曲线拟合的线性最小二乘法及其MATLAB程序.docx
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曲线拟合的线性最小二乘法及其MATLAB程序
1
曲线拟合的线性最小二乘法及其MATLA程序
例7.2.1
给出一组数据点(人,yj列入表7^2中,试用线性最小二乘法求拟合曲线,
并用(7.2),(7.3)和(7.4)式估计其误差,作出拟合曲线
表7-2例7.2.1的一组数据(x^yj
xi
-2.5-1.7-1.1-0.800.11.52.73.6
yi
-192.9-85.50-36.15-26.52-9.10-8.43-13.126.5068.04
解
(1)在MATLAB工作窗口输入程序
>>x=[-2.5-1.7-1.1-0.800.11.52.73.6];
y=[-192.9-85.50-36.15-26.52-9.10-8.43-13.126.50
68.04];
plot(x,y,'r*'),
legend('实验数据(xi,yi)')
xlabel('x'),ylabel('y'),
title('例7.2.1的数据点(xi,yi)的散点图’)
运行后屏幕显示数据的散点图(略)
(3)编写下列MATLAB程序计算f(x)在(xi,yi)处的函数值,即输入程序
>>symsa1a2a3a4
x=[-2.5-1.7-1.1-0.800.11.52.73.6];
fi=a1.*x.A3+a2.*x.A2+a3.*x+a4
运行后屏幕显示关于a1,a2,a3和a4的线性方程组
fi=[-125/8*a1+25/4*a2-5/2*a3+a4,
-4913/1000*a1+289/100*a2-17/10*a3+a4,
-1331/1000*a1+121/100*a2-11/10*a3+a4,-64/125*a1+16/25*a2-4/5*a3+a4,a4,1/1000*a1+1/100*a2+1/10*a3+a4,
27/8*a1+9/4*a2+3/2*a3+a4,19683/1000*a1+729/100*a2+27/10*a3+a4,5832/125*a1+324/25*a2+18/5*a3+a4]
编写构造误差平方和的MATLAB程序
>>y=[-192.9-85.50-36.15-26.52-9.10-8.43-13.126.50
68.04];
fi=[-125/8*a1+25/4*a2-5/2*a3+a4,
-4913/1000*a1+289/100*a2-17/10*a3+a4,
-1331/1000*a1+121/100*a2-11/10*a3+a4,
-64/125*a1+16/25*a2-4/5*a3+a4,a4,
1/1000*a1+1/100*a2+1/10*a3+a4,
27/8*a1+9/4*a2+3/2*a3+a4,
19683/1000*a1+729/100*a2+27/10*a3+a4,
5832/125*a1+324/25*a2+18/5*a3+a4];
fy=fi-y;fy2=fy.A2;J=sum(fy.A2)
运行后屏幕显示误差平方和如下
J=
(-125/8*a1+25/4*a2-5/2*a3+a4+1929/10)A2+(-4913/1000*a1+2
89/100*a2-17/10*a3+a4+171/2)A2+(-1331/1000*a1+121/100*a2-11/10*a3+a4+723/20)A2+(-64/125*a1+16/25*a2-4/5*a3+a4+663/25)A2+(a4+91/10)A2+(1/1000*a1+1/100*a2+1/10*a3+a4+843/100)A2+(27/8*a1+9/4*a2+3/2*a3+a4+328/25)A2+(19683/1000*a1+729/100*a2+27/10*a3+a4-13/2)A2+(5832/125*a1+324/25*a2+18/5*a3+a4-1701/25)A2
QI
为求a1,a2,a3,a4使J达到最小,只需利用极值的必要条件——-0(k=1,2,3,4),
得到关于a1,a2,a3,a4的线性方程组,这可以由下面的MATLAB程序完成,即输入程序
>>symsala2a3a4
J=(-125/8*a1+25/4*a2-5/2*a3+a4+1929/10)A2+(-4913/1000*a1+
289/100*a2-17/10*a3+a4...+171/2)A2+(-1331/1000*a1+121/100*a2-11/10*a3+a4+723/20)A2+(-64/125*a1+16/25*a2-4/5*a3+a4+663/25)A2+(a4+91/10)A2+(1/1000*a1+1/100*a2+1/10*a3+a4+843/100)A2+(27/8*a1+9/4*a2+3/2*a3+a4+328/25)A2+(19683/1000*a1+729/100*a2+27/10*a3+a4-13/2)A2+(5832/125*a1+324/25*a2+18/5*a3+a4-1701/25)A2;
Ja1=diff(J,a1);Ja2=diff(J,a2);Ja3=diff(J,a3);
Ja4=diff(J,a4);
Ja11=simple(Ja1),Ja21=simple(Ja2),Ja31=simple(Ja3),
Ja41=simple(Ja4),
运行后屏幕显示J分别对a1,a2,a3,a4的偏导数如下
Ja1仁
56918107/10000*a1+32097579/25000*a2+1377283/2500*a3+23667/250*a4-8442429/625
Ja21=
32097579/25000*a1+1377283/2500*a2+23667/250*a3+67*a4+767319/625
Ja31=1377283/2500*a1+23667/250*a2+67*a3+18/5*a4-232638/125
Ja41=23667/250*a1+67*a2+18/5*a3+18*a4+14859/25
解线性方程组Jan=0,Ja21=0,Ja31=0,Ja41=0,输入下列程序
>>A=[56918107/10000,32097579/25000,1377283/2500,
23667/250;32097579/25000,1377283/2500,23667/250,67;
1377283/2500,23667/250,67,18/5;23667/250,67,18/5,18];
B=[8442429/625,-767319/625,232638/125,-14859/25];
C=B/A,f=poly2sym(C)
运行后屏幕显示拟合函数f及其系数C如下
C=5.0911-14.19056.4102-8.2574
f=716503695845759/140737488355328*xA3-7988544102557579/562949953421312*xA2+1804307491277693/281474976710656*x-4648521160813215/562949953421312
故所求的拟合曲线为
32
f(x)=5.0911x-14.1905x6.4102x-8.2574.
