The Least Squares Regression Line | 最小二乘回归线

📚 The Least Squares Regression Line | 最小二乘回归线

In statistics, we often collect data that consists of pairs of observations on two variables. The least squares regression line is a fundamental tool for modelling the linear relationship between an explanatory variable and a response variable. By minimising the sum of the squares of the vertical distances from the data points to the line, we obtain the line of best fit that can be used for prediction and analysis.

在统计学中，我们经常收集两个变量的成对观测数据。最小二乘回归线是对解释变量和响应变量之间的线性关系进行建模的基本工具。通过最小化数据点到直线的垂直距离的平方和，我们得到可用于预测和分析的最佳拟合直线。

1. Introduction to Bivariate Data | 双变量数据简介

Bivariate data consist of measurements on two variables for each individual in a sample. For example, we might record the number of hours studied (x) and the test score (y) for a group of students. The variable x is often called the independent or explanatory variable, while y is the dependent or response variable.

双变量数据由样本中每个个体的两个变量的测量值组成。例如，我们可以记录一组学生的学习时间（x）和考试成绩（y）。变量 x 通常称为自变量或解释变量，而 y 为因变量或响应变量。

We are interested in exploring whether changes in x are associated with changes in y. A scatter plot is the first step to visualise this relationship.

我们关心 x 的变化是否与 y 的变化相关联。散点图是可视化这种关系的第一步。

2. Scatter Plots and Correlation | 散点图与相关性

A scatter plot displays each pair (x, y) as a point in the coordinate plane. The pattern of points suggests the direction, form, and strength of the relationship. If the points tend to rise from left to right, there is a positive correlation. If they fall, the correlation is negative. Points scattered randomly indicate little or no linear correlation.

散点图将每个数据对 (x, y) 显示为坐标平面上的一个点。点的形态指示了关系的方向、形式和强度。如果点从左到右呈上升趋势，则为正相关；若下降，则为负相关。点随机散布表明线性相关很弱或不存在。

The Pearson correlation coefficient r measures the strength of a linear association. Its value lies between −1 and 1. However, even when r indicates a strong correlation, we need a regression line to make predictions.

皮尔逊相关系数 r 度量线性关联的强度，其值介于 −1 和 1 之间。然而，即使 r 表明有强相关，我们仍需要一条回归线来进行预测。

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3. What is a Regression Line? | 什么是回归线？

egression of y on x, we treat x as the explanatory variable and y as the response. The equation is typically written as ŷ = a + b x, where ŷ denotes the predicted value of y for a given x.

回归线通过寻找最能描述 y 如何随 x 变化的直线来概括 x 和 y 的关系。在进行 y 对 x 的回归时，我们将 x 视为解释变量，y 为响应变量。方程通常写为 ŷ = a + b x，其中 ŷ 表示给定 x 的预测 y 值。

cted value of y when x = 0, although this may not always have a meaningful interpretation.

斜率 b 表示 x 每增加一个单位时 y 的估计变化量。截距 a 是 x = 0 时 y 的预测值，尽管它不一定总有实际意义。

4. The Least Squares Principle | 最小二乘原理

For any line we choose, there will be vertical deviations called residuals: e_i = y_i − ŷ_i. The least squares method selects a and b to minimise the sum of the squares of these residuals, that is, minimise Σ e_i² = Σ (y_i − a − b x_i)².

对于任何我们选择的直线，都会存在垂直偏差，称为残差：e_i = y_i − ŷ_i。最小二乘法选择 a 和 b 以使这些残差的平方和最小，即最小化 Σ e_i² = Σ (y_i − a − b x_i)²。

This criterion gives more weight to larger deviations and ensures that positive and negative residuals do not cancel. It leads to unique formulas for the best-fitting slope and intercept.

这一准则对较大偏差施加更大的权重，并确保正负残差不会互相抵消，从而得到唯一的最佳斜率与截距公式。

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