Mehod Advanage apply differen weighs o randomness and rend Disadvanage specify parameers, no simple Hol's Example : Invenory Demand Daa (α=., β=.7) Period Observed daa moohing of daa moohing of rend Forecas Tes e (m = ) 8.8 (m = ) (m = ) (m = ) (m = ) 9. (m = ) Analysis of errors from period o period Mean Error =.78 Mean Absolue Percenage Error =. Mean Absolue Error =.9 Theil s U-saisic =.78 Mean quare Error = 9.78
5 Winers' Muliplicaive Mehod uiable for linear rend and muliplicaive seasonaliy series Y = (β + β ) + ε β, β, may change slowly wih ime, and he seasonal variaion is increasing as he average level of he series β + β increases. Basic equaions for Winers' Muliplicaive Mehod A y + ( α )( A + T ) (.) = α T = β( A A ) + ( β) T = γ y + ( γ ) A F + m = ( A + mt ) + m is he lengh of seasonaliy, A represens he level of he series, T denoes he rend, and is he seasonal componen. (.) (.) (.) Winers' Muliplicaive Mehod Winers' Muliplicaive Mehod Winers' exponenial smoohing is an exension of Hol's linear exponenial smoohing. Winers' smoohing uses he equaions of Hol's model bu inroduces he seasonal index ino he formulas and includes an exra equaion ha is used o esimae seasonaliy. Equaions (.), (.) and (.) are used o obain esimaes of he presen level of he daa, he rend, and he forecas for some fuure period, + m. There is a sligh difference beween equaion (.) and (.9). In equaion (.), y is divided by. This removes he seasonal effecs which may exis in he original daa y. 7 The esimae of seasonaliy is given as an index, flucuaing around, and is calculaed wih equaion (.). The form of equaion (.) is similar o ha of all oher exponenial smoohing equaions i.e. a value in his case y /A is muliplied by a consan γ and is hen added o is previous smoohed esimae which has been muliplied by γ. Equaion (.) is similar o equaion (.) excep ha he esimae for he fuure period, + m, is muliplied by +m. This is he las seasonal index available and hence is used o readjus he forecas for seasonaliy. Muliplying he forecas by +m has he opposie effec of dividing y by in equaion (.). 8 Choice of a, b, g in Winers' Muliplicaive Mehod α is used o smooh randomness, β o smooh rend and γ o smooh seasonaliy. Usually β, γ are less han α. Choose α, β, γ which minimize ME or MAPE. Makridakis (P.8) evel : Trend : easonal : A = ( y + y y ) y + y y + y y T = y y y =, =,..., = A A A + y 9
6 Bowerman(P.-7). Calculae he iniial esimae of β (rend componen) by y y T m = ( m ) y, he average of observaions in year, measure he average level of he ime series in he middle of year ; y m, he average of observaions in year m, measure he average level of he ime series in he middle of year m; (m ) is he oal number of seasons elapsed beween he middle of year and he middle of year m. Bowerman(P.-7). Calculae he iniial esimae of β by A = y T The number of seasons ha have elapsed from he sar of year o he middle of year is /. o he iniial esimae of β which represens he average level of he ime series a ime is he average level of he ime series a he middle of year less he amoun his average level has changed from he sar of year o he middle of year. o he iniial esimae T is simply he change in average level per season from he middle of year o he middle of year m. Bowerman (P.-7). Calculae iniial esimae for each season occurring in years hrough m by Where y = y i + j T, =,,, m, he average of observaions in year i, measures he yi average level of he ime series in he middle of he year in which season occurs; j denoes he posiion of season wihin he year; and [(+ )/ j] measures he number of seasons ha season is from he middle of he year. The value of his expression is negaive if season occurs before he middle of he year and posiive if season occurs afer he middle of he year. o y i [( + ) j] T measure he average level of he ime series is season by eiher subracing he appropriae rend from he average level a midyear (season occurs before he middle of he year) or adding he appropriae rend o he average level a midyear (season occurs afer he middle of he year). Winers Muliplicaive Example : Quarerly Expors of a French Company(a =.8, b =. and gg =.) Period Acual evel Trend easonal Forecas Calculae he average seasonal index for each differen season m k j = m = j+ k for j=,,,. Normalize he iniial esimaes so ha hey add o j = j = j for j =,,, Analysis of errors from period o period Mean Error =.9 Mean Absolue Error = (m = ). 78. (m = ) 89. (m = ) 78.9 (m = ) 777. (m = ) 8. (m = ) Mean Absolue Percenage Error =. Theil s U-saisic =. Mean quare Error = 8.9
7 Winers' Addiive Mehod uiable for linear rend and addiive seasonaliy series Basic Equaions for Winers' Addiive Mehod A α[ y ] + ( α )( A T ) = + (.) y = (β + β ) + + ε T β( A A ) + ( β) T = (.7) β, β, may change slowly wih ime and he seasonal variaion is consan over ime. = γ [ y A ] + ( γ ) F + m = A + mt + + m (.8) (.9) The equaions for he addiive model can be obained from hose for muliplicaive model by replacing division operaions wih subr acion operaions and by replacing muliplicaion operaions wih addiive operaions. 7 8 Winers Addiive Example : eminar demand series (a =., b =. and g =.) Choice of a, b, g and Iniializaion in Winers Addiive Mehod Makridakis(P.9) ame as hose for muliplicaive mehod excep for seasonal indices, we use = y A, = y A,, = y A Bowerman(P.7) Compue he leas squares esimaes of he parameers in he dummy variable regression model y = β + β + β x, + β x, + K+ β ( ) x ( ), + independen error erms are assumed. ε y A A = T T = ( ) () Forecas
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