(4)编写下面的MATLAB程序估计其误差,并作出拟合曲线和数据的图形.输入程
序
>>xi=[-2.5-1.7-1.1-0.800.11.52.73.6];
y=[-192.9-85.50-36.15-26.52-9.10-8.43-13.126.5068.04];
n=length(xi);
f=5.0911.*xi.A3-14.1905.*xi.A2+6.4102.*xi-8.2574;
x=-2.5:
0.01:
3.6;
F=5.0911.*x.A3-14.1905.*x.A2+6.4102.*x-8.2574;
fy=abs(f-y);fy2=fy.A2;Ew=max(fy),
E仁sum(fy)/n,E2=sqrt((sum(fy2))/n)
plot(xi,y,'r*'),holdon,plot(x,F,'b-'),holdoff
legend('数据点(xi,yi)',拟合曲线y=f(x)'),
xlabel('x'),ylabel('y'),
title('例7.2.1的数据点(xi,yi)和拟合曲线y=f(x)的图形')
运行后屏幕显示数据(Xj,%)与拟合函数f的最大误差Ev,平均误差E和均方根误差E及其数据点(Xj,yj和拟合曲线y=f(x)的图形(略).
Ew=E1=E2=
3.10540.90341.2409
7.3函数m(x)的选取及其MATLA程序
例7.3.1给出一组实验数据点(x「%)的横坐标向量为
x=(-8.5,-8.7,-7.1,-6.8,-5.10,-4.5,-3.6,-3.4,-2.6,-2.5,-2.1,-1.5,-2.7,-3.6),
纵横坐标向量为y=(459.26,52.81,198.27,165.60,59.17,41.66,25.92,22.37,13.47,
12.87,11.87,6.69,14.87,24.22),试用线性最小二乘法求拟合曲线,并用(7.2),(7.3)
和(7.4)式估计其误差,作出拟合曲线.
解
(1)在MATLAB工作窗口输入程序
>>x=[-8.5,-8.7,-7.1,-6.8,-5.10,-4.5,-3.6,-3.4,-2.6,-2.5,-2.1,-1.5,-2.7,-3.6];
y=[459.26,52.81,198.27,165.60,59.17,41.66,25.92,
22.37,13.47,12.87,11.87,6.69,14.87,24.22];
plot(x,y,'r*'),legend('实验数据(xi,yi)')
xlabel('x'),ylabel('y'),
title('例7.3.1的数据点(xi,yi)的散点图’)
运行后屏幕显示数据的散点图(略)
(3)编写下列MATLAB程序计算f(x)在(xi,yi)处的函数值,即输入程序
>>symsab
x=[-8.5,-8.7,-7.1,-6.8,-5.10,-4.5,-3.6,-3.4,-2.6,-2.5,-2
.1,-1.5,-2.7,-3.6];fi=a.*exp(-b.*x)
运行后屏幕显示关于a和b的线性方程组
fi=[a*exp(17/2*b),a*exp(87/10*b),a*exp(71/10*b),
a*exp(34/5*b),a*exp(51/10*b),a*exp(9/2*b),a*exp(18/5*b),a*exp(17/5*b),a*exp(13/5*b),a*exp(5/2*b),a*exp(21/10*b),a*exp(3/2*b),a*exp(27/10*b),a*exp(18/5*b)]
编写构造误差平方和的MATLAB程序如下
>>y=[459.26,52.81,198.27,165.60,59.17,41.66,25.92,22.37,
13.47,12.87,11.87,6.69,14.87,24.22];
a*exp(71/10*b),
a*exp(9/2*b),a*exp(13/5*b),
a*exp(3/2*b),
a,b的线性方程组,
fi=[a*exp(17/2*b),a*exp(87/10*b),
a*exp(34/5*b),a*exp(51/10*b),
a*exp(18/5*b),a*exp(17/5*b),
a*exp(5/2*b),a*exp(21/10*b),
a*exp(27/10*b),a*exp(18/5*b)];
fy=fi-y;
fy2=fy.A2;
J=sum(fy.A2)
运行后屏幕显示误差平方和如下
J=
(a*exp(17/2*b)-22963/50)A2+(a*exp(87/10*b)-5281/100)A2+(a*exp(71/10*b)-19827/100)A2+(a*exp(34/5*b)-828/5)A2+(a*exp(51/10*b)-5917/100)A2+(a*exp(9/2*b)-2083/50)A2+(a*exp(18/5*b)-648/25F2+(a*exp(17/5*b)-2237/100)A2+(a*exp(13/5*b)-1347/100)A2+(a*exp(5/2*b)-1287/100)A2+(a*exp(21/10*b)-1187/100)A2+(a*exp(3/2*b)-669/100F2+(a*exp(27/10*b)-1487/100)A2+(a*exp(18/5*b)-1211/50)A
