Finding the ‘Heavenly Path’ of Data Points: A Linear Regression Walkthrough

Imagine a cloud of data points on a coordinate plane and draw a line that captures their overall trend—this line, expressed as y = a·x + b, represents the "Heavenly Path" (regression line) that the points tend to follow.

The slope a indicates how steep the line is, while the intercept b gives the value of y when x = 0. To evaluate how well a line fits the data, the article introduces a cost (or loss) function: the average of the squared differences between the actual

y</i> values and the predicted values <code>ŷ_i = a·x_i + b

. This is the mean‑squared‑error (MSE):

Minimizing the MSE requires computing its partial derivatives with respect to a and b. Using the chain rule, the gradient for the slope becomes -2·Σ (x_i·(y_i − (a·x_i + b))) and for the intercept -2·Σ (y_i − (a·x_i + b)). The article shows each differentiation step, including how terms that do not contain the variable disappear.

With the gradients, gradient descent updates the parameters iteratively:

Initialize a and b (randomly or at zero).

Compute gradients using the formulas above.

Update a ← a − α·∂J/∂a and b ← b − α·∂J/∂b, where α is the learning rate.

Repeat steps 2‑3 until the loss change is below a threshold or a maximum number of iterations is reached.

The article then provides a complete Python script that loads the data, computes the optimal a and b, plots the original points and the fitted line, and prints the equation and data ranges.

# 导入必要的库
import numpy as np
import matplotlib.pyplot as plt

# 使用之前计算的 a_orig 和 b_orig（线性回归的斜率和截距）

# 重新绘制图表
plt.figure(figsize=(12, 8))

# 绘制原始数据点
plt.scatter(x_orig, y_orig, color='blue', alpha=0.7, label='Data points')

# 计算拟合线的点
x_line = np.linspace(x_orig.min(), x_orig.max(), 100)
y_line = a_orig * x_line + b_orig

# 绘制拟合线
plt.plot(x_line, y_line, color='green', linewidth=3, linestyle='--', label='Fitted line')

# 设置图表标题和轴标签
plt.title("最终修正后的散点和拟合线", fontsize=16)
plt.xlabel('房屋面积 (平方米)', fontsize=12)
plt.ylabel('房屋价格 (万元)', fontsize=12)
plt.legend(fontsize=10)

# 设置合适的坐标轴范围
x_margin = (x_orig.max() - x_orig.min()) * 0.1
y_margin = (y_orig.max() - y_orig.min()) * 0.1
plt.xlim(x_orig.min() - x_margin, x_orig.max() + x_margin)
plt.ylim(min(y_orig.min(), y_line.min()) - y_margin, max(y_orig.max(), y_line.max()) + y_margin)

# 添加网格线
plt.grid(True, linestyle='--', alpha=0.7)

# 在图上添加拟合线方程
equation = f'y = {a_orig:.2f}x + {b_orig:.2f}'
plt.text(0.05, 0.95, equation, transform=plt.gca().transAxes, fontsize=12, verticalalignment='top')

# 显示图像
plt.show()

# 打印拟合线信息和原始数据范围
print(f"拟合线方程: {equation}")
print(f"最终修正后的拟合线起点: ({x_line[0]:.2f}, {y_line[0]:.2f})")
print(f"最终修正后的拟合线终点: ({x_line[-1]:.2f}, {y_line[-1]:.2f})")
print(f"原始数据 x 范围: ({x_orig.min():.2f}, {x_orig.max():.2f})")
print(f"原始数据 y 范围: ({y_orig.min():.2f}, {y_orig.max():.2f})")

Running this code visualizes the data, draws the regression line, and demonstrates how gradient descent converges to the optimal parameters, completing the "Heavenly Path" analogy.