Chapter
5
NonlinearRe
gression
Nonlinearregressionisamethodoffindinganonlinearmodeloftherelationship
betweenthe
dependentvariableandasetofindependentvariables.Unliketraditional
linearregression,whichisrestrictedtoestimatinglinearmodels,nonlinearregression
canestimatemodelswitharbitraryrelationshipsbetweenindependentanddependent
variables
.Thisisaccomplishedusingiterativeestimationalgorithms.Notethatthis
procedureisnotnecessaryforsimplepolynomialmodelsoftheformY=A+BX**2.
BydefiningW=X**2,wegetasimplelinearmodel,Y=A+BW,whichcanbe
estimate
dusingtraditionalmethodssuchastheLinearRegressionprocedure.
Example.Canpopulationbepredictedbasedontime?Ascatterplotshowsthatthere
seemstobeastrongrelationshipbetweenpopulationandtime,buttherelationshipis
nonlinea
r,soitrequiresthespecialestimationmethodsoftheNonlinearRegression
procedure.Bysettingupanappropriateequation,suchasalogisticpopulationgrowth
model,wecangetagoodestimateofthemodel,allowingustomakepredictions
aboutpo
pulationfortimesthatwerenotactuallymeasured.
Statistics.Foreachiteration:parameterestimatesandresidualsumofsquares.For
eachmodel:sumofsquaresforregression,residual,uncorrectedtotalandcorrected
total,
parameterestimates,asymptoticstandarderrors,andasymptoticcorrelation
matrixofparameterestimates.
Note: Constrainednonlinearregressionusesthealgorithmsproposedand
implementedinNPSOL®byGill,Murray,Saunders,andWrighttoestimatethe
modelp
arameters.
31
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32
Chapter5
NonlinearRegressionDataConsiderations
Data.Thedependentandindependentvariablesshouldbequantitative.Categorical
variables,suchasreligion,major,orregionofresidence,needtoberecodedtobinary
(dummy)variablesorothertypesofcontrastvariables.
Assumptions.Resultsarevalidonlyifyouhavespecifiedafunctionthataccurately
describestherelationshipbetweendependentandindependentvariables.
Additionally,thechoiceofgoodstartingvaluesisveryimportant.Evenifyou've
specifiedthecorrectfunctionalformofthemodel,ifyouusepoorstartingvalues,
yourmodelmayfailtoconvergeoryoumaygetalocallyoptimalsolutionrather
thanonethatisgloballyoptimal.
Relatedprocedures.Manymodelsthatappearnonlinearatfirstcanbetransformedto
alinearmodel,whichcanbeanalyzedusingtheLinearRegressionprocedure.Ifyou
areuncertainwhatthepropermodelshouldbe,theCurveEstimationprocedurecan
helptoidentifyusefulfunctionalrelationsinyourdata.
ObtainingaNonlinearRegressionAnalysis
E
Fromthemenuschoose:
Analyze
Regression
Nonlinear...
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33
NonlinearRegr
ession
Figure5-1
NonlinearRegressiondialogbox
E
Selectonenumericdependentvariablefromthelistofvariablesinyourworking
datafile.
E
Tobuildamodelexpression,entertheexpressionintheModelfieldorpaste
components(variables,parameters,functions)intothefield.
E
Identifyparametersinyourmodelbyclicking
Parameters
.
Asegmentedmodel(onethattakesdifferentformsindifferentpartsofitsdomain)
mustbespecifiedbyusingconditionallogicwithinthesinglemodelstatement.
ConditionalLogic(NonlinearRegression)
Youcanspecifyasegmentedmodelusingconditionallogic.Touseconditionallogic
withinamodelexpressionoralossfunction,youformthesumofaseriesofterms,
oneforeachcondition.Eachtermconsistsofalogicalexpression(inparentheses)
multipliedbytheexpressionthatshouldresultwhenthatlogicalexpressionistrue.
