商务与经济统计第18章A课件

StatisticsforBusiness

andEconomicsAndersonSweeneyWilliamsSlidesbyJohnLoucksSt.Edward’sUniversityStatisticsforBusiness

andEcChapter18,PartA

ForecastingQuantitativeApproachestoForecastingTimeSeriesPatternsForecastAccuracyMovingAveragesandExponentialSmoothingChapter18,PartA

ForecastingForecastingMethodsForecastingmethodscanbeclassifiedasqualitativeorquantitative.Suchmethodsareappropriatewhenhistoricaldataonthevariablebeingforecastareeithernotapplicableorunavailable.Qualitativemethods

generallyinvolvetheuseofexpertjudgmenttodevelopforecasts.Wewillfocusexclusivelyonquantitativeforecastingmethodsinthischapter.ForecastingMethodsForecastingForecastingMethodsQuantitativeforecastingmethodscanbeusedwhen:Insuchcases,aforecastcanbedevelopedusingatimeseriesmethodoracausalmethod.pastinformationaboutthevariablebeingforecastisavailable,theinformationcanbequantified,anditisreasonabletoassumethatthepatternofthepastwillcontinueintothefuture.ForecastingMethodsQuantitativQuantitativeForecastingMethodsQuantitativemethodsarebasedonananalysisofhistoricaldataconcerningoneormoretimeseries.Atimeseriesisasetofobservationsmeasuredatsuccessivepointsintimeoroversuccessiveperiodsoftime.Ifthehistoricaldatausedarerestrictedtopastvaluesoftheseriesthatwearetryingtoforecast,theprocedureiscalledatimeseriesmethod.Ifthehistoricaldatausedinvolveothertimeseriesthatarebelievedtoberelatedtothetimeseriesthatwearetryingtoforecast,theprocedureiscalledacausalmethod.QuantitativeForecastingMethoTimeSeriesMethodsTheobjectiveoftimeseriesanalysisistodiscoverapatterninthehistoricaldataortimeseriesandthenextrapolatethepatternintothefuture.Theforecastisbasedsolelyonpastvaluesofthevariableand/orpastforecasterrors.TimeSeriesMethodsTheobjectiCausalMethodsCausalforecastingmethodsarebasedontheassumptionthatthevariableweareforecastinghasacause-effectrelationshipwithoneormoreothervariables.Lookingatregressionanalysisasaforecastingtool,wecanviewthetimeseriesvaluethatwewanttoforecastasthedependentvariable.Ifwecanidentifyagoodsetofrelatedindependent,orexplanatory,variableswemaybeabletodevelopanestimatedregressionequationforforecastingthetimeseries.CausalMethodsCausalforecastiRegressionAnalysisBytreatingtimeastheindependentvariableandthetimeseriesasadependentvariable,regressionanalysiscanalsobeusedasatimeseriesmethod.Time-seriesregressionreferstotheuseofregressionanalysiswhenthesoleindependentvariableistime.Cross-sectionalregressionreferstotheuseofregressionanalysiswhentheindependentvariable(s)is(are)somethingotherthantime.RegressionAnalysisBytreatingForecastingMethodsForecastingMethodsQuantitativeQualitativeCausalTimeSeriesForecastingMethodsForecastingTimeSeriesPatternsAtimeseriesisasequenceofmeasurementstakeneveryhour,day,week,month,quarter,year,oratanyotherregulartimeinterval.Thepatternofthedataisanimportantfactorinunderstandinghowthetimeserieshasbehavedinthepast.Ifsuchbehaviorcanbeexpectedtocontinueinthefuture,wecanuseittoguideusinselectinganappropriateforecastingmethod.TimeSeriesPatternsAtimeserTimeSeriesPlotAtimeseriesplotisagraphicalpresentationoftherelationshipbetweentimeandthetimeseriesvariable.Timeisonthehorizontalaxis,andthetimeseriesvaluesareshownontheverticalaxis.Ausefulfirststepinselectinganappropriateforecastingmethodistoconstructatimeseriesplot.TimeSeriesPlotAtimeseriesTimeSeriesPatternsThecommontypesofdatapatternsthatcanbeidentifiedwhenexaminingatimeseriesplotinclude:HorizontalTrendSeasonalCyclicalTrend&SeasonalTimeSeriesPatternsThecommonTimeSeriesPatternsHorizontalPatternAhorizontalpatternexistswhenthedatafluctuatearoundaconstantmean.Changesinbusinessconditionscanoftenresultinatimeseriesthathasahorizontalpatternshiftingtoanewlevel.Achangeinthelevelofthetimeseriesmakesitmoredifficulttochooseanappropriateforecastingmethod.TimeSeriesPatternsHorizontalTimeSeriesPatternsTrendPatternAtimeseriesmayshowgradualshiftsormovementstorelativelyhigherorlowervaluesoveralongerperiodoftime.Trendisusuallytheresultoflong-termfactorssuchaschangesinthepopulation,demographics,technology,orconsumerpreferences.Asystematicincreaseordecreasemightbelinearornonlinear.Atrendpatterncanbeidentifiedbyanalyzingmultiyearmovementsinhistoricaldata.TimeSeriesPatternsTrendPattTimeSeriesPatternsSeasonalpatternsarerecognizedbyseeingthesamerepeatingpatternofhighsandlowsoversuccessiveperiodsoftimewithinayear.SeasonalPatternAseasonalpatternmightoccurwithinaday,week,month,quarter,year,orsomeotherintervalnogreaterthanayear.Aseasonalpatterndoesnotnecessarilyrefertothefourseasonsoftheyear(spring,summer,fall,andwinter).TimeSeriesPatternsSeasonalpTimeSeriesPatternsSometimeseriesincludeacombinationofatrendandseasonalpattern.TrendandSeasonalPatternInsuchcasesweneedtouseaforecastingmethodthathasthecapabilitytodealwithbothtrendandseasonality.Timeseriesdecompositioncanbeusedtoseparateordecomposeatimeseriesintotrendandseasonalcomponents.TimeSeriesPatternsSometimeTimeSeriesPatternsAcyclicalpatternexistsifthetimeseriesplotshowsanalternatingsequenceofpointsbelowandabovethetrendlinelastingmorethanoneyear.CyclicalPatternOften,thecyclicalcomponentofatimeseriesisduetomultiyearbusinesscycles.Businesscyclesareextremelydifficult,ifnotimpossible,toforecast.Inthischapterwedonotdealwithcyclicaleffectsthatmaybepresentinthetimeseries.TimeSeriesPatternsAcyclicalSelectingaForecastingMethodTheunderlyingpatterninthetimeseriesisanimportantfactorinselectingaforecastingmethod.Thus,atimeseriesplotshouldbeoneofthefirstthingsdevelopedwhentryingtodeterminewhatforecastingmethodtouse.Ifweseeahorizontalpattern,thenweneedtoselectamethodappropriateforthistypeofpattern.Ifweobserveatrendinthedata,thenweneedtouseamethodthathasthecapabilitytohandletrendeffectively.SelectingaForecastingMethodForecastAccuracyMeasuresofforecastaccuracyareusedtodeterminehowwellaparticularforecastingmethodisabletoreproducethetimeseriesdatathatarealreadyavailable.Byselectingthemethodthathasthebestaccuracyforthedataalreadyknown,wehopetoincreasethelikelihoodthatwewillobtainbetterforecastsforfuturetimeperiods.Measuresofforecastaccuracyareimportantfactorsincomparingdifferentforecastingmethods.ForecastAccuracyMeasuresoffForecastAccuracyThekeyconceptassociatedwithmeasuringforecastaccuracyisforecasterror.Apositiveforecasterrorindicatestheforecastingmethodunderestimatedtheactualvalue.ForecastError=ActualValue-ForecastAnegativeforecasterrorindicatestheforecastingmethodoverestimatedtheactualvalue.ForecastAccuracyThekeyconceForecastAccuracy Asimplemeasureofforecastaccuracyisthemean

oraverageoftheforecasterrors.Becausepositiveand

negativeforecasterrorstendtooffsetoneanother,the

meanerrorislikelytobesmall.Thus,themeanerror

isnotaveryusefulmeasure.

