Question

最小二乘法用matlab怎么实现（求具体程序）

Answer 1

%离散试验数据点的多项式曲线拟合

function A=multifit(X,Y,m)
%A--输出的拟合多项式的系数
N=length(X);
M=length(Y);
if(N ~= M)
disp('数据点坐标不匹配！');
return;
end

c(1:(2*m+1))=0;
b(1:(m+1))=0;

for j=1:(2*m+1) %求出c和b
for k=1:N
c(j)=c(j)+X(k)^(j-1);
if(j<(m+2))
b(j)=b(j)+Y(k)*X(k)^(j-1);
end
end
end

C(1,:)=c(1:(m+1));
for s=2:(m+1)
C(s,:)=c(s:(m+s));
end

A=b'\C; %直接求解法求出拟合系数

%离散试验数据点的线性最小二乘拟合

function [a,b]=LZXEC(x,y)
if(length(x) == length(y))
n = length(x);
else
disp('x和y的维数不相等！');
return;
end %维数检查

A = zeros(2,2);
A(2,2) = n;
B = zeros(2,1);
for i=1:n
A(1,1) = A(1,1) + x(i)*x(i);
A(1,2) = A(1,2) + x(i);
B(1,1) = B(1,1) + x(i)*y(i);
B(2,1) = B(2,1) + y(i);
end
A(2,1) = A(1,2);

s = A\B;
a = s(1);
b = s(2);

%离散试验数据点的正交多项式最小二乘拟合

function a=ZJZXEC(x,y,m)
if(length(x) == length(y))
n = length(x);
else
disp('x和y的维数不相等！');
return;
end %维数检查

syms v;
d = zeros(1,m+1);
q = zeros(1,m+1);
alpha = zeros(1,m+1);
for k=0:m
px(k+1)=power(v,k);
end %x的幂多项式
B2 = [1];
d(1) = n;
for l=1:n
q(1) = q(1) + y(l);
alpha(1) = alpha(1) + x(l);
end
q(1) = q(1)/d(1);
alpha(1) = alpha(1)/d(1);
a(1) = q(1);
B1 = [-alpha(1) 1];
for l=1:n
d(2) = d(2) + (x(l)-alpha(1))^2;
q(2) = q(2) + y(l)*(x(l)-alpha(1));
alpha(2) = alpha(2) + x(l)*(x(l)-alpha(1))^2;
end
q(2) = q(2)/d(2);
alpha(2) = alpha(2)/d(2);
a(1) = a(1)+q(2)*(-alpha(1));
a(2) = q(2);
beta = d(2)/d(1);

for i=3:(m+1)
B = zeros(1,i);
B(i) = B1(i-1);
B(i-1) = -alpha(i-1)*B1(i-1)+B1(i-2);
for j=2:i-2
B(j) = -alpha(i-1)*B1(j)+B1(j-1)-beta*B2(j);
end
B(1) = -alpha(i-1)*B1(1)-beta*B2(1);
BF = B*transpose(px(1:i));
for l=1:n
Qx = subs(BF,'v',x(l));
d(i) = d(i) + (Qx)^2;
q(i) = q(i) + y(l)*Qx;
alpha(i) = alpha(i) + x(l)*(Qx)^2;
end
alpha(i) = alpha(i)/d(i);
q(i) = q(i)/d(i);
beta = d(i)/d(i-1);
for k=1:i-1
a(k) = a(k)+q(i)*B(k);
end
a(i) = q(i)*B(i);
B2 = B1;
B1 = B;
end

举第一个运行结果的例子吧！
>> X=[3,4,5,6,7,8,9];
>> Y=[2.01,2.98,3.50,5.02,5.47,6.02,7.05];
>> multifit(X,Y,7)

ans =

Columns 1 through 3

0.153705817445571 1.28812217846988 10.9204180096604

Columns 4 through 6

93.4341699408163 805.369642330511 6984.42872632458

Columns 7 through 8

60878.714442185 532900.1377831

Answer 2

最小二乘法在matlab语言中就是最简单的函数拟合。