Forexample,considerasegmentedmodelthatequals0forX<=0,Xfor0<X<1,and
1forX>=1.Theexpressionforthisis:
(X<=0)*0+(X>0&X<1)*X+(X>=1)*1.
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34
Chapter5
Thelogicalexpressionsinparenthesesallevaluateto1(true)or0(false).Therefore:
IfX<=0,theabovereducesto1*0+0*X+0*1=0.
If0<X<1,itreducesto0*0+1*X+0*1=X.
IfX>=1,itreducesto0*0+0*X+1*1=1.
Morecomplicatedexamplescanbeeasilybuiltbysubstitutingdifferentlogical
expressionsandoutcomeexpressions.Rememberthatdoubleinequalities,suchas
0<X<1,mustbewrittenascompoundexpressions,suchas(X>0&X<1).
Stringvariablescanbeusedwithinlogicalexpressions:
(city='NewYork')*costliv+(city='DesMoines')*0.59*costliv
Thisyieldsoneexpression(thevalueofthevariablecostliv)forNewYorkersand
another(59%ofthatvalue)forDesMoinesresidents. Stringconstantsmustbe
enclosedinquotationmarksorapostrophes,asshownhere.
NonlinearRegressionParameters
Figure5-2
NonlinearRegressionParametersdialogbox
ParametersarethepartsofyourmodelthattheNonlinearRegressionprocedure
estimates. Parameterscanbeadditiveconstants,multiplicativecoefficients,
exponents,orvaluesusedinevaluatingfunctions.Allparametersthatyouhave
definedwillappear(withtheirinitialvalues)ontheParameterslistinthemain
dialogbox.
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35
NonlinearRegr
ession
Name.Youmustspecifyanameforeachparameter.Thisnamemustbeavalid
SPSSvariablenameandmustbethenameusedinthemodelexpressioninthemain
dialogbox.
StartingValue.Allowsyoutospecifyastartingvaluefortheparameter,preferably
ascloseaspossibletotheexpectedfinalsolution.Poorstartingvaluescanresultin
failuretoconvergeorinconvergenceonasolutionthatislocal(ratherthanglobal)or
isphysicallyimpossible.
Usestartingvaluesfrompreviousanalysis.Ifyouhavealreadyrunanonlinear
regressionfromthisdialogbox,youcanselectthisoptiontoobtaintheinitialvalues
ofparametersfromtheirvaluesinthepreviousrun.Thispermitsyoutocontinue
searchingwhenthealgorithmisconvergingslowly.(Theinitialstartingvalueswill
stillappearontheParameterslistinthemaindialogbox.)
Note:Thisselectionpersistsinthisdialogboxfortherestofyoursession.Ifyou
changethemodel,besuretodeselectit.
NonlinearRegressionCommonModels
Thetablebelowprovidesexamplemodelsyntaxformanypublishednonlinear
regressionmodels.Amodelselectedatrandomisnotlikelytofityourdatawell.
Appropriatestartingvaluesfortheparametersarenecessary,andsomemodelsrequire
constraintsinordertoconverge.
