5.3 Calculating Least Squares Linear Regression

一、核心概念 / Core Concepts

1. 最小二乘法 / Least Squares Method

最小二乘法是一种数学优化技术，通过最小化误差的平方和来寻找数据的最佳函数匹配。

The least squares method is a mathematical optimization technique that finds the best function match for the data by minimizing the sum of the squares of the errors.

目标函数 / Objective Function：最小化残差平方和 Minimize the sum of squared residuals

\[S = \sum_{i=1}^{n} (y_i - \hat{y}_i)^2\]

2. 回归直线 / Regression Line

回归直线是对自变量和因变量之间线性关系的数学描述，其形式为：

The regression line is a mathematical description of the linear relationship between the independent variable and the dependent variable, with the form:

\[\hat{y} = a + bx\]

其中，\(a\) 是截距（intercept），\(b\) 是斜率（slope）。

Where \(a\) is the intercept and \(b\) is the slope.

3. 关键统计量 / Key Statistics

离均差平方和 (Sum of Squared Deviations) \(S_{xx}\)：衡量自变量离散程度的指标。
The sum of squared deviations of the independent variable from its mean.
离均差乘积和 (Sum of Products of Deviations) \(S_{xy}\)：衡量自变量和因变量协同变化程度的指标。
The sum of products of deviations of the independent and dependent variables from their means.
回归系数 (Regression Coefficients)：包括斜率 \(b\) 和截距 \(a\)，描述自变量与因变量之间的关系强度和方向。
Include slope \(b\) and intercept \(a\), describing the strength and direction of the relationship between the independent and dependent variables.

二、重要公式 / Important Formulas

回归分析核心公式 / Core Formulas for Regression Analysis

1. \(S_{xx}\) 的计算公式 / Formula for \(S_{xx}\)

\[S_{xx} = \sum_{i=1}^{n} (x_i - \bar{x})^2 = \sum x_i^2 - \frac{(\sum x_i)^2}{n}\]

自变量的离均差平方和，衡量自变量的离散程度。

Sum of squared deviations of the independent variable, measuring the spread of the independent variable.

2. \(S_{xy}\) 的计算公式 / Formula for \(S_{xy}\)

\[S_{xy} = \sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y}) = \sum x_i y_i - \frac{(\sum x_i)(\sum y_i)}{n}\]

自变量与因变量的离均差乘积和，衡量它们之间的线性关系强度。

Sum of products of deviations of the independent and dependent variables, measuring the strength of their linear relationship.

3. 斜率 \(b\) 的计算公式 / Formula for Slope \(b\)

\[b = \frac{S_{xy}}{S_{xx}}\]

斜率表示自变量每变化一个单位时，因变量的平均变化量。

The slope represents the average change in the dependent variable for each unit change in the independent variable.

4. 截距 \(a\) 的计算公式 / Formula for Intercept \(a\)

\[a = \bar{y} - b\bar{x}\]

截距表示当自变量为0时，因变量的预测值。

The intercept represents the predicted value of the dependent variable when the independent variable is 0.

5. 回归方程 / Regression Equation

\[\hat{y} = a + bx\]

用于根据自变量的值预测因变量的值。

Used to predict the value of the dependent variable based on the value of the independent variable.

三、计算步骤 / Calculation Steps

最小二乘线性回归计算步骤 / Steps for Calculating Least Squares Linear Regression

步骤1：确定样本量 \(n\)
Step 1: Determine sample size \(n\)
步骤2：计算必要的总和
Step 2: Calculate necessary sums
计算 \(\sum x_i\)、\(\sum y_i\)、\(\sum x_i^2\)、\(\sum y_i^2\) 和 \(\sum x_i y_i\)
Calculate \(\sum x_i\), \(\sum y_i\), \(\sum x_i^2\), \(\sum y_i^2\), and \(\sum x_i y_i\)
步骤3：计算平均值
Step 3: Calculate means
\(\bar{x} = \frac{\sum x_i}{n}\)，\(\bar{y} = \frac{\sum y_i}{n}\)
\(\bar{x} = \frac{\sum x_i}{n}\), \(\bar{y} = \frac{\sum y_i}{n}\)
步骤4：计算 \(S_{xx}\)
Step 4: Calculate \(S_{xx}\)
\(S_{xx} = \sum x_i^2 - \frac{(\sum x_i)^2}{n}\)
步骤5：计算 \(S_{xy}\)
Step 5: Calculate \(S_{xy}\)
\(S_{xy} = \sum x_i y_i - \frac{(\sum x_i)(\sum y_i)}{n}\)
步骤6：计算斜率 \(b\)
Step 6: Calculate slope \(b\)
\(b = \frac{S_{xy}}{S_{xx}}\)
步骤7：计算截距 \(a\)
Step 7: Calculate intercept \(a\)
\(a = \bar{y} - b\bar{x}\)
步骤8：写出回归方程
Step 8: Write the regression equation
\(\hat{y} = a + bx\)

四、计算示例 / Calculation Example

步骤 / Step	计算内容 / Calculation Content	结果 / Result
1	样本量 \(n\)	\(n = 5\)
2	\(\sum x_i\)	25
	\(\sum y_i\)	116
	\(\sum x_i^2\)	151
	\(\sum y_i^2\)	2928
	\(\sum x_i y_i\)	658
3	\(\bar{x} = \frac{25}{5}\)	5
3	\(\bar{y} = \frac{116}{5}\)	23.2
4	\(S_{xx} = 151 - \frac{25^2}{5}\)	26
5	\(S_{xy} = 658 - \frac{25 \times 116}{5}\)	78
6	\(b = \frac{78}{26}\)	3
7	\(a = 23.2 - 3 \times 5\)	8.2
8	回归方程 / Regression Equation	\(\hat{y} = 8.2 + 3x\)

五、重要提示 / Important Tips

计算与应用注意事项 / Notes on Calculation and Application

线性关系假设：确保自变量和因变量之间存在线性关系，否则回归分析可能不适用。
Linear Relationship Assumption: Ensure there is a linear relationship between the independent and dependent variables; otherwise, regression analysis may not be applicable.
异常值影响：异常值可能严重影响回归系数的估计，应在分析前检查并处理异常值。
Outlier Impact: Outliers can significantly affect regression coefficient estimates; they should be checked and handled before analysis.
回归直线的性质：回归直线总是通过点 \((\bar{x}, \bar{y})\)，这可以作为计算正确性的检查方法。
Property of Regression Line: The regression line always passes through the point \((\bar{x}, \bar{y})\), which can be used as a method to check calculation correctness.
避免外推：回归方程通常只在原始数据范围内有效，避免对超出范围的值进行预测。
Avoid Extrapolation: Regression equations are usually only valid within the range of the original data; avoid making predictions for values outside this range.
计算精度：在手动计算时，保留足够的小数位数以确保结果准确性。
Calculation Precision: During manual calculations, retain sufficient decimal places to ensure result accuracy.