Lecture 4: Exponential Smoothing for Trended and Seasonal Time Series
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1 NATCOR: Forecasing & Predicive Anayics Lecure 4: Exponenia Smoohing for Trended and Seasona Time Series John Boyan Lancaser Cenre for Forecasing Deparmen of Managemen Science
2 Srucure of his Lecure Par I Par II Par III Opimising parameers and iniia vaues Roing origin evauaion (ou-of-sampe Predicion Inervas Par IV Par V Par VI Trend Forecasing Seasona Forecasing Mehod Seecion Side 2 NATCOR Exp Smooh: Trend and Seasonaiy
3 Par I: Opimising parameers and iniia vaues Side 3 NATCOR Exp Smooh: Trend and Seasonaiy
4 Parameer Esimaion for SES Saes (A Forecas (F SKU A - SES α = Saes (A Forecas (F SKU A - SES α = Saes Saes Week Week MSE(in-sampe = 27.3 < MSE(in-sampe = 35.6 This parameer is beer, resuing in ower in-sampe MSE Based on his idea we can opimise parameers (for any exponenia smoohing mehod or generay. Side 4 NATCOR Exp Smooh: Trend and Seasonaiy
5 MSE as a funcion of apha We can cacuae he in-sampe MSE for various vaues of apha and idenify he vaue ha gives he owes error MSE Minimum In-sampe MSE α = Apha For more compex mehods, invoving rend and seasonaiy, we wi need o opimise severa parameers simuaneousy. Side 5 NATCOR Exp Smooh: Trend and Seasonaiy
6 MSE: funcion of Apha & Iniia Leve The same principe can be appied o choose he Iniia Leve (= Iniia Forecas for SES. Now we vary boh iniia forecas and smoohing parameer and joiny opimise boh Saes (A Forecas (F SKU B - SES 4000 Saes Week Side 6 Join opimisaion can be done in R Leve = 1697 Apha = NATCOR Exp Smooh: Trend and Seasonaiy
7 Is opimisaion aways he bes? No! MSE is very reacive o exreme errors, which may disor he error surface ha we are searching on for he opima vaues. Sampe imiaions as he number of parameers o opimise increases (especiay for seasona modes. Opimisaion is done in-sampe. The correaion beween insampe error and ou-of-sampe may be ow. Opimisaion is (ypicay done using 1 MSE. In pracice we forecas for onger horizons. MSE may no reae we o a company s financia objecives. Opimisaion is very usefu for auomaion, bu human expers may override idenified parameers if hey vioae heory or objecives. Side 7 NATCOR - Exp Smooh: Trend and Seasonaiy
8 Par II: Roing Origin Evauaion (Ou-of-sampe Side 8 NATCOR Exp Smooh: Trend and Seasonaiy
9 The hisorica observaions can be spi ino wo subses: In-sampe: In- and Ou-of-Sampe used for mode buiding and parameerisaion. Ou-of-sampe: used for mode evauaion. No used in buiding he mode and no seen by he mode. Used o simuae rue forecass, insead of waiing for new unobserved vaues, o evauae he forecasing performance of aernaive forecasing mehods In-sampe Ou-of-sampe 550 Unis Side Monh NATCOR Exp Smooh: Trend and Seasonaiy
10 Remarks on Singe Origin Evauaion Use o fi he mode In-sampe Use o evauae he mode Ou-of-sampe (Hodou Origin Forecass Limiaions: 1. Forecas suscepibe o srange origins. 2. One forecas per horizon 3. Averaging over differen horizons no recommended. Side 10 NATCOR Exp Smooh: Trend and Seasonaiy
11 Roing Origin Evauaion. In-sampe Ou-of-sampe Ro origin In-sampe (increased Origin Ou-of-sampe Ro origin In-sampe (increased Origin Ou-of-sampe Origin Coninue roing uni ou-of-sampe shorer han forecas horizon.
12 Visuaising Roing Origin Forecass Apha = 0.20 Apha = Side 12 In-sampe Ou-of-sampe In-sampe Ou-of-sampe Back dos are forecas origins Visuaising he roing origin forecass makes i easier o appreciae he imporance of smooh forecass ha fier noise NATCOR Exp Smooh: Trend and Seasonaiy
13 Remarks on Roing Origin Evauaion Hodou: 5 Hodou: 10 Forecas Fixed Origin Roing Origin Fixed Origin Roing Origin Horizon Number of forecass Number of forecass Number of forecass Number of forecass Provides more forecass per origin 2. Overcomes imiaions of fixed origin evauaion Provides more hisory per horizon for same ou-of-sampe Does no need o average over horizons Can overcome srange origins or arges Side 13 NATCOR Exp Smooh: Trend and Seasonaiy
14 Par III: Predicion Inervas Side 14 NATCOR Exp Smooh: Trend and Seasonaiy
15 Predicion Inervas SKU C SKU A 80% and 90% predicion inervas UK Android Marke Share US Air Passengers Observe ha he predicion inervas vary wih he series, mehod and horizon. Side 15 We have more confidence o forecass wih igh PIs.
