1 曲线拟合的线性最小二乘法及其MATLAB程序
例7.2.1 给出一组数据点(xi,yi)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(xi,yi) xi yi -2.5 -1.7 -1.1 -0.8 0 0.1 1.5 2.7 3.6 -192.9 -85.50 -36.15 -26.52 -9.10 -8.43 -13.12 6.50 68.04 解 （1）在MATLAB工作窗口输入程序
>> x=[-2.5 -1.7 -1.1 -0.8 0 0.1 1.5 2.7 3.6];
y=[-192.9 -85.50 -36.15 -26.52 -9.10 -8.43 -13.12 6.50 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
>> syms a1 a2 a3 a4
x=[-2.5 -1.7 -1.1 -0.8 0 0.1 1.5 2.7 3.6]; fi=a1.*x.^3+ a2.*x.^2+ 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 -13.12 6.50
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.^2; J=sum(fy.^2)
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)^2+(-4913/1000*a1+2
89/100*a2-17/10*a3+a4+171/2)^2+(-1331/1000*a1+121/100*a2-11/10*a3+a4+723/20)^2+(-64/125*a1+16/25*a2-4/5*a3+a4+663/25)^2+(a4+91/10)^2+(1/1000*a1+1/100*a2+1/10*a3+a4+843/100)^2+(27/8*a1+9/4*a2+3/2*a3+a4+328/25)^2+(19683/1000*a1+729/100*a2+27/10*a3+a4-13/2)^2+(5832/125*a1+324/25*a2+18/5*a3+a4-1701/25)^2
为求a1,a2,a3,a4使J达到最小，只需利用极值的必要条件
?J?0 (k?1,2,3,4)，?ak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
>> syms a1 a2 a3 a4
J=(-125/8*a1+25/4*a2-5/2*a3+a4+1929/10)^2+(-4913/1000*a1+
289/100*a2-17/10*a3+a4...+171/2)^2+(-1331/1000*a1+121/100*a2-11/10*a3+a4+723/20)^2+(-64/125*a1+16/25*a2-4/5*a3+a4+663/25)^2+(a4+91/10)^2+(1/1000*a1+1/100*a2+1/10*a3+a4+843/100)^2+(27/8*a1+9/4*a2+3/2*a3+a4+328/25)^2+(19683/1000*a1+729/100*a2+27/10*a3+a4-13/2)^2+(5832/125*a1+324/25*a2+18/5*a3+a4-1701/25)^2;
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
Ja11=
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Ja11 =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 -14.1905 6.4102 -8.2574 f=716503695845759/140737488355328*x^3 -7988544102557579/562949953421312*x^2 +1804307491277693/281474976710656*x
-4648521160813215/562949953421312 故所求的拟合曲线为
f(x)?5.0911x3?14.1905x2?6.4102x?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 -0.8 0 0.1 1.5 2.7 3.6];
y=[-192.9 -85.50 -36.15 -26.52 -9.10 -8.43 -13.12 6.50 68.04]; n=length(xi);
f=5.0911.*xi.^3-14.1905.*xi.^2+6.4102.*xi -8.2574; x=-2.5:0.01: 3.6;
F=5.0911.*x.^3-14.1905.*x.^2+6.4102.*x -8.2574; fy=abs(f-y); fy2=fy.^2; Ew=max(fy), E1=sum(fy)/n, E2=sqrt((sum(fy2))/n)
plot(xi,y,'r*'), hold on, plot(x,F,'b-'), hold off 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(xi,yi)与拟合函数f的最大误差Ew，平均误差E1和均方根误差E2及其数据点(xi,yi)和拟合曲线y=f(x)的图形（略）.
Ew = E1 = E2 =
3.105 4 0.903 4 1.240 9
7.3 函数rk(x)的选取及其MATLAB程序
例7.3.1 给出一组实验数据点(xi,yi)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
>> syms a b
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];
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)]; fy=fi-y; fy2=fy.^2; J=sum(fy.^2)
运行后屏幕显示误差平方和如下
J =
(a*exp(17/2*b)-22963/50)^2+(a*exp(87/10*b)-5281/100)^2+(
a*exp(71/10*b)-19827/100)^2+(a*exp(34/5*b)-828/5)^2+(a*exp(51/10*b)-5917/100)^2+(a*exp(9/2*b)-2083/50)^2+(a*exp(18/5*b)-648/25)^2+(a*exp(17/5*b)-2237/100)^2+(a*exp(13/5*b)-1347/100)^2+(a*exp(5/2*b)-1287/100)^2+(a*exp(21/10*b)-1187/100)^2+(a*exp(3/2*b)-669/100)^2+(a*exp(27/10*b)-1487/100)^2+(a*exp(18/5*b)-1211/50)^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a,b的线性方程组，这可以由下面的MATLAB程序完成，即输入程序
>> syms a b
J=(a*exp(17/2*b)-22963/50)^2+(a*exp(87/10*b)-5281/100)^2
+(a*exp(71/10*b)-19827/100)^2+(a*exp(34/5*b)-828/5)^2+(a*exp(51/10*b)-5917/100)^2+(a*exp(9/2*b)-2083/50)^2+(a*exp(18/5*b)-648/25)^2+(a*exp(17/5*b)-2237/100)^2+(a*exp(13/5*b)-1347/100)^2+(a*exp(5/2*b)-1287/100)^2+(a*exp(21/10*b)-1187/100)^2+(a*exp(3/2*b)-669/100)^2+(a*exp(27/10*b)-1487/100)^2+(a*exp(18/5*b)-1211/50
曲线拟合的线性最小二乘法及其MATLAB程序
