这里用到的还是最小二乘方法,和上一次这篇文章原理差不多。
就是首先构造最小二乘函数,然后对每一个系数计算偏导,构造矩阵乘法形式,最后解方程组。
比如有一个二次曲面:z=ax^2+by^2+cxy+dx+ey+f
首先构造最小二乘函数,然后计算系数偏导(我直接手写了):
解方程组(下图中A矩阵后面求和符号我就没写了啊),然后计算C:
代码如下:
clear all;
close all;
clc;
a=2;b=2;c=-3;d=1;e=2;f=30; %系数
n=1:0.2:20;
x=repmat(n,96,1);
y=repmat(n',1,96);
z=a*x.^2+b*y.^2+c*x.*y+d*x+e*y +f; %原始模型
surf(x,y,z)
N=100;
ind=int8(rand(N,2)*95+1);
X=x(sub2ind(size(x),ind(:,1),ind(:,2)));
Y=y(sub2ind(size(y),ind(:,1),ind(:,2)));
Z=z(sub2ind(size(z),ind(:,1),ind(:,2)))+rand(N,1)*20; %生成待拟合点,加个噪声
hold on;
plot3(X,Y,Z,'o');
A=[N sum(Y) sum(X) sum(X.*Y) sum(Y.^2) sum(X.^2);
sum(Y) sum(Y.^2) sum(X.*Y) sum(X.*Y.^2) sum(Y.^3) sum(X.^2.*Y);
sum(X) sum(X.*Y) sum(X.^2) sum(X.^2.*Y) sum(X.*Y.^2) sum(X.^3);
sum(X.*Y) sum(X.*Y.^2) sum(X.^2.*Y) sum(X.^2.*Y.^2) sum(X.*Y.^3) sum(X.^3.*Y);
sum(Y.^2) sum(Y.^3) sum(X.*Y.^2) sum(X.*Y.^3) sum(Y.^4) sum(X.^2.*Y.^2);
sum(X.^2) sum(X.^2.*Y) sum(X.^3) sum(X.^3.*Y) sum(X.^2.*Y.^2) sum(X.^4)];
B=[sum(Z) sum(Z.*Y) sum(Z.*X) sum(Z.*X.*Y) sum(Z.*Y.^2) sum(Z.*X.^2)]';
C=inv(A)*B;
z=C(6)*x.^2+C(5)*y.^2+C(4)*x.*y+C(3)*x+C(2)*y +C(1); %拟合结果
mesh(x,y,z)
结果如下,深色曲面是原模型,浅色曲面是用噪声数据拟合的模型:
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