Residuals · 残差
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| residual/rɪˈsɪdʒuːəl/ | 残差 | cán chà |
| residual plot/rɪˈsɪdʒuːəl plɒt/ | 残差图 | cán chà tú |
How wrong is each prediction?
- A residual 残差 is the leftover: observed minus predicted, $y - \hat{y}$.
- It's the vertical gap between a data point and the regression line.
- Positive residual: the point sits above the line (model underpredicted).
- Negative residual: the point sits below the line (model overpredicted).
每次预测错了多少？
- 残差是剩下的部分：观测值减去预测值，$y - \hat{y}$。
- 它是数据点与回归线之间的竖直间隙。
- 正残差：点位于线上方（模型低估了）。
- 负残差：点位于线下方（模型高估了）。
The residual plot
- A residual plot 残差图 graphs the residuals ($y$-axis) against $x$ (or against $\hat{y}$).
- It magnifies the leftover pattern the eye misses on the original scatterplot.
- Each point's height is how far off that prediction was.
- The horizontal line at residual $=0$ is "a perfect prediction."
残差图
- 残差图把残差（$y$ 轴）对 $x$（或对 $\hat{y}$）作图。
- 它放大了那些在原散点图上肉眼错过的剩余模式。
- 每个点的高度，就是那次预测偏离了多远。
- 残差 $=0$ 处的水平线代表“完美预测”。
Is a line appropriate?
- Good sign: residuals scattered randomly around zero — no leftover pattern → a line fits well.
- Bad sign: a curve or U-shape in the residuals → the true form is non-linear.
- Also watch for a fan shape (spread growing), which signals unequal variability.
- The residual plot is the deciding test of whether linear regression was the right choice.
直线合适吗？
- 好迹象：残差随机地散布在零附近——没有剩余模式 → 直线拟合得好。
- 坏迹象：残差里出现曲线或 U 形 → 真实形状是非线性的。
- 还要留意扇形（散布逐渐变大），它预示变异性不均等。
- 残差图是判定线性回归是否为正确选择的决定性检验。
Over- and under-prediction
- Where residuals are negative, the model overpredicts (guessed too high).
- Where residuals are positive, the model underpredicts (guessed too low).
- A stretch of same-sign residuals means the line is systematically off there.
- Random signs, small sizes → the model is doing its job.
高估与低估
- 残差为负的地方，模型高估了（猜得太高）。
- 残差为正的地方，模型低估了（猜得太低）。
- 一段同号残差意味着直线在那里系统性地偏了。
- 符号随机、大小很小 → 模型在正常工作。
Sign convention trips people up: a residual is observed − predicted. A positive residual means the actual value is above the line, so the model underpredicted it. And a curved residual plot is the clearest evidence that a straight-line model is wrong, even if the original scatterplot looked "close enough."
符号约定容易把人绊倒：残差是观测 − 预测。正残差意味着实际值在线上方，所以模型低估了它。而弯曲的残差图是直线模型错误的最清晰证据，即便原散点图看起来“足够接近”。
A house is predicted at $300$ but actually sells for $330$ (thousands of dollars).
- Residual $= 330 - 300 = +30$ — positive.
- The point sits above the line; the model underpredicted this house.
- If many nearby points share a positive residual, the line is biased low there.
一套预测为 $300$、实际售出 $330$（单位：千美元）的房子。
- 残差 $= 330 - 300 = +30$——正。
- 点位于线上方；模型低估了这套房子。
- 如果附近许多点都是正残差，直线在那里就偏低了。
A residual is $y - \hat{y}$ (observed − predicted): positive = point above the line (underpredicted), negative = below (overpredicted). A residual plot with random scatter around $0$ supports a linear model; a curved pattern means the relationship is non-linear and a line is not appropriate.
残差是 $y - \hat{y}$（观测 − 预测）：正 = 点在线上方（低估），负 = 在下方（高估）。残差图若在 $0$ 附近呈随机散布，支持线性模型；若呈弯曲模式，则说明关系是非线性的，直线不合适。
Residuals = vertical gaps to the line · 残差 = 到直线的竖直间隙
Each point's vertical distance from the line is its residual. · 每个点到直线的竖直距离就是它的残差。
A house predicted at 300 sold for 330 (thousands). What is the residual? · 一套预测为 300、实际售出 330（千元）的房子。残差是多少？
Residual = observed − predicted = 330 − 300 = 30. · 残差 = 观测 − 预测 = 330 − 300 = 30。
A positive residual tells you the point is... and the model... · 正残差告诉你点在……而模型……
Observed > predicted → point above line → model underpredicted. · 观测 > 预测 → 点在线上方 → 模型低估。
What does a residual plot with an obvious curved (U-shaped) pattern tell you? · 残差图呈明显的弯曲（U 形）模式说明了什么？
A leftover curve means the true form isn't linear. · 剩余的曲线意味着真实形状不是线性的。
A residual plot with points scattered randomly around zero suggests a linear model is appropriate. · 残差图的点随机散布在零附近，说明线性模型是合适的。
Random scatter, no pattern → a line fits well. · 随机散布、没有模式 → 直线拟合得好。
A residual is computed as observed minus ___. · 残差的计算是观测值减去 ___。
Residual = y − y-hat = observed − predicted. · 残差 = y − y-hat = 观测 − 预测。