Statistics 简明教程
Statistics - Quadratic Regression Equation
采用二次回归法计算出最适合给定数据集的抛物线方程。其形式如下：
Quadratic regression is deployed to figure out an equation of the parabola which can best fit the given set of data. It is of following form:
最小二乘法可用于找出二次回归方程。在此方法中，我们找出 a、b 和 c 的值，使每个给定点 (${x_i, y_i}$) 和抛物线方程 (${ y = ax^2 + bx + c}$) 之间的垂直距离平方最小。抛物线曲线的矩阵方程由下式给出：
Least square method can be used to find out the Quadratic Regression Equation. In this method, we find out the value of a, b and c so that squared vertical distance between each given point (${x_i, y_i}$) and the parabola equation (${ y = ax^2 + bx + c}$) is minimal. The matrix equation for the parabolic curve is given by:
Correlation Coefficient, r
相关系数 r 决定了二次方程可以多好地拟合给定数据。如果 r 接近 1，则拟合效果良好。可以使用以下公式计算 r。
Correlation coefficient, r determines how good a quardratic equation can fit the given data. If r is close to 1 then it is good fit. r can be computed by following formula.
通常，使用二次回归计算器来计算二次回归方程。
Generally, quadratic regression calculators are used to compute the quadratic regression equation.
Example
Problem Statement:
Problem Statement:
计算以下数据的二次回归方程。检查其最佳拟合度。
Compute the quadratic regression equation of following data. Check its best fitness.
x |
-3 |
-2 |
-1 |
0 |
1 |
2 |
3 |
y |
7.5 |
3 |
0.5 |
1 |
3 |
6 |
14 |
Solution:
Solution:
在计算器上输入 x 和 y 值来计算二次回归。上述点的最佳二次回归方程如下：
Compute a quadratic regression on calculator by putting the x and y values. The best fit quadratic equation for above points comes as
要检查最佳拟合度，请绘制图形。
To check the best fitness, plot the graph.
因此，数据相关系数 r 的值为 0.99420，接近 1。因此，二次回归方程最适合。
So the value of Correlation Coefficient, r for the data is 0.99420 and is close to 1. Hence quadratic regression equation is best fit.