matlab擬合最小二乘法

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最小二乘法擬合

一、繪制離散點圖

x=[0510152025303540455055];

y=[01.27*10^-42.16*10^-42.86*10^-43.44*10^-43.87*10^-44.15*10^-44.37*10^-4

4.51*10^-44.58*10^-44.02*10^-44.64*10^-4];

plot(x,y,'r*');

legend('試驗數據(xi,yi)');

xlabel('x'),ylabel('y'),title('數據點(xi,yi)的散點圖')

運行如圖

二、編寫程序計算在x處的函數值

symsa1a2a3a4

x=[0510152025303540455055];

fi=a1.*x.^3+a2.*x.^2+a3.*x+a4

運行后顯示a1、a2、a3、a4的方程組

fi=

[a4,125*a1+25*a2+5*a3+a4,1000*a1+100*a2+10*a3+a4,3375*a1+225*a2+15*a3+a4,8000*a1+400*a2+20*a3+a4,15625*a1+625*a2+25*a3+a4,27000*a1+900*a2+

30*a3+a4,42875*a1+1225*a2+35*a3+a4,64000*a1+1600*a2+40*a3+a4,91125*a1+

2025*a2+45*a3+a4,125000*a1+2500*a2+50*a3+a4,166375*a1+3025*a2+55*a3+a4]

三、編寫構造平方和的matlab程序

y=[01.27*10^-42.16*10^-42.86*10^-43.44*10^-43.87*10^-44.15*10^-44.37*10^-4

4.51*10^-44.58*10^-44.02*10^-44.64*10^-4];

fi=[a4,125*a1+25*a2+5*a3+a4,1000*a1+100*a2+10*a3+a4,3375*a1+225*a2+15*a3+a4,8000*a1+400*a2+20*a3+a4,15625*a1+625*a2+25*a3+a4,27000*a1+900*a2+30*a3+a4,42875*a1+1225*a2+35*a3+a4,64000*a1+1600*a2+40*a3+a4,91125*a1+2025*a2+45*a3+a4,125000*a1+2500*a2+50*a3+a4,166375*a1+3025*a2+55*a3+a4]fy=fi-y;

fy2=fy.^2;

J=sum(fy.^2)

運行的如下結果

fi=

[a4,125*a1+25*a2+5*a3+a4,1000*a1+100*a2+10*a3+a4,3375*a1+225*a2+15*a3+a4,8000*a1+400*a2+20*a3+a4,15625*a1+625*a2+25*a3+a4,27000*a1+900*a2+30*a3+a4,42875*a1+1225*a2+35*a3+a4,64000*a1+1600*a2+40*a3+a4,91125*a1+2025*a2+45*a3+a4,125000*a1+2500*a2+50*a3+a4,166375*a1+3025*a2+55*a3+a4]

J=

58593218750*a1^2+2386725000*a1*a2+49967500*a1*a3+1089000*a1*a4-(274377591928296252150123*a1)/576460752303423488000+24983750*a2^2+1089000*a2*a3+25300*a2*a4-(31331074233255294718193*a2)/2882303761517117440000+12650*a3^2+660*a3*a4-(7819978335372091569501*a3)/28823037615171174400000+12*a4^2-(2298349019433749545307*a4)/288230376151711744000000+520374483464852566590953249225508026224249/332306998946228968225951765070086144000000000000

四、求a1、a2、a3、a4使J達到最小，分別對a1、a2、a3、a4求偏導數，使之等于0程序如下

symsa1a2a3a4

J=58593218750*a1^2+2386725000*a1*a2+49967500*a1*a3+1089000*a1*a4-(274377591928296252150123*a1)/576460752303423488000+24983750*a2^2+1089000*a2*a3+25300*a2*a4-(31331074233255294718193*a2)/2882303761517117440000+12650*a3^2+660*a3*a4-(7819978335372091569501*a3)/28823037615171174400000+12*a4^2-(2298349019433749545307*a4)/288230376151711744000000+520374483464852566590953249225508026224249/332306998946228968225951765070086144000000000000

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)

運行得

Ja11=

117186437500*a1+2386725000*a2+49967500*a3+1089000*a4-1903877/4000

Ja21=

2386725000*a1+49967500*a2+1089000*a3+25300*a4-217403/20000

Ja31=

49967500*a1+1089000*a2+25300*a3+660*a4-27131/100000

Ja41=

1089000*a1+25300*a2+660*a3+24*a4-3987/500000

解線性方程組Ja11=0，Ja21=0，Ja31=0，Ja41=0，輸入以下程序

A=[117186437500,2386725000,49967500,1089000;

2386725000,49967500,1089000,25300;

49967500,1089000,25300,660;

1089000,25300,660,24];

B=[274377591928296252150123/576460752303423488000,31331074233255294718193/2882303761517117440000,7819978335372091569501/28823037615171174400000,

2298349019433749545307/288230376151711744000000];

C=B/A,f=poly2sym(C)

運行即可得

C=

1.0e-004*

0.0000-0.00520.26340.0178

f=

(4154371348005165*x^3)/1208925819614629174706176-(307872452451165*x^2)/590295810358705651712+(971769044862535*x)/36893488147419103232+4212074133985165/2361183241434822606848

五、編寫下面的MATLAB程序估計其誤差，并作出擬合曲線和數據的圖形.輸入程序xi=[0510152025303540455055];

y=[01.27*10^-42.16*10^-42.86*10^-43.44*10^-43.87*10^-44.15*10^-44.37*10^-4

4.51*10^-44.58*10^-44.02*10^-44.64*10^-4];

n=length(xi);

f=4154371348005165./1208925819614629174706176.*xi.^3307872452451165./590295810358705651712.*xi.^2

971769044862535./36893488147419103232.*xi

4212074133985165./2361183241434822606848.;

x=0:1:55;

F=4154371348005165./1208925819614629174706176.*x.^3307872452451165./590295810358705651712.*x.^2

971769044862535./36893488147419103232.*x

4212074133985165./2361183241434822606848.;

fy=abs(f-y);

fy2=fy.^2;

Ew=max(fy),E1=sum(fy)/n,E2=sqrt((sum(fy2))/n)

plot(xi,y,