Thismeasureavoidstheproblemofpositiveandnegativeerrorsoffsettingoneanother.Itisthemeanoftheabsolutevaluesoftheforecasterrors.MeanErrorMeanAbsoluteError(MAE)ForecastAccuracy AsimplForecastAccuracy Thisisanothermeasurethatavoidstheproblemofpositiveandnegativeerrorsoffsettingoneanother.Itistheaverageofthesquaredforecasterrors.

ThesizeofMAEandMSEdependuponthescaleofthedata,soitisdifficulttomakecomparisonsfordifferenttimeintervals.Tomakesuchcomparisonsweneedtoworkwithrelativeorpercentageerrormeasures.TheMAPEistheaverageoftheabsolute

percentageerrorsoftheforecasts.MeanSquaredError(MSE)MeanAbsolutePercentageError(MAPE)ForecastAccuracy ThisisForecastAccuracyTodemonstratethecomputationofthesemeasuresofforecastaccuracywewillintroducethesimplestofforecastingmethods.Thenaïveforecastingmethodusesthemostrecentobservationinthetimeseriesastheforecastforthenexttimeperiod.Ft+1=ActualValueinPeriodtForecastAccuracyTodemonstrat

SalesofComfortbrandheadachemedicineforthepast10weeksatRoscoDrugsareshownbelow.Example:RoscoDrugs12345678910110115125120125120130115110130WeekWeekSalesSalesForecastAccuracy

IfRoscousesthenaïveforecastmethodtoforecastsalesforweeks2–10,whataretheresultingMAE,MSE,andMAPEvalues? SalesofComfortbrandh12345678910110115125120125120130115110130125120130115110125120WeekSalesNaïveForecast-510-15-520-55ForecastErrorAbsoluteErrorSquaredError51015520805525100125254008502525Abs.%Error4.177.6913.044.5515.3865.354.174.00ForecastAccuracy110115510510100254.358.00Total1110125125120WeekSalesNaïve-MovingAveragesNaiveForecastAccuracyMovingAveragesNaiveForecastMovingAveragesandExponentialSmoothingNowwediscussthreeforecastingmethodsthatareappropriateforatimeserieswithahorizontalpattern:ExponentialSmoothingWeightedMovingAveragesMovingAveragesTheyarecalledsmoothingmethodsbecausetheirobjectiveistosmoothouttherandomfluctuationsinthetimeseries.Theyaremostappropriateforshort-rangeforecasts.MovingAveragesandExponentia Themovingaveragesmethodusestheaverageofthemostrecentkdatavaluesinthetimeseries.Astheforecastforthenextperiod.MovingAverageswhere:Ft+1=forecastofthetimeseriesforperiodt+1 Eachobservationinthemovingaveragecalculationreceivesthesameweight. ThemovingaveragesmethoduMovingAveragesThetermmovingisusedbecauseeverytimeanewobservationbecomesavailableforthetimeseries,itreplacestheoldestobservationintheequation.Asaresult,theaveragewillchange,ormove,asnewobservationsbecomeavailable.MovingAveragesThetermmovingMovingAveragesIfmorepastobservationsareconsideredrelevant,thenalargervalueofkisbetter.Asmallervalueofkwilltrackshiftsinatimeseriesmorequicklythanalargervalueofk.Tousemovingaveragestoforecast,wemustfirstselecttheorderk,ornumberoftimeseriesvalues,tobeincludedinthemovingaverage.MovingAveragesIfmorepastob

