Logistic曲线拟合方法研究

:1002-1566(2002)01-0041-06Logistic殷祚云(,510520):Logistic模型具有广泛的实用性本文推导了用三点法估计该模型中参数K值的公式,并提出了估计K值的新方法一四点法和拐点法用3种方法求出K值后,再用线性化回归获得另外两个参数ar,应用实例研究表明:3种方法都可得到较高拟合精度,其中以四点法最优而且,以这些方法得到的参数估计值作为初始值进行非线性回归,易获得3个参数的最优估计:Logistic曲线;三点法;四点法;拐点法;回归:O212:A,Logistic,LogisticLogisticS,,[1~13]1Logistic:dNdt=rN(1-NK)(1):N=K1+ea-rt(2)(1)(2),N(biomass)(growth)();t();r,(instantaneousgrowthrate);K,(carryingcapacity);e;a(2)SLogistic1.1K1.1.1K,K=2N1N2N3-N22(N1+N3)N1N3-N22,2t2=t1+t3(3)(t1,N1)(t2,N2)(t3,N3)[2,8]:(2):41Logistic:2001-01-03lnK-NN=a-rt(4)(t1,N1)(t2,N2)(t3,N3),,:lnK-N1N1=a-rt1(4a)lnK-N2N2=a-rt2(4b)lnK-N3N3=a-rt3(4c)2t2=t1+t3,2(4b)=(4a)+(4c),a,r,:2lnK-N2N2=lnK-N1N1+lnK-N3N3K0,(3)1.1.2,4K,K:K=N1N4(N2+N3)-N2N3(N1+N4)N1N4-N2N3,t2+t3=t1+t4(5)(t1,N1)(t4,N4),(t2,N2)(t3,N3)1.1.3(1)dNdt=rN(1-NK),(2)N=K1+ea-rt:d2Ndt2=rdNdt(1-2NK)(6)d2Ndt2=0,rdNdt0,:N=K2(1)(2),N=K/2,Logistic(2)S,S,(),K:K=2Nm(7)(7),NmN(t=tm),()(tm,Nm)NmdN/dt0,t|!t|,dN/dt∀!N/!t,|!t|,!N/!t()!N/!t,!N/!tNNm(1),max{(N2-N1)/(t2-t1),(N3-N2)/(t3-t2),#,(Nn-Nn-1)/(tn-tn-1)}∃(tm,Nm)(7)K42211200211(tm,Nm)tN!t!N!N/!tmax(!N/!t)t1N1%%%t2N2t2-t1N2-N1(N2-N1)/(t2-t1)t3N3t3-t2N3-N2(N3-N2)/(t3-t2)tmNmtm-tm-1Nm-Nm-1(Nm-Nm-1)/(tm-tm-1)tnNntn-tn-1Nn-Nn-1(Nn-Nn-1)/(tn-tn-1)1.21.2.1Logistic,[12]Logistic(2),(4)lnK-NN=a-rty=lnK-NN(K3),:y=a-rta,r,,1.2.2Logistic,Levenberg-Marquardt,Logistic,K,a,r,()R2[14]22.11[2](2)2t(&)N!t!N!N/!tmax(!N/!t)10120%%%1322331025034216480325750858∋20829434890872∋2389030611020412,(1985),!N/!t(2),(1)N,Logistic(43LogisticS)[3]2,16&20&!N/!t!N/!t,|!t|,,(16,480)(20,829)!N/!t=0872N=8292K,K,NNNmLogistic,,(7):K=2480829=126162K33LogisticKKestimatemethodregressionmethodKarR2iterationtimes103385523010315909844three-pointlinear9639461866038930995112nonlinear95254583680373609892four-pointlinear9639461866038930995112nonlinear126162474840255909736yieldingpointlinear9639461866038930995119nonlineart=101622&(t=22&,N=88[2]);t=10132023&2.22(1)[1](4)4t(d)N!t!N!N/!tmax(!N/!t)1001%%%20041003003301010060064024101401450461022022605310070077055100200280561001001a)t61;b)12,,(1984),4,,!N/!t(40,24)(50,46),N4421120021NNm,(7):K=22446=66453K55LogisticKKestimatemethodregressionmethodKarR2iterationtimes56845504660121009900three-pointlinear5632660507014710997512nonlinear56305533690132009971four-pointlinear5632660507014710997512nonlinear66453440150087309304yieldingpointlinear5632660507014710997517nonlineart=104070d;t=10405080d,K:(1)3K(,)Kar(),,|!t|K;(2)3KKar:,K;(3)R2:,,,,(1097,2093),4Logistic3(1)K,4,3(2)K,1,,,()(),[2],;,|!t||!t|,K,|!t|,K45Logistic3K,,(3)Logistic,,Logistic,[13]Logistic,,,,(4),Logistic3,,3[][1],,.0618[J].,1984,3:59~63.[2],.[M].:,1985,445~449.[3],.[M].:,1987,267~273.[4],.Logistic[J].,1993,8(3):81~86.[5],,.[J].,1995,15(1):66~71.[6],.[J].,1995,15(2):214~219.[7].[J].,1995.14(2):70~75.[8],,.,,1997,16(1):6~12.[9],.Logistic[J].,1998,11(5):537~541.[10],.[J].,1998,34(4):123~128.[11],.Logistic[J].,1999,19(1):48-51.[12]SwedaT,KoideT.Applicabilityofgrowthequationsofthegrowthoftreesinstemradius(I):Applicationtowhitespruce.J.Jap.For.Sci.,1981,61:321~329.[13].[M].:,1996,206~226.[14].[M].:,1990,48~50.StudyonthefittingmethodsoflogisticcurveYINZuoyun(GuangdongForestResearchInstitute,Guangzhou510520,China)Abstract:Logisticcurvehasfar-rangingpracticability.TheformulaestimatingparameterKofthemodelwith3-pointmethodwasdeduced.Inthispaper,theothertwonewmethodsestimatingKvalueoftheLogisticcurveequation,4-pointmethodandyieldingpointmethodwereputforward.AfterKvaluewasevaluated,theothertwocoefficientsa,rwereestimatedwithlinearregression.Withtwoquotedexamplesdiscussed,theresultsindicatedthatthreemethodsestimatingKvalueallmakegoodfittingprecision,especially4-pointmethodbest.Moreover,Nonlinearregressionapplyingabove-mentionedestimatevaluesofK,a,rasstartingpoints,easilygivesthesecoefficientsbestevaluation.Keyword:logisticcurve;3-pointmethod;4-pointmethod;yieldingpointmethod;regression4621120021