Table5-1
Examplemodelsyntax
Name
Modelexpression
AsymptoticRegression
b1+b2*exp(b3*x)
AsymptoticRegression
b1–(b2*(b3**x))
Density
(b1+b2*x)**(–1/b3)
Gauss
b1*(1–b3*exp(–b2*x**2))
Gompertz
b1*exp(–b2*exp(–b3*x))
Johnson-Schumacher
b1*exp(–b2/(x+b3))
Log-Modified
(b1+b3*x)**b2
Log-Logistic
b1–ln(1+b2*exp(–b3*x))
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36
Chapter5
Name
Modelexpression
MetcherlichLawofDiminishing
Returns
b1+b2*exp(–b3*x)
MichaelisMenten
b1*x/(x+b2)
Morgan-Mercer-Florin
(b1*b2+b3*x**b4)/(b2+x**b4)
Peal-Reed
b1/(1+b2*exp(–(b3*x+b4*x**2+b5*x**3)))
RatioofCubics
(b1+b2*x+b3*x**2+b4*x**3)/(b5*x**3)
RatioofQuadratics
(b1+b2*x+b3*x**2)/(b4*x**2)
Richards
b1/((1+b3*exp(–b2*x))**(1/b4))
Verhulst
b1/(1+b3*exp(–b2*x))
VonBertalanffy
(b1**(1–b4)–b2*exp(–b3*x))**(1/(1–b4))
Weibull
b1–b2*exp(–b3*x**b4)
YieldDensity
(b1+b2*x+b3*x**2)**(–1)
NonlinearRegressionLossFunction
Figure5-3
NonlinearRegressionLossFunctiondialogbox
Thelossfunctioninnonlinearregressionisthefunctionthatisminimizedbythe
algorithm.Selecteither
Sumofsquaredresiduals
tominimizethesumofthesquared
residualsor
User-definedlossfunction
tominimizeadifferentfunction.
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NonlinearRegr
ession
Ifyouselect
User-definedlossfunction
,youmustdefinethelossfunctionwhosesum
(acrossallcases)shouldbeminimizedbythechoiceofparametervalues.
ɸ
MostlossfunctionsinvolvethespecialvariableRESID_,whichrepresentsthe
residual.(ThedefaultSumofsquaredresidualslossfunctioncouldbeentered
explicitlyas
RESID_**2
.)Ifyouneedtousethepredictedvalueinyourloss
function,itisequaltothedependentvariableminustheresidual.
ɸ
Itispossibletospecifyaconditionallossfunctionusingconditionallogic.
YoucaneithertypeanexpressionintheUser-definedlossfunctionfieldorpaste
componentsoftheexpressionintothefield.Stringconstantsmustbeenclosedin
quotationmarksorapostrophes,andnumericconstantsmustbetypedinAmerican
format,withthedotasadecimaldelimiter.
NonlinearRegressionParameterConstraints
Figure5-4
NonlinearRegressionParameterConstraintsdialogbox
Aconstraintisarestrictionontheallowablevaluesforaparameterduringthe
iterativesearchforasolution.Linearexpressionsareevaluatedbeforeastepistaken,
soyoucanuselinearconstraintstopreventstepsthatmightresultinoverflows.
Nonlinearexpressionsareevaluatedafterastepistaken.
38
Chapter5
Eachequationorinequalityrequiresthefollowingelements:
ɸ
Anexpressioninvolvingatleastoneparameterinthemodel.Typetheexpression
orusethekeypad,whichallowsyoutopastenumbers,operators,orparentheses
intotheexpression.Youcaneithertypeintherequiredparameter(s)alongwith
therestoftheexpressionorpastefromtheParameterslistattheleft.Youcannot
useordinaryvariablesinaconstraint.
ɸ
Oneofthethreelogicaloperators<=,=,or>=.
ɸ
Anumericconstant,towhichtheexpressioniscomparedusingthelogical
operator. Typetheconstant.NumericconstantsmustbetypedinAmerican
format,withthedotasadecimaldelimiter.
NonlinearRegressionSaveNewVariables
Figure5-5
NonlinearRegressionSaveNewVariablesdialogbox
Youcansaveanumberofnewvariablestoyouractivedatafile.Availableoptionsare
Predictedvalues,Residuals,Derivatives,andLossfunctionvalues.Thesevariables
canbeusedinsubsequentanalysestotestthefitofthemodelortoidentifyproblem
cases.
ɸ
PredictedValues.Savespredictedvalueswiththevariablenamepred_.
ɸ
Residuals.Savesresidualswiththevariablenameresid.
ɸ
Derivatives.Onederivativeissavedforeachmodelparameter.Derivativenames
arecreatedbyprefixing'd.'tothefirstsixcharactersofparameternames.
