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二次回帰計算機
放物線の最小二乗近似を段階的に求める
入力内容
$$$\left\{\left(1, 0\right), \left(4, 5\right), \left(6, 2\right), \left(7, 1\right), \left(3, -3\right)\right\}$$$ に対する最小二乗近似の放物線を求めよ。
解答
観測数は $$$n = 5$$$ です。
以下の表を作成してください：
| $$$x$$$ | $$$y$$$ | $$$x y$$$ | $$$x^{2}$$$ | $$$x^{2} y$$$ | $$$x^{3}$$$ | $$$x^{4}$$$ | $$$y^{2}$$$ | |
| $$$1$$$ | $$$0$$$ | $$$0$$$ | $$$1$$$ | $$$0$$$ | $$$1$$$ | $$$1$$$ | $$$0$$$ | |
| $$$4$$$ | $$$5$$$ | $$$20$$$ | $$$16$$$ | $$$80$$$ | $$$64$$$ | $$$256$$$ | $$$25$$$ | |
| $$$6$$$ | $$$2$$$ | $$$12$$$ | $$$36$$$ | $$$72$$$ | $$$216$$$ | $$$1296$$$ | $$$4$$$ | |
| $$$7$$$ | $$$1$$$ | $$$7$$$ | $$$49$$$ | $$$49$$$ | $$$343$$$ | $$$2401$$$ | $$$1$$$ | |
| $$$3$$$ | $$$-3$$$ | $$$-9$$$ | $$$9$$$ | $$$-27$$$ | $$$27$$$ | $$$81$$$ | $$$9$$$ | |
| $$$\sum$$$ | $$$21$$$ | $$$5$$$ | $$$30$$$ | $$$111$$$ | $$$174$$$ | $$$651$$$ | $$$4035$$$ | $$$39$$$ |
$$$a = \frac{(n(\sum x^2y)-(\sum x^2)(\sum y))(n(\sum x^2)-(\sum x)^2)-(n(\sum xy)-(\sum x)(\sum y))(n(\sum x^3)-(\sum x^2)(\sum x)))}{(n(\sum x^4)-(\sum x^2)^2)(n(\sum x^2)-(\sum x)^2)-(n(\sum x^3)-(\sum x^2)(\sum x))^2} = \frac{\left(5 \cdot 174 - \left(111\right)\cdot \left(5\right)\right)\cdot \left(5 \cdot 111 - 21^{2}\right) - \left(5 \cdot 30 - \left(21\right)\cdot \left(5\right)\right)\cdot \left(5 \cdot 651 - \left(111\right)\cdot \left(21\right)\right)}{\left(5 \cdot 4035 - 111^{2}\right)\cdot \left(5 \cdot 111 - 21^{2}\right) - \left(5 \cdot 651 - \left(111\right)\cdot \left(21\right)\right)^{2}} = - \frac{3}{22}$$$
$$$b = \frac{(n(\sum xy)-(\sum x)(\sum y))(n(\sum x^4)-(\sum x^2)^2)-(n(\sum x^2y)-(\sum x^2)(\sum y))(n(\sum x^3)-(\sum x^2)(\sum x)))}{(n(\sum x^4)-(\sum x^2)^2)(n(\sum x^2)-(\sum x)^2)-(n(\sum x^3)-(\sum x^2)(\sum x))^2} = \frac{\left(5 \cdot 30 - \left(21\right)\cdot \left(5\right)\right)\cdot \left(5 \cdot 4035 - 111^{2}\right) - \left(5 \cdot 174 - \left(111\right)\cdot \left(5\right)\right)\cdot \left(5 \cdot 651 - \left(111\right)\cdot \left(21\right)\right)}{\left(5 \cdot 4035 - 111^{2}\right)\cdot \left(5 \cdot 111 - 21^{2}\right) - \left(5 \cdot 651 - \left(111\right)\cdot \left(21\right)\right)^{2}} = \frac{3}{2}$$$
$$$c = \frac{(\sum y)-b(\sum x)-a(\sum x^2)}{n} = \frac{5 - \left(\frac{3}{2}\right)\cdot \left(21\right) - \left(- \frac{3}{22}\right)\cdot \left(111\right)}{5} = - \frac{25}{11}$$$
最小二乗近似の放物線は $$$y = a x^{2} + b x + c$$$ です。
したがって、最小二乗近似の放物線は $$$y = - \frac{3 x^{2}}{22} + \frac{3 x}{2} - \frac{25}{11}$$$ です。
解答
最小二乗近似の放物線は $$$y = - \frac{3 x^{2}}{22} + \frac{3 x}{2} - \frac{25}{11}\approx - 0.136363636363636 x^{2} + 1.5 x - 2.272727272727273$$$A です。