IfRoscoDrugsusesa3-periodmovingaverageto

forecastsales,whataretheforecastsforweeks4-11?Example:RoscoDrugsMovingAverages12345678910110115125120125120130115110130WeekWeekSalesSales IfRoscoDrugsusesa3-1234567891011110115125120125120130115110130WeekSalesMovingAverages123.3121.7125.0121.7118.3118.3116.7120.03MAForecast(110+115+125)/31110WeekSalesMovingAverages12MovingAverages12345678910110115125120125120130115110130123.3121.7125.0121.7118.3116.7120.0WeekSales3MAForecast-3.38.3-10.0-11.75.0ForecastErrorAbsoluteErrorSquaredError3.38.310.011.711.710.8968.89100.00136.89136.89489.4510.8925.00Abs.%Error2.756.388.7010.649.0044.222.754.00TotalMovingAverages1110123.3116.71MovingAverages3-MAForecastAccuracyThe3-weekmovingaverageapproachprovidedmoreaccurateforecaststhanthenaïveapproach.MovingAverages3-MAForecastAWeightedMovingAveragesWeightedMovingAveragesThemorerecentobservationsaretypicallygivenmoreweightthanolderobservations.Forconvenience,theweightsshouldsumto1.Tousethismethodwemustfirstselectthenumberofdatavaluestobeincludedintheaverage.Next,wemustchoosetheweightforeachofthedatavalues.WeightedMovingAveragesWeightWeightedMovingAveragesAnexampleofa3-periodweightedmovingaverage(3WMA)is:3WMA=.2(110)+.3(115)+.5(125)=119MostrecentofthethreeobservationsWeights(.2,.3,and.5)sumto1WeightedMovingAveragesWeightedMovingAveragesAnexaExponentialSmoothingThismethodisaspecialcaseofaweightedmovingaveragesmethod;weselectonlytheweightforthemostrecentobservation.Theweightsfortheotherdatavaluesarecomputedautomaticallyandbecomesmallerastheobservationsgrowolder.Theexponentialsmoothingforecastisaweightedaverageofalltheobservationsinthetimeseries.Thetermexponentialsmoothingcomesfromtheexponentialnatureoftheweightingschemeforthehistoricalvalues.ExponentialSmoothingThismethExponentialSmoothingExponentialSmoothingForecastFt+1=aYt+(1–a)Ftwhere:Ft+1=forecastofthetimeseriesforperiodt+1Yt=actualvalueofthetimeseriesinperiodtFt=forecastofthetimeseriesforperiodta=smoothingconstant(0<

a

<1)andlet:F2

=Y1(toinitiatethecomputations)ExponentialSmoothingExponentiWithsomealgebraicmanipulation,wecanrewriteFt+1=aYt+(1–a)Ftas:ExponentialSmoothingExponentialSmoothingForecastFt+1=Ft+a(Yt–Ft)WeseethatthenewforecastFt+1isequaltothepreviousforecastFtplusanadjustment,whichisatimesthemostrecentforecasterror,Yt–Ft.Withsomealgebraicmanipulati

IfRoscoDrugsusesexponentialsmoothingtoforecastsales,whichvalueforthesmoothingconstant,.1or.8,givesbetterforecasts?Example:RoscoDrugsExponentialSmoothing12345678910110115125120125120130115110130WeekWeekSalesSales IfRoscoDrugsusesexponentExponentialSmoothingUsingSmoothingConstantValue

=.1 F2=Y1=110 F3=.1Y2+.9F2=.1(115)+.9(110)=110.5 F4=.1Y3+.9F3=.1(125)+.9(110.5)=111.95 F5=.1Y4+.9F4=.1(120)+.9(111.95)=112.76 F6=.1Y5+.9F5=.1(125)+.9(112.76)=113.98 F7=.1Y6+.9F6=.1(120)+.9(113.98)=114.58 F8=.1Y7+.9F7=.1(130)+.9(114.58)=116.12 F9=.1Y8+.9F8=.1(115)+.9(116.12)=116.01 F10=.1Y9+.9F9=.1(110)+.9(116.01)=115.41ExponentialSmoothingUsingSmoExponentialSmoothingUsingSmoothingConstantValue