ɸ
LossFunctionValues.Thisoptionisavailableifyouspecifyyourownloss
function.Thevariablenameloss_isassignedtothevaluesofthelossfunction.
39
NonlinearRegr
ession
NonlinearRegressionOptions
Figure5-6
NonlinearRegressionOptionsdialogbox
Optionsallowyoutocontrolvariousaspectsofyournonlinearregressionanalysis:
Bootstr
apEstimates.Amethodofestimatingthestandarderrorofastatisticusing
repeatedsamplesfromtheoriginaldataset. Thisisdonebysampling(with
replacement)togetmanysamplesofthesamesizeastheoriginaldataset.The
nonlin
earequationisestimatedforeachofthesesamples.Thestandarderrorofeach
parameterestimateisthencalculatedasthestandarddeviationofthebootstrapped
estimates.Parametervaluesfromtheoriginaldataareusedasstartingvaluesforeach
bootst
rapsample.Thisrequiresthesequentialquadraticprogrammingalgorithm.
EstimationMethod.Allowsyoutoselectanestimationmethod,ifpossible.(Certain
choicesinthisorotherdialogboxesrequirethesequentialquadraticprogramming
algor
ithm.)AvailablealternativesincludeSequentialquadraticprogrammingand
Levenberg-Marquardt.
ɸ
SequentialQuadraticProgramming.Thismethodisavailableforconstrainedand
uncon
strainedmodels.Sequentialquadraticprogrammingisusedautomatically
ifyouspecifyaconstrainedmodel,auser-definedlossfunction,orbootstrapping.
YoucanenternewvaluesforMaximumiterationsandSteplimit,andyoucan
40
Chapter5
changetheselectioninthedrop-downlistsforOptimalitytolerance,Function
precision,andInfinitestepsize.
ɸ
Levenberg-Marquardt.Thisisthedefaultalgorithmforunconstrainedmodels.
TheLevenberg-Marquardtmethodisnotavailableifyouspecifyaconstrained
model,auser-definedlossfunction,orbootstrapping.Youcanenternewvalues
forMaximumiterations,andyoucanchangetheselectioninthedrop-downlists
forSum-of-squaresconvergenceandParameterconvergence.
InterpretingNonlinearRegressionResults
Nonlinearregressionproblemsoftenpresentcomputationaldifficulties:
ɸ
Thechoiceofinitialvaluesfortheparametersinfluencesconvergence.Tryto
chooseinitialvaluesthatarereasonableand,ifpossible,closetotheexpected
finalsolution.
ɸ
Sometimesonealgorithmperformsbetterthantheotheronaparticularproblem.
IntheOptionsdialogbox,selecttheotheralgorithmifitisavailable.(Ifyou
specifyalossfunctionorcertaintypesofconstraints,youcannotusethe
Levenberg-Marquardtalgorithm.)
ɸ
Wheniterationstopsonlybecausethemaximumnumberofiterationshas
occurred,the“final”modelisprobablynotagoodsolution.Select
Usestarting
valuesfrompreviousanalysis
intheParametersdialogboxtocontinuetheiteration
or,betteryet,choosedifferentinitialvalues.
ɸ
Modelsthatrequireexponentiationoforbylargedatavaluescancauseoverflows
orunderflows(numberstoolargeortoosmallforthecomputertorepresent).
Sometimesyoucanavoidthesebysuitablechoiceofinitialvaluesorbyimposing
constraintsontheparameters.
NLRCommandAdditionalFeatures
TheSPSScommandlanguagealsoallowsyouto:
ɸ
Nameafilefromwhichtoreadinitialvaluesforparameterestimates.
ɸ
Specifymorethanonemodelstatementandlossfunction.Thismakesiteasierto
specifyasegmentedmodel.
ɸ
Supplyyourownderivativesratherthanusethosecalculatedbytheprogram.
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