16 Require Error Disribuion Daa SES We can use he error disribuion o formuae predicion inervas of he forecasing mehods. Probabiiy Error PDF e =A -F σ-2σ -σ µ σ 2σ 3σ 99.7% 95.5% 68.2% x 10-3 Side 16 NATCOR Exp Smooh: Trend and Seasonaiy
17 Which Errors? Three Approaches 1. Mode Errors (Irreguar Componens Requires ransaion of esimaes of he variance of mode errors (epsion erms, using a modeing framework, o esimaes of Predicion Inervas. This approach is adoped in es (R package and is based on anaysis in Hyndman e a (2008. Or Forecas Errors Two possibiiies o cacuae Predicion Inerva (PI : 2. In-sampe MSE 3. Ou-of-sampe MSE Side 17 NATCOR Exp Smooh: Trend and Seasonaiy
18 PI based on Ou-of-Sampe MSE We can use ou-of-sampe MSE: h = F h ± za / 2se = F h ± PI z h 2 a / MSE h This can be obained by cacuaing he roing origin ou-of-sampe MSE of he reevan forecas horizon: Advanages over oher approaches Genuiney ou-of-sampe forecas errors. Aows for parameers or mode ou-of-sampe being differen from parameers or mode in-sampe. Disadvanage over oher approaches Can be inaccurae if sma ou-of-sampe (es se. Side 18 NATCOR Exp Smooh: Trend and Seasonaiy
19 Par IV: Trend Forecasing Side 19 NATCOR Exp Smooh: Trend and Seasonaiy
20 Principes of Ho s Mehod Smooh he eve AND Smooh he rend Aows for differen raes of smoohing for eve and rend. Originay proposed in 1957 (Research Memorandum reprined in Inernaiona Journa of Forecasing in Impemened in mos commercia forecasing sofware packages. Forms he basis for more sophisicaed smoohing mehods. Side 20 NATCOR Exp Smooh: Trend and Seasonaiy
21 Ho s Mehod b y = αy h = = β ( (1 α( 1 hb 1 b 1 (1 β b 1 Firs equaion updaes he eve, using he aes observaion and he previous eve, adjused by he previous rend (b erm. Second equaion updaes he rend, using he aes difference in eves (oca rend and he previous smoohed rend. Third equaion provide a forecas h-seps-ahead by adding h imes he updaed rend o he updaed eve. Side 21 NATCOR Exp Smooh: Trend and Seasonaiy
22 Iniiaising Leve and Trend The iniia eve and rend are usuay opimised, aong wih he smoohing parameers. (As for SES in Par I of his ecure. However, some simpe empirica cacuaions incude: Iniia eve Firs acua observaion Simpe average of k firs observaions Iniia rend Difference of he firs wo observaions (Y 2 - Y 1 (Y 3 - Y 1 /2 ec Side 22 NATCOR Exp Smooh: Trend and Seasonaiy
23 Ho s Mehod: Linear Exrapoaion Does a sraigh ine forecas makes sense? WhDo any facors indicae ha he forecas shoud be differen o a sraigh ine? If as sraigh ine does no make sense, hen a damped rend mehod may be used insead Unis Daa Trend EXSM True Trend Monh Side 23 NATCOR Exp Smooh: Trend and Seasonaiy
24 Par V: Seasona Forecasing Side 24 NATCOR Exp Smooh: Trend and Seasonaiy
25 Need o Fier Noise Daa January January PDF Unis Probabiiy Jan96 Jan97 Jan98 Jan99 Jan00 Jan01 Jan02 Jan03 Jan04 Jan05 Jan06 Jan07 Jan08 Jan09 Jan10 Monh 100 Observed True Simiar o eve or rend, he seasona componen is affeced by noise, which needs o be fiered The deviaion around he rue January is due o noise Side 25 NATCOR Exp Smooh: Trend and Seasonaiy
26 Winers Mehod (Addiive Seasonaiy Side 26 NATCOR Exp Smooh: Trend and Seasonaiy Firs equaion updaes he eve, using he difference beween he aes observaion and previous seasona effec, adjused by he previous eve. Second equaion updaes he season, using he difference in aes observaions and eves, and he previous seasona. Third equaion provide a forecas h-seps-ahead by adding mos recen seasona o he aes eve. h m h m m s y s y s s y = = = 1 (1 ( (1 ( γ γ α α
27 Winers Mehod (Muipicaive Seasonaiy Side 27 NATCOR Exp Smooh: Trend and Seasonaiy These equaions work on he same basis as for addiive seasonaiy. Again, he seasona componen is smoohed hrough he gamma parameer (smoohing consan. h m h m m s y s y s s y = = = 1 (1 (1 γ γ α α
