第1讲 插值、曲线拟合与回归分析

#了。3.1多元线性回归y二p+px+pxHFpx+£8〜N(0,G2)01122nnSyntaxb=regress(y,X)[b,bint,r,rint,stats]=regress(y,X)[b,bint,r,rint,stats]=regress(y,X,alpha)1）原始数据仃x11x仃x11x=:、1xn1x)1mry]1xnm其中解释变量第1列的系数是p。02)p=regress(y,x)3.2非线性回归非线性模型：y=c+（0.49-c）*e—c2（x-8）观察数据如下：11xyxyxyxy80.49140.43240.42320.480.49160.44240.4340.4100.48160.43240.4360.41100.47160.43260.41360.38100.48180.46260.4380.4100.47180.45260.41380.4120.46200.42280.41400.39120.46200.42280.4420.39120.45200.43300.4120.43220.41300.4140.45220.41300.38140.43220.4320.41使用nlinfit命令f=@(c,x)c(1)+(0.49-c(1))*exp(-c(2)*(x(:,1)-8));c0=[0.1,0.01];%赋参数初值c=nlinfit(x(:,1),x(:,2),f,c0)运行结果：c=0.3901400079615900.101632683988975使用lsqcurvefit命令f=@(c,x)c(1)+(0.49-c(1))*exp(-c(2)*(x(:,1)-8));c=lsqcurvefit(f,[0.1,0.01],x(:,1),x(:,2))运行结果：c=0.3901400079615900.101632683988975注意：lsqcurve与nlinfit结合使用，lsqcurve给出结果不确定，与初值有很大关系，往往给出满足默认精度要求的结果；nlinfit对初值要求高，如果初值与真值偏离很大，会提示不能运行。实际中，可以把lsqcurve给出结果作为nlinfit的初值。还可以用eviews软件和stata软件计算。

例2007赛题：中国人口预测生育率（r岁育龄妇女千人中生育的比例）曲线可用如下模型刻画：r_q丫(r一15)a_ie_o9«r9«r(a)r为了拟合出模型中参数，则对函数进行简化：h=c(r_15)c2e_c3(r-i5)r=15~49r1Matlab程序：f=@(c,x)c(l)*(xT5)."c(2).*exp(-c(3)*(xT5));x=[15:49]';y=[0.02,0.88,1.16,2.83,15.23,33.05,97.53,161.11,200.7,176.2,179.37,148.09,112.65,97.64,79.58,59.66,59.65,48.57,35.18,27.23,16.83,13.67,7.21,6.51,5.68,3.89,5.97,2.01,0.29,1.5,1.37,0.41,0.45,1.23,0.64;]';c=nlinfit(x,y,f,[0.02117.50510.8248])y2=c(1)*(xT5)."c(2).*exp(-c(3)*(xT5));plot(x,y,'r',x,y2,'o')运行结果：c=0.02117.50510.8248Eviews软件计算：（1）输入x，y序列；（2）在equation窗口里输入y=c(l)*(x-15Fc(2)*exp(-c(3)*(x-15))3）运行结果如下：DependentVariable:YMethod:LeastSquaresDate:05/23/09Time:11:07Sample:135Includedobservations:35Convergenceachievedafter92iterations①忙⑵*EXP(-C(3)*(X-15))VariableCoefficientStd.Errort-StatisticProb.c⑴0.021145O.O140S51.4205S90.1651c⑵7.5050600.56096713.373010.0000C(3)0.8243070.05871414.047950.0000R-squared0.969175Meandependentvar45.02829AdjustedR-squared0.967248S.D.dependentvar61.69093S.E.ofregression11.16443Akaikeinfocriterion7.745160SumsquarEdresid3983.650Schwarzcriterion7.070483Logl