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INTRODUCTION TO FORECASTING

(PART 2)AMAT 167

Techniques for Trend

EXAMPLEOFTRENDS

Inourdiscussion,wewillfocusonlineartrend…

buthereareexamplesofnon-lineartrends:

EXAMPLEOFTRENDS

Ifyouwanttostudyothertechniquesforfindingtrends,studysignalanalysis.

LINEARTRENDEQUATION

LINEARTRENDEQUATION

HOWTOCOMPUTEFORa ANDb FROMDATA

• Wecanusetheformulainlinearregression• MSExcelalsohasafeatureforautomaticallycomputingtheparameters• InMSExcel,youcanalsocomputeforR2.

TREND-ADJUSTEDEXPONENTIALSMOOTHING(DOUBLESMOOTHING)

• Thisisusedwhendatavaryaroundanaverageorhavesteporgradualchanges.• Ifaseriesexhibitstrend,andsimplesmoothingisusedonit,theforecastswillalllagthetrend:Ifthedataareincreasing,eachforecastwillbetoolow;ifdecreasing,eachforecastwillbetoohigh.

TREND-ADJUSTEDEXPONENTIALSMOOTHING(DOUBLESMOOTHING)

Thetrend-adjustedforecast(TAF)is

Estimatefortheslope

EXAMPLE

Period Actual(At) Tt St TAFt Error1 7002 724 24 7003 720 10 724 724 -44 728 10 731.6 734 -65 740 9.28 740.96 741.6 -1.66 742 9.088 746.944 750.24 -8.247 758 8.0992 756.8192 756.032 1.9688 750 8.33536 758.95104 764.9184 -14.91849 770 6.545152 768.37184 767.2864 2.713610 775 6.870784 774.950195 774.916992 0.08300811 FUTURE 781.820979

alpha=0.4beta=0.3

EXAMPLE

690700710720730740750760770780790

0 2 4 6 8 10 12

Actual(At) TAFt SimpleExponentialSmoothing

Techniques for Seasonality

EXAMPLEOFSEASONALITY

• weathervariations(e.g.,salesofwinterandsummersportsequipment)• vacationsorholidays(e.g.,airlinetravel,greetingcardsales,visitorsattouristandresortcenters)• daily,weekly,monthly,andotherregularlyrecurringpatternsindata(e.g.,rushhourtrafficoccurstwiceaday—incominginthemorningandoutgoinginthelateafternoon)

EXPRESSIONOFSEASONALITY

• Iftheseriestendstovaryaroundanaveragevalue,thenseasonalityisexpressedintermsofthataverage(oramovingaverage).• Iftrendispresent,seasonalityisexpressedintermsofthetrendvalue.

MODELSOFSEASONALITY

expressed as a quantity (e.g., 20 units), which is added to or subtracted from the series average

expressed as a percentage of the average (or trend) amount (e.g., 1.10), which is then used to multiply the value of a series

WEWILLUSETHEMULTIPLICATIVEMODEL(COMMONLYUSED)

• Seasonalpercentagesinthemultiplicativemodelarereferredtoasseasonalrelativesorseasonalindexes.• SupposethattheseasonalrelativeforthequantityoftoyssoldinMayatastoreis1.20.Thisindicatesthattoysalesforthatmonthare20percentabovethemonthlyaverage.• Aseasonalrelativeof.90forJulyindicatesthatJulysalesare90percentofthemonthlyaverage.

REMARKS

• Knowledgeoftheextentofseasonalityinatimeseriescanenableonetoremoveseasonalityfromthedata(i.e.,toseasonallyadjustdata)inordertodiscernotherpatternsorthelackofpatternsintheseries.• Seasonalrelativesareusedintwodifferentwaysinforecasting.Onewayistodeseasonalize data;theotherwayistoincorporateseasonalityinaforecast.

DESEASONALIZING THEDATA

• Wewanttoremovetheseasonalcomponentfromthedatainordertogetaclearerpictureofthenonseasonal (e.g.,trend)components.• Deseasonalizing dataisaccomplishedbydividingeachdatapointbyitscorrespondingseasonalrelative(e.g.,divideNovemberdemandbytheNovemberrelative,divideDecemberdemandbytheDecemberrelative,andsoon).

