Worked example

示例讲解

Slope, intercept and the variables

斜率、截距与变量

The line of best fit through the sunflower's measurements takes the form height = intercept + slope × weeks. Click each part to see what it means and what it works out to for this data set.

穿过向日葵测量数据的最佳拟合直线形式为 height = intercept + slope × weeks(高度 = 截距 + 斜率 × 周数)。点击每个部分,查看它的含义以及在这组数据中的具体数值。

Try it yourself

动手试一试

Fitting a line with numpy

用 numpy 拟合一条直线

The slope and intercept above weren't guessed, they come from an actual calculation: the line that minimises the total squared distance between itself and every measured point. This runs that calculation on the sunflower's six weekly measurements, then uses the result to predict a seventh value the club hasn't measured yet.

上面的斜率和截距并非凭空猜测,而是来自一次真实的计算:找到一条使自身与每一个测量点之间的总平方距离最小的直线。这里对向日葵的六周测量数据执行这项计算,然后用结果预测俱乐部尚未测量过的第七个数值。

This prints a slope of about 4.34, an intercept of about 7.8, and an R-squared of about 0.99, an almost perfect fit. Try changing one height value to something that breaks the pattern (say, 34 to 5) and re-running: the R-squared drops sharply, and the week-10 prediction swings with it, exactly the kind of thing a single bad measurement can do to a small data set.

这会打印出约 4.34 的斜率、约 7.8 的截距,以及约 0.99 的 R 平方值,几乎是完美拟合。试着把某个高度值改成会打破规律的数字(例如把 34 改成 5)再重新运行:R 平方值会大幅下降,第 10 周的预测值也会随之剧烈波动,这正是一次糟糕的测量会对一个小数据集造成的影响。

Worked example

示例讲解

Fitting and plotting with scikit-learn

用 scikit-learn 拟合并绘图

The calculation above was done by hand, one NumPy line at a time, to show exactly where the slope and intercept come from. In practice, most real work uses scikit-learn's built-in LinearRegression instead, and matplotlib to actually see the line against the data. This fits the same six sunflower measurements and plots the result.

上面的计算是逐行手动用 NumPy 完成的,目的是准确展示斜率和截距的来源。而在实际工作中,大多数情况会改用 scikit-learn 内置的 LinearRegression,并用 matplotlib 实际查看这条线与数据的对比。这里对同样的六个向日葵测量值进行拟合并绘制结果。

import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression

# The same six sunflower measurements, fitted with scikit-learn instead of by hand
weeks = np.array([1, 2, 3, 4, 5, 6]).reshape(-1, 1)
height = np.array([12, 17, 20, 26, 29, 34])

model = LinearRegression()
model.fit(weeks, height)

print("Slope:", round(model.coef_[0], 2))
print("Intercept:", round(model.intercept_, 2))
print("R-squared:", round(model.score(weeks, height), 2))

# Predict the height at week 10, the same value calculated by hand earlier
week10 = np.array([[10]])
print("Predicted height at week 10:", round(model.predict(week10)[0], 2))

# Plot the six measurements against the fitted line of best fit
plt.scatter(weeks, height, color="green", label="Measured height")
line_x = np.array([[0], [10]])
line_y = model.predict(line_x)
plt.plot(line_x, line_y, color="orange", label="Line of best fit")
plt.xlabel("Weeks since planting")
plt.ylabel("Height (cm)")
plt.title("Sunflower growth: linear regression")
plt.legend()
plt.show()

Running this prints the same values as the numpy version above, Slope: 4.34, Intercept: 7.8, R-squared: 0.99, Predicted height at week 10: 51.23, since LinearRegression fits a line the same way (minimising squared distance) as the manual calculation. The plot window then opens showing the six green points with the orange line of best fit running through them. This isn't wired into this page's own in-browser console: between scikit-learn and matplotlib, that's a large amount of supporting library to download just for one page, so try this one in your own Python setup instead (pip install scikit-learn numpy matplotlib).

运行这段代码会打印出与上面 numpy 版本相同的数值,Slope: 4.34、Intercept: 7.8、R-squared: 0.99、Predicted height at week 10: 51.23,因为 LinearRegression 拟合直线的方式(最小化平方距离)与手动计算完全相同。随后会打开一个绘图窗口,显示六个绿色数据点以及穿过它们的橙色最佳拟合直线。这里没有接入本页自身的浏览器内控制台:scikit-learn 与 matplotlib 加起来是相当大量的支持库,仅为一个页面下载并不划算,因此请在你自己的 Python 环境中尝试(pip install scikit-learn numpy matplotlib)。