2
为求a,b使J达到最小,只需利用极值的必要条件,得到关于这可以由下面的MATLAB程序完成,即输入程序
>>symsab
J=(a*exp(17/2*b)-22963/50)A2+(a*exp(87/10*b)-5281/100)A2+(a*exp(71/10*b)-19827/100)A2+(a*exp(34/5*b)-828/5)A2+(a*exp(51/10*b)-5917/100)A2+(a*exp(9/2*b)-2083/50)A2+(a*exp(18/5*b)-648/25)A2+(a*exp(17/5*b)-2237/100)A2+(a*exp(13/5*b)-1347/100F2+(a*exp(5/2*b)-1287/100)A2+(a*exp(21/10*b)-1187/100)A2+(a*exp(3/2*b)-669/100)A2+(a*exp(27/10*b)-1487/100F2+(a*exp(18/5*b)-1211/50
)A2;
Ja=diff(J,a);Jb=diff(J,b);
Ja仁simple(Ja),Jb仁simple(Jb),
运行后屏幕显示J分别对a,b的偏导数如下
Ja1=
2*a*exp(3*b)+2*a*exp(17*b)+2*a*exp(87/5*b)+2*exp(68/5*b)*a+2*exp(9*b)*a+2*a*exp(34/5*b)-669/50*exp(3/2*b)-1487/50*exp(27/10*b)-2507/25*exp(18/5*b)-22963/25*exp(17/2*b)-5281/50*exp(87/10*b)-19827/50*exp(71/10*b)-2237/50*exp(17/5*b)-1656/5*exp(34/5*b)-1347/50*exp(13/5*b)-5917/50*exp(51/10*b)-1287/50*exp(5/2*b)-2083/25*exp(9/2*b)-1187/50*exp(21/10*b)+4*a*exp(36/5*b)+2*a*exp(26/5*b)+2*a*exp(71/5*b)+2*a*exp(51/5*b)+2*a*exp(5*b)+2*a*exp(21/5*b)+2*a*exp(27/5*b)
Jb1=
1/500*a*(2100*a*exp(21/10*b)A2+8500*a*exp(17/2*b)A2+6800*a*exp(34/5*b)A2-10035*exp(3/2*b)-40149*exp(27/10*b)-180504*exp(18/5*b)-3903710*exp(17/2*b)-459447*exp(87/10*b)-1407717*exp(71/10*b)-76058*exp(17/5*b)-1126080*exp(34/5*b)-35022*exp(13/5*b)-301767*exp(51/10*b)-32175*exp(5/2*b)-187470*exp(9/2*b)-24927*exp(21/10*b)+7100*a*exp(71/10*b)A2+5100*a*exp(51/10*b)A2+4500*a*exp(9/2*b)A2+7200*a*exp(18/5*b)A2+3400*a*exp(17/5*b)A2+2600*a*exp(13/5*b)A2+2500*a*exp(5/2*b)A2+1500*a*exp(3/2*b)A2+2700*a*exp(27/10*b)A2+8700*a*exp(87/10*b)A2)
用解二元非线性方程组的牛顿法的MATLAB^序求解线性方程组Jai=0,Jbi=0,得
a=b=
2.81100.5816
故所求的拟合曲线(7.13)为
f(X)=2.8110e^.5816x.(7.14)
(4)根据(7.2),(7.3),(7.4)和(7.14)式编写下面的MATLAB程序估计其误差,并做出拟合曲线和数据的图形.输入程序
>>xi=[-8.5-8.7-7.1-6.8-5.10-4.5-3.6-3.4-2.6-2.5
-2.1-1.5-2.7-3.6];
y=[459.2652.81198.27165.6059.1741.6625.9222.37
13.4712.8711.876.6914.8724.22];
n=length(xi);f=2.8110.*exp(-0.5816.*xi);x=-9:
0.01:
-1;
F=2.8110.*exp(-0.5816.*x);fy=abs(f-y);fy2=fy.A2;
Ew=max(fy),
E仁sum(fy)/n,E2=sqrt((sum(fy2))/n),plot(xi,y,'r*'),holdon
plot(x,F,'b-'),holdoff,
legend('数据点(xi,yi)','拟合曲线y=f(x)')
xlabel('x'),ylabel('y'),
title('例7.3.1的数据点(xi,yi)和拟合曲线y=f(x)的图形')
运行后屏幕显示数据(xi,yi)与拟合函数f的最大误差Ew=390.1415,平均误差
E1=36.9422和均方根误差