📝 题目
2．已知一组实验数据为 $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \cdots,\left(x_{n}, y_{n}\right)$ ．现若假定经验公式是
$$ y=a x^{2}+b x+c $$
试按最小二乘法建立 $a, b, c$ 应满足的三元一次方程组．
💡 答案与解析
[AI解答]
[AI解答]
我们已知实验数据点 $(x_i, y_i), i=1,2,\dots,n$，并假设经验公式为 $$ y = a x^2 + b x + c $$ 根据最小二乘法原理，要使误差平方和 $$ S(a,b,c) = \displaystyle\sum_{i=1}^{n} \left[ y_i - (a x_i^2 + b x_i + c) \right]^2 $$ 达到最小，则需满足 $$ \frac{\partial S}{\partial a} = 0,\quad \frac{\partial S}{\partial b} = 0,\quad \frac{\partial S}{\partial c} = 0. $$
分别求偏导数：
1. 对 $a$ 求偏导： $$ \frac{\partial S}{\partial a} = -2 \displaystyle\sum_{i=1}^{n} \left[ y_i - (a x_i^2 + b x_i + c) \right] x_i^2 = 0 $$ 整理得： $$ a \displaystyle\sum_{i=1}^{n} x_i^4 + b \displaystyle\sum_{i=1}^{n} x_i^3 + c \displaystyle\sum_{i=1}^{n} x_i^2 = \displaystyle\sum_{i=1}^{n} x_i^2 y_i $$
2. 对 $b$ 求偏导： $$ \frac{\partial S}{\partial b} = -2 \displaystyle\sum_{i=1}^{n} \left[ y_i - (a x_i^2 + b x_i + c) \right] x_i = 0 $$ 整理得： $$ a \displaystyle\sum_{i=1}^{n} x_i^3 + b \displaystyle\sum_{i=1}^{n} x_i^2 + c \displaystyle\sum_{i=1}^{n} x_i = \displaystyle\sum_{i=1}^{n} x_i y_i $$
3. 对 $c$ 求偏导： $$ \frac{\partial S}{\partial c} = -2 \displaystyle\sum_{i=1}^{n} \left[ y_i - (a x_i^2 + b x_i + c) \right] = 0 $$ 整理得： $$ a \displaystyle\sum_{i=1}^{n} x_i^2 + b \displaystyle\sum_{i=1}^{n} x_i + c n = \displaystyle\sum_{i=1}^{n} y_i $$
因此，$a, b, c$ 满足的三元一次方程组（正规方程组）为：
$$ \begin{cases} \displaystyle a \sum_{i=1}^{n} x_i^4 + b \sum_{i=1}^{n} x_i^3 + c \sum_{i=1}^{n} x_i^2 = \sum_{i=1}^{n} x_i^2 y_i \$$1em] \displaystyle a \sum_{i=1}^{n} x_i^3 + b \sum_{i=1}^{n} x_i^2 + c \sum_{i=1}^{n} x_i = \sum_{i=1}^{n} x_i y_i \$$1em] \displaystyle a \sum_{i=1}^{n} x_i^2 + b \sum_{i=1}^{n} x_i + c n = \sum_{i=1}^{n} y_i \end{cases} $$
难度：★★☆☆☆