计算最小二乘线性回归
本节将详细介绍如何计算最小二乘线性回归方程。最小二乘法是一种用于寻找最佳拟合直线的数学方法，它通过最小化实际观测值与预测值之间的残差平方和来确定回归直线的参数。
This section will detail how to calculate the least squares linear regression equation. Least squares is a mathematical method used to find the best-fitting line by minimizing the sum of squared residuals between actual observed values and predicted values.
通过学习本节内容，您将掌握如何手动计算回归系数、理解计算过程背后的数学原理，以及如何解释计算结果的实际意义。这些技能对于进行数据分析和统计预测至关重要。
By studying this section, you will learn how to manually calculate regression coefficients, understand the mathematical principles behind the calculation process, and how to interpret the practical significance of calculation results. These skills are essential for data analysis and statistical prediction.
衡量观测值与回归直线预测值之间差异的总和，是最小二乘法优化的目标函数。
A measure of the total difference between observed values and values predicted by the regression line, which is the objective function optimized by least squares method.
包括斜率(b)和截距(a)，决定了回归直线的位置和方向，反映了自变量对因变量的影响程度。
Includes slope (b) and intercept (a), which determine the position and direction of the regression line, reflecting the extent of influence of the independent variable on the dependent variable.
协方差衡量两个变量的共同变化程度，方差衡量单个变量的离散程度，它们是计算回归系数的基础。
Covariance measures the degree of joint variation between two variables, while variance measures the dispersion of a single variable. They form the basis for calculating regression coefficients.
\[b = \frac{S_{xy}}{S_{xx}}\]
\[a = \bar{y} - b\bar{x}\]
\[S_{xx} = \sum (x_i - \bar{x})^2\]
\[S_{xy} = \sum (x_i - \bar{x})(y_i - \bar{y})\]