EXAMPLE

Period Quarter Sales QuarterRelative(given) DeseasonalizedSales1 1 158.4 1.2 1322 2 153 1.1 139.09090913 3 110 0.75 146.66666674 4 146.3 0.95 1545 1 192 1.2 1606 2 187 1.1 1707 3 132 0.75 1768 4 173.8 0.95 182.9473684

EXAMPLE

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0 2 4 6 8 10

Sales DeseasonalizedSales

INCORPORATINGSEASONALITYINAFORECAST

Steps:1. Obtaintrendestimatesfordesiredperiodsusingatrend

equation.2. Addseasonalitytothetrendestimatesbymultiplyingthesetrend

estimatesbythecorrespondingseasonalrelative(e.g.,multiplytheNovembertrendestimatebytheNovemberseasonalrelative,multiplytheDecembertrendestimatebytheDecemberseasonalrelative,andsoon).

EXAMPLE

y=3.3274x+141.59R²=0.08656

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100

150

200

250

0 2 4 6 8 10

Sales Linear(Sales)

Note:Youcanalsousethetrendlinegeneratedbythedeseasonalizeddatawhichisy=7.3473x +124.53

R² =0.99837

EXAMPLE

Period Quarter Sales QuarterRelative(given) SeasonalizedSales1 1 158.4 1.22 2 153 1.13 3 110 0.754 4 146.3 0.955 1 192 1.26 2 187 1.17 3 132 0.758 4 173.8 0.959 1 171.5366 1.2 205.84392

(forecastusingtrendline)

COMPUTINGSEASONALRELATIVESUSINGSIMPLEAVERAGEMETHOD

• Whenthedatahaveastationarymean(i.e.,variationaroundanaverage),theSAmethodworksquitewell.• Itcanbeusedtoobtainfairlygoodvaluesofseasonalrelativesaslongasthevariations(seasonalandrandom)aroundthetrendlinearelargerelativetotheslopeoftheline.• AnotherapproachistheCenteredMovingAverage:Thisapproacheffectivelyaccountsforanytrend(linearorcurvilinear)thatmightbepresentinthedata.

COMPUTINGSEASONALRELATIVESUSINGSIMPLEAVERAGEMETHOD

EXAMPLE

STEP1 STEP2Season Week1 Week2 Week3 SeasonAverage SAIndexTues 67 60 64 63.66666667 0.88955422Wed 75 73 76 74.66666667 1.04324684Thurs 82 85 87 84.66666667 1.1829674Fri 98 99 96 97.66666667 1.36460413Sat 90 86 88 88 1.22954092Sun 36 40 44 40 0.55888224Mon 55 52 50 52.33333333 0.73120426

OVERALLAVERAGE= 71.57142857

NOTES ABOUT ASSOCIATIVE/CASUAL FORECASTING: LINEAR REGRESSION

WARNING!

Wewillnotdiscusslinearregressionanymore.However,pleasetakenoteofthefollowing(youknowtheseifyouhavetakenSTAT101,STAT166etc).Useofsimpleregressionanalysisimpliesthatcertainassumptionshavebeensatisfied,suchas• Variationsaroundthelinearerandom.Iftheyarerandom,nopatternssuchascyclesortrendsshouldbeapparentwhenthelineanddataareplotted.

• Deviationsaroundtheaveragevalue(i.e.,theline)shouldbenormallydistributed.

Moreover,oneneedsaconsiderableamountofdatatoestablishthelinearrelationship—inpractice,20ormoreobservations.

Monitoring the ForecastUsing Control Chart

CONTROLCHART

DETECTNONRANDOMNESS,EXAMPLES:

CREATINGCONTROLCHARTS

• Controlchartsarebasedontheassumptionthatwhenerrors(Actual-Forecast)arerandom,theywillbedistributedaccordingtoanormaldistributionaroundameanofzero.• Recallthatforanormaldistribution,approximately95.5percentofthevalues(errorsinthiscase)canbeexpectedtofallwithinlimitsof0±2s (i.e.,0±2standarddeviations),andapproximately99.7percentofthevaluescanbeexpectedtofallwithin±3s ofzero.

• Weapproximates, 𝑠 ≈ 𝑀𝑆𝐸�

CREATINGCONTROLCHARTS

Forexample,z canbe2or3.

DETECTINGBIAS

A value of zero would be ideal; limits of ±4 or ±5 are often used for a range of acceptable values of the tracking signal.

DETECTINGBIAS(ALTERNATIVE)

MAD that is updated and smoothed (SMAD) using exponential smoothing