28 Principes of Ho-Winers Mehod Smooh he eve AND Smooh he rend AND Smooh he seasonaiy Aows for differen raes of smoohing for eve (apha, rend (bea and seasonaiy (gamma. Impemened in mos commercia forecasing sofware packages. Side 28 NATCOR Exp Smooh: Trend and Seasonaiy
29 Ho- Winers Mehod (Addiive Seasonaiy Side 29 NATCOR Exp Smooh: Trend and Seasonaiy Combines he Ho s and Winers (addiive seasonaiy mehods. Aows for hree differen eves of smoohing. Common o observe boh rend and seasonaiy ogeher h m h m m s hb y s y s b b b s y = = = = (1 ( (1 ( ( (1 ( γ γ β β α α
30 Ho- Winers Mehod (Muipicaive Seasonaiy Side 30 NATCOR Exp Smooh: Trend and Seasonaiy Combines he Ho s and Winers (muipicaive seasonaiy mehods. Again, aows for hree differen eves of smoohing. h m h m m s hb y s y s b b b s y = = = = ( (1 (1 ( ( ( γ γ β β α α
31 Forecasing Compeiions Two famous Forecasing Compeiions: M1 Compeiion ( rea series. M3 Compeiion ( rea series. Up o as 18 periods wihhed from compeiors o ensure genuine ou of sampe resus. Concusions Simper mehods (eg exponenia smoohing ofen compeiive wih more sophisicaed mehods. Performance depends on horizon and choice of accuracy meric. Averages of differen mehods can perform we. Side 31 NATCOR Exp Smooh: Trend and Seasonaiy
32 Par VI: Mehod Seecion for Exponenia Smoohing Side 32 NATCOR Exp Smooh: Trend and Seasonaiy
33 Famiy of Exponenia Smoohing Modes There are 30 Exponenia Smoohing Modes 15 wih addiive errors 15 wih muipicaive errors Each se of 15 modes comprise combinaions of: 5 ypes of rend 3 ypes of seasonaiy Forecas mehods do no differ beween addiive and muipicaive error modes (provided he same parameer vaues are used. Formuae for Predicion Inervas do differ. Side 33 NATCOR Exp Smooh: Trend and Seasonaiy
34 Famiy of Exponenia Smoohing Mehods
35 Seecing Exponenia Smoohing Mehod Temping o simpy compare in-sampe MSEs for differen Exponenia Smoohing mehods. This fais o ake ino accoun he number of parameers being esimaed. There is a risk of over-fiing. Side 35 NATCOR Exp Smooh: Trend and Seasonaiy
36 Over-Fiing Suppose we fi on a saionary ime series a Trend-Seasona Exponenia Smoohing mode: Trend seasona inerpreed noise as rend and seasonaiy! Trend seasona exaggeraed noise in-sampe! Daa Trend-Seasona SES NATCOR Exp Smooh: Trend and Seasonaiy
37 Comparison of Mehods using Informaion Crieria Le p = number of parameers (incuding Iniia Vaues used in he mode. Informaion Crieria penaize according o size of p Le n = sampe size afer aowing for he observaions used o esabish saring vaues. Akaike s Informaion Crierion Sma-sampe correced Bayesian Informaion Crierion AIC = n( MSE 2 p / n 2 p AICc = n( MSE n p 1 BIC = n( MSE p n( n / n The mode wih he owes Informaion Crierion is seeced. es has AIC, AICc and BIC; AICc is he defau. Side 37 NATCOR Exp Smooh: Trend and Seasonaiy
38 Summary Can opimize over whoe se of parameers (incuding iniia vaues. Roing Origins make more efficien use of he ou-ofsampe (es or hod-ou se. Predicion Inervas can be cacuaed using ou-of-sampe MSE for fuure forecass. Exponenia smoohing can be exended nauray for rend and/or seasona daa. Exponenia Smoohing mehods can be compared using Informaion Crieria which penaise he number of parameers. Side 38 NATCOR Exp Smooh: Trend and Seasonaiy
39 References Exponenia Smoohing Mehods Ho CC (2004 Forecasing seasonas and rends by exponeniay weighed moving averages. In J Forecasing, 20, Goodwin P (2010 The Ho-Winers approach o exponenia smoohing: 50 years od and going srong. Foresigh, Fa 2010, Forecasing Compeiions Makridakis S e a (1982 The accuracy of exrapoaion (ime series mehods: resus of a forecasing compeiion. J Forecasing, 1, Makridakis S and Hibon M (2000 The M3 Compeiion: resus, concusions and impicaions, In J Forecasing, 16, Side 39 NATCOR Exp Smooh: Trend and Seasonaiy
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