Applied Business Forecasting and Planning

AppliedBusinessForecastingandPlanningMOVINGAVERAGESANDEXPONENTIALSMOOTHINGIntroductionThischapterintroducesmodelsapplicabletotimeseriesdatawithseasonal,trend,orbothseasonalandtrendcomponentandstationarydata.Forecastingmethodsdiscussedinthischaptercanbeclassifiedas:Averagingmethods.EquallyweightedobservationsExponentialSmoothingmethods.Unequalsetofweightstopastdata,wheretheweightsdecayexponentiallyfromthemostrecenttothemostdistantdatapoints.Allmethodsinthisgrouprequirethatcertainparameterstobedefined.Theseparameters(withvaluesbetween0and1)willdeterminetheunequalweightstobeappliedtopastdata.IntroductionAveragingmethodsIfatimeseriesisgeneratedbyaconstantprocesssubjecttorandomerror,thenmeanisausefulstatisticandcanbeusedasaforecastforthenextperiod.Averagingmethodsaresuitableforstationarytimeseriesdatawheretheseriesisinequilibriumaroundaconstantvalue(theunderlyingmean)withaconstantvarianceovertime.IntroductionExponentialsmoothingmethodsThesimplestexponentialsmoothingmethodisthesinglesmoothing(SES)methodwhereonlyoneparameterneedstobeestimatedHolt’smethodmakesuseoftwodifferentparametersandallowsforecastingforserieswithtrend.Holt-Winters’methodinvolvesthreesmoothingparameterstosmooththedata,thetrend,andtheseasonalindex.AveragingMethodsTheMeanUsestheaverageofallthehistoricaldataastheforecastWhennewdatabecomesavailable,theforecastfortimet+2isthenewmeanincludingthepreviouslyobserveddataplusthisnewobservation.Thismethodisappropriatewhenthereisnonoticeabletrendorseasonality.tiitytF11111211tiitytFAveragingMethodsThemovingaveragefortimeperiodtisthemeanofthe“k”mostrecentobservations.Theconstantnumberkisspecifiedattheoutset.Thesmallerthenumberk,themoreweightisgiventorecentperiods.Thegreaterthenumberk,thelessweightisgiventomorerecentperiods.MovingAveragesAlargekisdesirablewhentherearewide,infrequentfluctuationsintheseries.Asmallkismostdesirablewhentherearesuddenshiftsinthelevelofseries.Forquarterlydata,afour-quartermovingaverage,MA(4),eliminatesoraveragesoutseasonaleffects.MovingAveragesFormonthlydata,a12-monthmovingaverage,MA(12),eliminateoraveragesoutseasonaleffect.Equalweightsareassignedtoeachobservationusedintheaverage.Eachnewdatapointisincludedintheaverageasitbecomesavailable,andtheoldestdatapointisdiscarded.MovingAveragesAmovingaverageoforderk,MA(k)isthevalueofkconsecutiveobservations.Kisthenumberoftermsinthemovingaverage.Themovingaveragemodeldoesnothandletrendorseasonalityverywellalthoughitcandobetterthanthetotalmean.tktiitkttttttykFKyyyyyF11121111)(ˆExample:WeeklyDepartmentStoreSalesTheweeklysalesfigures(inmillionsofdollars)presentedinthefollowingtableareusedbyamajordepartmentstoretodeterminetheneedfortemporarysalespersonnel.Period(t)Sales(y)15.324.435.445.855.664.875.685.695.4106.5115.1125.8135146.2155.6166.7175.2185.5195.8205.1215.8226.7235.2246255.8Example:WeeklyDepartmentStoreSalesWeeklySales012345678051015202530WeeksSalesSales(y)Example:WeeklyDepartmentStoreSalesUseathree-weekmovingaverage(k=3)forthedepartmentstoresalestoforecastfortheweek24and26.Theforecasterroris9.538.57.62.53)(ˆ21222324yyyy1.9.56ˆ242424yyeExample:WeeklyDepartmentStoreSalesTheforecastfortheweek26is7.532.568.53ˆ23242526yyyyExample:WeeklyDepartmentStoreSalesPeriod(t)Sales(y)forecast15.324.435.445.85.03333355.65.264.85.675.65.485.65.33333395.45.333333106.55.533333115.15.833333125.85.6666671355.8146.25.3155.65.666667166.75.6175.26.166667185.55.833333195.85.8205.15.5215.85.466667226.75.566667235.25.8666672465.9255.85.9666675.666667RMSE=0.63WeeklySalesForecasts012345678051015202530WeeksSalesSales(y)forecastExponentialSmoothingMethodsThismethodprovidesanexponentiallyweightedmovingaverageofallpreviouslyobservedvalues.Appropriatefordatawithnopredictableupwardordownwardtrend.Theaimistoestimatethecurrentlevelanduseitasaforecastoffuturevalue.SimpleExponentialSmoothingMethodFormally,theexponentialsmoothingequationisforecastforthenextperiod.=smoothingconstant.yt=observedvalueofseriesinperiodt.=oldforecastforperiodt.TheforecastFt+1isbasedonweightingthemostrecentobservationytwithaweightandweightingthemostrecentforecastFtwithaweightof1-tttFyF)1(11tFtFSimpleExponentialSmoothingMethodTheimplicationofexponentialsmoothingcanbebetterseenifthepreviousequationisexpandedbyreplacingFtwithitscomponentsasfollows:121111)1()1(])1()[1()1(tttttttttFyyFyyFyFSimpleExponentialSmoothingMethodIfthissubstitutionprocessisrepeatedbyreplacingFt-1byitscomponents,Ft-2byitscomponents,andsoontheresultis:Therefore,Ft+1istheweightedmovingaverageofallpastobservations.11332211)1()1()1()1(yyyyyFttttttSimpleExponentialSmoothingMethodThefollowingtableshowstheweightsassignedtopastobservationsfor=0.2,0.4,0.6,0.8,0.9Weightassignedto0.20.40.60.80.9Yt0.20.40.60.80.9Yt-10.2(1-0.2)0.4(1-0.4)0.6(1-0.6)Yt-20.2(1-0.2)20.4(1-0.4)20.6(1-0.6)2Yt-30.2(1-0.2)30.4(1-0.4)30.6(1-0.6)3Yt-40.2(1-0.2)40.4(1-0.4)40.6(1-0.6)4Yt-50.2(1-0.2)50.4(1-0.4)50.6(1-0