最小二乘拟合曲线

最小二乘拟合曲线

给定逼近函数列 { φ k } k = 0 n \left\{\varphi_k \right\}_{k=0}^{n} {φk​}k=0n​,和样本点列$(x_i,y_i),i=0,1,2,3,...,m$,参数向量方程组如下给出：

$$\mathbf{Ga}=\mathbf{d}$$

其中:

$$\mathbf{G}=\left(\begin{array}{cccc}({\phi }_0，{\phi }_0)& ({\phi }_0，{\phi }_1)& \cdots & ({\phi }_0，{\phi }_n)\\ ({\phi }_1，{\phi }_0)& ({\phi }_1，{\phi }_1)& \cdots & ({\phi }_1，{\phi }_n)\\ ⋮& ⋮& \cdots & ⋮\\ ({\phi }_n，{\phi }_0)& ({\phi }_n，{\phi }_1)& \cdots & ({\phi }_n，{\phi }_n)\end{array}\right)$$

$$\mathbf{a}=\left(\begin{array}{c}{a}_0\\ a_1\\ ⋮\\ a_n\end{array}\right)$$

$$\mathbf{d}=\left(\begin{array}{c}{d}_0\\ d_1\\ ⋮\\ d_n\end{array}\right)$$

Remark

• $$({\phi }_j,{\phi }_k)=\sum _{i=0}^m\omega (x_i){\phi }_j(x_i){\phi }_k(x_i)$$

• $$d_k=(f,{\phi }_k)=\sum _{i=0}^n\omega (x_i)f(x_i){\phi }_j(x_i)$$

例

已知一组实验数据如表，求它的拟合曲线。

$$x_i$$	1	2	3	4	5
$$f(x_i)$$	4	4.5	6	8	8.5
$$\omega (x_i)$$	2	1	3	1	1

解

令 $S_1(x)=a_0+a_{1}x$，这里 $m=4，n=1$，${\phi }_0(x)=1，{\phi }_1(x)=x$

故
$$\begin{array}{rl}({\phi }_0,{\phi }_0)& =\sum _{i=0}^4{\omega }_i=8\\ ({\phi }_0,{\phi }_1)& =({\phi }_1,{\phi }_0)=\sum _{i=0}^4{\omega }_i{x}_i=22\\ ({\phi }_1,{\phi }_1)& =\sum _{i=0}^4{\omega }_i{x}_i^2=74\\ ({\phi }_0,f)& =\sum _{i=0}^4{\omega }_i{f}_i=47\\ ({\phi }_1,f)& =\sum _{i=0}^4{\omega }_i{x}_i{f}_i=145.5\end{array}$$

带入法方程得到：

$$\left(\begin{array}{c}{a}_0\\ a_1\end{array}\right)=\left(\begin{array}{c}2.5648\\ 1.2037\end{array}\right)$$