Time Series Modeling for Trend Analysis and Forecasting Wheat Production of India

Here, Z _t-1 , …, Z _t-p are values of past series with lag 1, …,

includingȱ investigationȱ ofȱ residualȱ plotsȱ forȱ ACFȱ

p respectively.

andȱPACF,ȱHistogram-NormalityȱandȱRandomnessȱ

testsȱ ofȱ residualsȱ i.e.,ȱ Shapiro-Wilkȱ andȱ Runȱ testsȱ

Modeling using ARIMA methodology consists of

wereȱappliedȱinȱtheȱpresentȱstudy.ȱTheȱmodelȱwithȱ

four steps viz. model identification, model estimation,

minimumȱvaluesȱofȱRMSE,ȱMAPE,ȱMAE,ȱMSE,ȱAIC,ȱ

diagnostic checking and forecasting (Sankar 2011).

SBCȱandȱwithȱhighȱR-squaredȱvalueȱwasȱconsideredȱ

asȱ anȱ appropriateȱ modelȱ forȱ forecastingȱ (Shafaqatȱ

Model identification and estimation

2012).

ModelȱidentificationȱbyȱARIMAȱ(p,ȱd,ȱq)ȱisȱbasedȱonȱtheȱ Parametric Regression model

conceptȱofȱtime-domainȱanalysisȱi.e.ȱautocorrelationȱ

Otherȱ thanȱ ARIMAȱ model,ȱ parametricȱ regressionȱ

functionȱ(ACF)ȱandȱpartialȱautocorrelationȱfunctionȱ

modelsȱ likeȱ Linear,ȱ Quadratic,ȱ Exponential,ȱ Powerȱ

(PACF).ȱTheȱACFȱandȱPACFȱplayȱvitalȱroleȱforȱtheȱ

andȱ Logarithmicȱ modelsȱ haveȱ beenȱ appliedȱ forȱ

internalȱstructureȱofȱtheȱanalyzedȱseries.ȱTheȱACFȱatȱ

modelingȱofȱwheatȱproduction.ȱTheȱmodelsȱareȱgivenȱ

lagȱkȱofȱtheȱy _t ȱseriesȱdenotesȱtheȱlinearȱcorrelationȱ

byȱEq.ȱ8ȱthroughȱEq.ȱ12:

coefficientȱbetweenȱy _t ȱandȱy _t-kȱ ,ȱcalculatedȱforȱkȱ=ȱ0,ȱ

1,ȱ2,ȱ3,..ȱandȱsoȱonȱasȱshownȱinȱEq.ȱ7.ȱTheȱPACFȱwasȱ (i)ȱLinear:ȱ

Z _t ȱ=ȱaȱ+ȱbtȱ+ȱe _tȱ

(8)

calculatedȱasȱtheȱlinearȱcorrelationȱbetweenȱy _t ȱandȱy _t-k ,ȱ

(ii)ȱQuadratic:ȱ

Z ȱ=ȱaȱ+ȱbtȱ+ȱct 2ȱ +ȱe

t

tȱ

(9)

controllingȱforȱpossibleȱeffectsȱofȱlinearȱrelationshipsȱ

amongȱvaluesȱatȱintermediateȱlags.

(iii)ȱExponential:ȱ Z _t ȱ=ȱaȱExpȱ(bt)ȱ+ȱe _tȱ

(10)

cov( y t , y t − k )

(iv)ȱPower:ȱ

Z ȱ=ȱat bȱ +ȱe

t

tȱ

(11)

ρ _k =

var( y t )var( y t − k )

ȱ

....(7)

(v)ȱLogarithmic:ȱ

Z ȱ=ȱaȱ+ȱbȱln(t)ȱ+ȱe

t

tȱ

(12)

Inȱ thisȱ presentȱ study,ȱ Augmentedȱ Dickeyȱ Fullerȱ whereȱ a,ȱ b,ȱ tȱ andȱ e _t ȱ representȱ constant,ȱ regressionȱ

(ADF)ȱ testȱ hasȱ beenȱ usedȱ toȱ findȱ unitȱ rootȱ inȱ theȱ coefficient,ȱ timeȱ andȱ errorȱ termȱ respectivelyȱ inȱ theȱ

timeȱ seriesȱ dataȱ ofȱ variableȱ underȱ considerationȱ models.

(DickeyȱandȱFullerȱ1979).ȱForȱidentificationȱofȱdataȱ

stationarity,ȱlineȱgraphȱhasȱbeenȱappliedȱtoȱrepresentȱ Exponential smoothing

theȱgraphicalȱbehaviorȱofȱobservationȱatȱlevel,ȱfirstȱ

Inȱ additionȱ toȱ aboveȱ models,ȱ Holtȱ (Doubleȱ

differenceȱandȱsoȱonȱ(GaynorȱandȱKirkpatrickȱ1994).ȱ

exponentialȱsmoothing)ȱmethodȱhasȱbeenȱemployedȱ

Onceȱtheȱorderȱofȱdifferencingȱhasȱbeenȱdiagnosed,ȱ

forȱ modelingȱ ofȱ non-seasonalȱ timeȱ seriesȱ wheatȱ

theȱdifferencedȱunivariateȱtimeȱseriesȱcanȱbeȱanalyzedȱ

productionȱdataȱwithȱtrends.ȱTheȱmodelȱisȱexpressedȱ

byȱtheȱmethodȱofȱtime-domain.

byȱ twoȱ equationsȱ toȱ dealȱ withȱ oneȱ forȱ Levelȱ (α)ȱ

Afterȱidentificationȱofȱtheȱappropriateȱpȱandȱqȱvaluesȱ andȱotherȱforȱTrendȱ(β)ȱasȱshownȱinȱEq.ȱ13ȱandȱ14ȱ

forȱtheȱmodel,ȱtheȱparameterȱofȱtheȱautoregressiveȱ respectively.ȱαȱandȱβȱcanȱassumeȱvaluesȱfromȱ0ȱtoȱ

andȱ movingȱ averageȱ termsȱ haveȱ beenȱ estimated.ȱ 1ȱwhereasȱoptimumȱvaluesȱofȱtheseȱtwoȱparametersȱ

StandardȱstatisticalȱpackageȱSASȱwasȱusedȱtoȱestimateȱ haveȱ beenȱ estimatedȱ byȱ minimizingȱ theȱ MSEȱ overȱ

relevantȱparametersȱusingȱiterativeȱprocedure.

observationsȱofȱdataȱset.

Diagnostic checking

L t = α y t + ( 1 − α ) ( L t − 1 + b t − 1 )

(13)

Theȱ estimatedȱ modelȱ wasȱ checkedȱ toȱ verifyȱ ifȱ itȱ

b = β ( L t − L t − 1 ) + (1 − β ) b t − 1

t

(14)

adequatelyȱrepresentsȱtheȱseriesȱorȱnotȱfurther.ȱForȱ

whereȱL ȱdenotesȱestimateȱofȱtheȱlevelȱandȱb ȱisȱtheȱ

t

t

evaluatingȱtheȱadequacyȱofȱARIMAȱprocess,ȱvariousȱ

trendȱ(slope)ȱofȱtheȱseriesȱatȱtimeȱt.

reliabilityȱstatisticsȱhaveȱbeenȱused.ȱDiagnosticȱchecksȱ

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