=.8 F2=

=110 F3=.8(115)+.2(110)=114 F4=.8(125)+.2(114)=122.80 F5=.8(120)+.2(122.80)=120.56 F6=.8(125)+.2(120.56)=124.11 F7=.8(120)+.2(124.11)=120.82 F8=.8(130)+.2(120.82)=128.16 F9=.8(115)+.2(128.16)=117.63 F10=.8(110)+.2(117.63)=111.53ExponentialSmoothingUsingSmo12345678910110115125120125120130115110130WeekSales113.98114.58116.12116.01115.41111.95112.76a=.1Forecast110.00110.50ExponentialSmoothing(a=.1)6.0215.42-1.12-6.0114.598.0512.24ForecastErrorAbsoluteErrorSquaredError6.0215.421.126.0114.5982.958.0512.2436.25237.731.2636.12212.87974.2264.80149.94Abs.%Error5.0211.860.975.4611.2266.986.719.795.0014.505.0014.50210.2525.004.3511.60Total1110WeekSales113.98111.95112.7ExponentialSmoothing(a=.1)ForecastAccuracyExponentialsmoothing(witha=.1)providedlessaccurateforecaststhanthe3-MAapproach.ExponentialSmoothing(a=.1)12345678910110115125120125120130115110130124.11120.82128.16117.63111.53122.80120.56WeekSalesa=.8Forecast110.00114.00ExponentialSmoothing(a=.8)-4.119.18-13.16-7.6318.47-2.204.44ForecastErrorAbsoluteErrorSquaredError4.119.1813.167.6318.47416.9184.23173.3058.26341.27847.527.8419.71Abs.%Error3.437.0611.446.9414.2161.611.833.555.0011.005.0011.00121.0025.004.358.80Total1110124.11122.80120.56WeekSaleExponentialSmoothing(a=.8)ForecastAccuracyExponentialsmoothing(witha=.8)provided

moreaccurateforecaststhanESwitha=.1,butlessaccuratethanthemovingaverage(withk=3).ExponentialSmoothing(a=.8)EndofChapter18,PartAEndofChapter18,PartAStatisticsforBusiness

andEconomicsAndersonSweeneyWilliamsSlidesbyJohnLoucksSt.Edward’sUniversityStatisticsforBusiness

andEcChapter18,PartA

ForecastingQuantitativeApproachestoForecastingTimeSeriesPatternsForecastAccuracyMovingAveragesandExponentialSmoothingChapter18,PartA

ForecastingForecastingMethodsForecastingmethodscanbeclassifiedasqualitativeorquantitative.Suchmethodsareappropriatewhenhistoricaldataonthevariablebeingforecastareeithernotapplicableorunavailable.Qualitativemethods

