Linear Regression | 线性回归

📚 Linear Regression | 线性回归

Linear regression is a fundamental statistical tool in the Edexcel A-Level Mathematics specification. It allows you to model the linear relationship between two variables, make predictions, and assess the strength of association.

线性回归是 Edexcel A-Level 数学大纲中的基础统计工具。它可以帮助你对两个变量之间的线性关系建模、进行预测，并评估关联强度。

1. What Is Linear Regression? | 什么是线性回归？

In a bivariate data set, one variable is treated as the explanatory variable x and the other as the response variable y. Linear regression finds the straight line that best describes how y changes with x, assuming the scatter diagram shows a roughly linear pattern.

在双变量数据集中，一个变量作为解释变量 x，另一个作为响应变量 y。线性回归寻找一条最佳描述 y 随 x 变化规律的直线，前提是散点图大致呈线性趋势。

The regression line of y on x is used when we want to predict values of y from known values of x. It is not the same as the regression line of x on y.

y 对 x 的回归直线用于从已知 x 值预测 y 值。它与 x 对 y 的回归直线不是同一条线。

2. The Least Squares Regression Line | 最小二乘回归直线

The best-fitting straight line is obtained by the method of least squares. We minimise the sum of the squares of the vertical residuals, where a residual is the difference between an observed y value and the predicted y value from the line.

最佳拟合直线通过最小二乘法获得。我们要最小化纵向残差的平方和，其中残差是实际观测值 y 与直线预测值之差。

For the regression line of y on x, the predicted value is written as ŷ = a + bx, and the residual for each point is eᵢ = yᵢ − ŷᵢ. Minimising Σeᵢ² gives the least squares estimates for a and b.

对于 y 对 x 的回归直线，预测值记为 ŷ = a + bx，每个点的残差为 eᵢ = yᵢ − ŷᵢ。最小化 Σeᵢ² 可以得到 a 和 b 的最小二乘估计。

ŷ = a + bx

3. Key Formulae for Edexcel | Edexcel 关键公式

Given n pairs of data (xᵢ, yᵢ), the summary quantities Sxx, Syy and Sxy are calculated first. These quantities measure the variability of x, the variability of y, and the joint variability of x and y.

给定 n 对数据 (xᵢ, yᵢ)，首先计算汇总量 Sxx、Syy 和 Sxy。这些量分别衡量 x 的变异、y 的变异以及 x 和 y 的联合变异。

Sxx = Σx² − (Σx)² ÷ n

Syy = Σy² − (Σy)² ÷ n

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