Kalkylator för kvadratisk regression
Hitta bäst anpassade parabler steg för steg
Kalkylatorn hittar det andragradspolynom som bäst passar den givna uppsättningen av parvisa data med hjälp av minsta kvadratmetoden, med steg som visas.
Relaterade kalkylatorer: Kalkylator för linjär regression, Kalkylator för kubisk regression
Din inmatning
Hitta den bäst passande parabeln för $$$\left\{\left(1, 0\right), \left(4, 5\right), \left(6, 2\right), \left(7, 1\right), \left(3, -3\right)\right\}$$$.
Lösning
Antalet observationer är $$$n = 5$$$.
Generera följande tabell:
| $$$x$$$ | $$$y$$$ | $$$x y$$$ | $$$x^{2}$$$ | $$$x^{2} y$$$ | $$$x^{3}$$$ | $$$x^{4}$$$ | $$$y^{2}$$$ | |
| $$$1$$$ | $$$0$$$ | $$$0$$$ | $$$1$$$ | $$$0$$$ | $$$1$$$ | $$$1$$$ | $$$0$$$ | |
| $$$4$$$ | $$$5$$$ | $$$20$$$ | $$$16$$$ | $$$80$$$ | $$$64$$$ | $$$256$$$ | $$$25$$$ | |
| $$$6$$$ | $$$2$$$ | $$$12$$$ | $$$36$$$ | $$$72$$$ | $$$216$$$ | $$$1296$$$ | $$$4$$$ | |
| $$$7$$$ | $$$1$$$ | $$$7$$$ | $$$49$$$ | $$$49$$$ | $$$343$$$ | $$$2401$$$ | $$$1$$$ | |
| $$$3$$$ | $$$-3$$$ | $$$-9$$$ | $$$9$$$ | $$$-27$$$ | $$$27$$$ | $$$81$$$ | $$$9$$$ | |
| $$$\sum$$$ | $$$21$$$ | $$$5$$$ | $$$30$$$ | $$$111$$$ | $$$174$$$ | $$$651$$$ | $$$4035$$$ | $$$39$$$ |
$$$a = \frac{(n(\sum x^2y)-(\sum x^2)(\sum y))(n(\sum x^2)-(\sum x)^2)-(n(\sum xy)-(\sum x)(\sum y))(n(\sum x^3)-(\sum x^2)(\sum x)))}{(n(\sum x^4)-(\sum x^2)^2)(n(\sum x^2)-(\sum x)^2)-(n(\sum x^3)-(\sum x^2)(\sum x))^2} = \frac{\left(5 \cdot 174 - \left(111\right)\cdot \left(5\right)\right)\cdot \left(5 \cdot 111 - 21^{2}\right) - \left(5 \cdot 30 - \left(21\right)\cdot \left(5\right)\right)\cdot \left(5 \cdot 651 - \left(111\right)\cdot \left(21\right)\right)}{\left(5 \cdot 4035 - 111^{2}\right)\cdot \left(5 \cdot 111 - 21^{2}\right) - \left(5 \cdot 651 - \left(111\right)\cdot \left(21\right)\right)^{2}} = - \frac{3}{22}$$$
$$$b = \frac{(n(\sum xy)-(\sum x)(\sum y))(n(\sum x^4)-(\sum x^2)^2)-(n(\sum x^2y)-(\sum x^2)(\sum y))(n(\sum x^3)-(\sum x^2)(\sum x)))}{(n(\sum x^4)-(\sum x^2)^2)(n(\sum x^2)-(\sum x)^2)-(n(\sum x^3)-(\sum x^2)(\sum x))^2} = \frac{\left(5 \cdot 30 - \left(21\right)\cdot \left(5\right)\right)\cdot \left(5 \cdot 4035 - 111^{2}\right) - \left(5 \cdot 174 - \left(111\right)\cdot \left(5\right)\right)\cdot \left(5 \cdot 651 - \left(111\right)\cdot \left(21\right)\right)}{\left(5 \cdot 4035 - 111^{2}\right)\cdot \left(5 \cdot 111 - 21^{2}\right) - \left(5 \cdot 651 - \left(111\right)\cdot \left(21\right)\right)^{2}} = \frac{3}{2}$$$
$$$c = \frac{(\sum y)-b(\sum x)-a(\sum x^2)}{n} = \frac{5 - \left(\frac{3}{2}\right)\cdot \left(21\right) - \left(- \frac{3}{22}\right)\cdot \left(111\right)}{5} = - \frac{25}{11}$$$
Den bäst anpassade parabeln är $$$y = a x^{2} + b x + c$$$.
Således är parabeln för bästa anpassning $$$y = - \frac{3 x^{2}}{22} + \frac{3 x}{2} - \frac{25}{11}$$$.
Svar
Den bäst anpassade parabeln är $$$y = - \frac{3 x^{2}}{22} + \frac{3 x}{2} - \frac{25}{11}\approx - 0.136363636363636 x^{2} + 1.5 x - 2.272727272727273.$$$A