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1、20Non-LinearRegressionSometimeswehaveamechanisticmodelfortherelationshipbetweenyandx,andwewanttoestimatetheparametersandstandarderrorsoftheparametersofaspecificnon-linearequationfromdata.Somefrequentlyusednon-linearmodelsareshowninTable20.1.Whatwemeaninth
2、iscaseby‘non-linear’isnotthattherelationshipiscurved(itwascurvedinthecaseofpolynomialregressions,butthesewerelinearmodels),butthattherelationshipcannotbelinearizedbytransformationoftheresponsevariableortheexplanatoryvariable(orboth).Hereisanexample:itsho
3、wsjawbonelengthasafunctionofageindeer.Theoryindicatesthattherelationshipisanasymptoticexponentialwiththreeparameters:y=a−be−cx.InR,themaindifferencebetweenlinearmodelsandnon-linearmodelsisthatwehavetotellRtheexactnatureoftheequationaspartofthemodelformul
4、awhenweusenon-linearmodelling.Inplaceoflmwewritenls(thisstandsfor‘non-linearleastsquares’).Then,insteadofy~x,wewritey~a-b*exp(-c*x)tospellouttheprecisenon-linearmodelwewantRtofittothedata.TheslightlytediousthingisthatRrequiresustospecifyinitialguessesfort
5、hevaluesoftheparametersa,bandc(note,however,thatsomecommonnon-linearmodelshave‘self-starting’versionsinRwhichbypassthisstep;seep.728).Letusplotthedatatoworkoutsensiblestartingvalues.Italwayshelpsincaseslikethistoworkouttheequation’s‘behaviouratthelimits’
6、–thatistosay,tofindthevaluesofywhenx=0andwhenx=∞(p.258).Forx=0,wehaveexp(–0)whichis1,and1×b=b,soy=a–b.Forx=∞,wehaveexp(–∞)whichis0,and0×b=0,soy=a.Thatistosay,theasymptoticvalueofyisa,andtheinterceptisa–b.deer<-read.table("c:\temp\jaws.txt",header=T)atta
7、ch(deer)names(deer)[1]"age""bone"plot(age,bone,pch=21,col="purple",bg="green")TheRBook,SecondEdition.MichaelJ.Crawley.©2013JohnWiley&Sons,Ltd.Published2013byJohnWiley&Sons,Ltd.716THERBOOKTable20.1.Usefulnon-linearfunctions.NameEquationAsymptoticfunctions
8、axMichaelis–Menteny=1+bx2-parameterasymptoticexponentialy=a(1−e−bx)3-parameterasymptoticexponentialy=a−be−cxS-shapedfunctionsea+bx2-parameterlogisticy=1+ea+bxa3-parameterlogisticy=1+be−cxb−a4-parameterlogisticy=a+1+e(c−x)/