generallyinvolvetheuseofexpertjudgmenttodevelopforecasts.Wewillfocusexclusivelyonquantitativeforecastingmethodsinthischapter.ForecastingMethodsForecastingForecastingMethodsQuantitativeforecastingmethodscanbeusedwhen:Insuchcases,aforecastcanbedevelopedusingatimeseriesmethodoracausalmethod.pastinformationaboutthevariablebeingforecastisavailable,theinformationcanbequantified,anditisreasonabletoassumethatthepatternofthepastwillcontinueintothefuture.ForecastingMethodsQuantitativQuantitativeForecastingMethodsQuantitativemethodsarebasedonananalysisofhistoricaldataconcerningoneormoretimeseries.Atimeseriesisasetofobservationsmeasuredatsuccessivepointsintimeoroversuccessiveperiodsoftime.Ifthehistoricaldatausedarerestrictedtopastvaluesoftheseriesthatwearetryingtoforecast,theprocedureiscalledatimeseriesmethod.Ifthehistoricaldatausedinvolveothertimeseriesthatarebelievedtoberelatedtothetimeseriesthatwearetryingtoforecast,theprocedureiscalledacausalmethod.QuantitativeForecastingMethoTimeSeriesMethodsTheobjectiveoftimeseriesanalysisistodiscoverapatterninthehistoricaldataortimeseriesandthenextrapolatethepatternintothefuture.Theforecastisbasedsolelyonpastvaluesofthevariableand/orpastforecasterrors.TimeSeriesMethodsTheobjectiCausalMethodsCausalforecastingmethodsarebasedontheassumptionthatthevariableweareforecastinghasacause-effectrelationshipwithoneormoreothervariables.Lookingatregressionanalysisasaforecastingtool,wecanviewthetimeseriesvaluethatwewanttoforecastasthedependentvariable.Ifwecanidentifyagoodsetofrelatedindependent,orexplanatory,variableswemaybeabletodevelopanestimatedregressionequationforforecastingthetimeseries.CausalMethodsCausalforecastiRegressionAnalysisBytreatingtimeastheindependentvariableandthetimeseriesasadependentvariable,regressionanalysiscanalsobeusedasatimeseriesmethod.Time-seriesregressionreferstotheuseofregressionanalysiswhenthesoleindependentvariableistime.Cross-sectionalregressionreferstotheuseofregressionanalysiswhentheindependentvariable(s)is(are)somethingotherthantime.RegressionAnalysisBytreatingForecastingMethodsForecastingMethodsQuantitativeQualitativeCausalTimeSeriesForecastingMethodsForecastingTimeSeriesPatternsAtimeseriesisasequenceofmeasurementstakeneveryhour,day,week,month,quarter,year,oratanyotherregulartimeinterval.Thepatternofthedataisanimportantfactorinunderstandinghowthetimeserieshasbehavedinthepast.Ifsuchbehaviorcanbeexpectedtocontinueinthefuture,wecanuseittoguideusinselectinganappropriateforecastingmethod.TimeSeriesPatternsAtimeserTimeSeriesPlotAtimeseriesplotisagraphicalpresentationoftherelationshipbetweentimeandthetimeseriesvariable.Timeisonthehorizontalaxis,andthetimeseriesvaluesareshownontheverticalaxis.Ausefulfirststepinselectinganappropriateforecastingmethodistoconstructatimeseriesplot.TimeSeriesPlotAtimeseriesTimeSeriesPatternsThecommontypesofdatapatternsthatcanbeidentifiedwhenexaminingatimeseriesplotinclude:HorizontalTrendSeasonalCyclicalTrend&SeasonalTimeSeriesPatternsThecommonTimeSeriesPatternsHorizontalPatternAhorizontalpatternexistswhenthedatafluctuatearoundaconstantmean.Changesinbusinessconditionscanoftenresultinatimeseriesthathasahorizontalpatternshiftingtoanewlevel.Achangeinthelevelofthetimeseriesmakesitmoredifficulttochooseanappropriateforecastingmethod.TimeSeriesPatternsHorizontalTimeSeriesPatternsTrendPatternAtimeseriesmayshowgradualshiftsormovementstorelativelyhigherorlowervaluesoveralongerperiodoftime.Trendisusuallytheresultoflong-termfactorssuchaschangesinthepopulation,demographics,technology,orconsumerpreferences.Asystematicincreaseordecreasemightbelinearornonlinear.Atrendpatterncanbeidentifiedbyanalyzingmultiyearmovementsinhistoricaldata.TimeSeriesPatternsTrendPattTimeSeriesPatternsSeasonalpatternsarerecognizedbyseeingthesamerepeatingpatternofhighsandlowsoversuccessiveperiodsoftimewithinayear.SeasonalPatternAseasonalpatternmightoccurwithinaday,week,month,quarter,year,orsomeotherintervalnogreaterthanayear.Aseasonalpatterndoesnotnecessarilyrefertothefourseasonsoftheyear(spring,summer,fall,andwinter).TimeSeriesPatternsSeasonalpTimeSeriesPatternsSometimeseriesincludeacombinationofatrendandseasonalpattern.TrendandSeasonalPatternInsuchcasesweneedtouseaforecastingmethodthathasthecapabilitytodealwithbothtrendandseasonality.Timeseriesdecompositioncanbeusedtoseparateordecomposeatimeseriesintotrendandseasonalcomponents.TimeSeriesPatternsSometimeTimeSeriesPatternsAcyclicalpatternexistsifthetimeseriesplotshowsanalternatingsequenceofpointsbelowandabovethetrendlinelastingmorethanoneyear.CyclicalPatternOften,thecyclicalcomponentofatimeseriesisduetomultiyearbusinesscycles.Businesscyclesareextremelydifficult,ifnotimpossible,toforecast.Inthischapterwedonotdealwithcyclicaleffectsthatmaybepresentinthetimeseries.TimeSeriesPatternsAcyclicalSelectingaForecastingMethodTheunderlyingpatterninthetimeseriesisanimportantfactorinselectingaforecastingmethod.Thus,atimeseriesplotshouldbeoneofthefirstthingsdevelopedwhentryingtodeterminewhatforecastingmethodtouse.Ifweseeahorizontalpattern,thenweneedtoselectamethodappropriateforthistypeofpattern.Ifweobserveatrendinthedata,thenweneedtouseamethodthathasthecapabilitytohandletrendeffectively.SelectingaForecastingMethodForecastAccuracyMeasuresofforecastaccuracyareusedtodeterminehowwellaparticularforecastingmethodisabletoreproducethetimeseriesdatathatarealreadyavailable.Byselectingthemethodthathasthebestaccuracyforthedataalreadyknown,wehopetoincreasethelikelihoodthatwewillobtainbetterforecastsforfuturetimeperiods.Measuresofforecastaccuracyareimportantfactorsincomparingdifferentforecastingmethods.ForecastAccuracyMeasuresoffForecastAccuracyThekeyconceptassociatedwithmeasuringforecastaccuracyisforecasterror.Apositiveforecasterrorindicatestheforecastingmethodunderestimatedtheactualvalue.ForecastError=ActualValue-ForecastAnegativeforecasterrorindicatestheforecastingmethodoverestimatedtheactualvalue.ForecastAccuracyThekeyconceForecastAccuracy Asimplemeasureofforecastaccuracyisthemean

oraverageoftheforecasterrors.Becausepositiveand

negativeforecasterrorstendtooffsetoneanother,the

meanerrorislikelytobesmall.Thus,themeanerror

isnotaveryusefulmeasure.

Thismeasureavoidstheproblemofpositiveandnegativeerrorsoffsettingoneanother.Itisthemeanoftheabsolutevaluesoftheforecasterrors.MeanErrorMeanAbsoluteError(MAE)ForecastAccuracy AsimplForecastAccuracy Thisisanothermeasurethatavoidstheproblemofpositiveandnegativeerrorsoffsettingoneanother.Itistheaverageofthesquaredforecasterrors.

ThesizeofMAEandMSEdependuponthescaleofthedata,soitisdifficulttomakecomparisonsfordifferenttimeintervals.Tomakesuchcomparisonsweneedtoworkwithrelativeorpercentageerrormeasures.TheMAPEistheaverageoftheabsolute

percentageerrorsoftheforecasts.MeanSquaredError(MSE)MeanAbsolutePercentageError(MAPE)ForecastAccuracy ThisisForecastAccuracyTodemonstratethecomputationofthesemeasuresofforecastaccuracywewillintroducethesimplestofforecastingmethods.Thenaïveforecastingmethodusesthemostrecentobservationinthetimeseriesastheforecastforthenexttimeperiod.Ft+1=ActualValueinPeriodtForecastAccuracyTodemonstrat

SalesofComfortbrandheadachemedicineforthepast10weeksatRoscoDrugsareshownbelow.Example:RoscoDrugs12345678910110115125120125120130115110130WeekWeekSalesSalesForecastAccuracy

IfRoscousesthenaïveforecastmethodtoforecastsalesforweeks2–10,whataretheresultingMAE,MSE,andMAPEvalues? SalesofComfortbrandh12345678910110115125120125120130115110130125120130115110125120WeekSalesNaïveForecast-510-15-520