Comparing Competing Models · 比较相互竞争的模型
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| linear model/ˈlɪnɪə ˈmɒdl/ | 线性模型 | xiàn xìng mó xíng |
| quadratic model/kwɒˈdrætɪk ˈmɒdl/ | 二次模型 | èr cì mó xíng |
| exponential model/ˌekspəˈnenʃl ˈmɒdl/ | 指数模型 | zhǐ shù mó xíng |
| residuals/rɪˈsɪdʒuːəlz/ | 残差 | cán chà |
Three curves, one data set
- The same scatter of points could be fit by a line, a parabola, or an exponential.
- Each will pass near the data, but only one captures its true shape.
- Picking the wrong family gives predictions that drift badly.
- So how do we decide which model actually deserves our trust?
三条曲线,一组数据
- 同样一堆散点,可以用直线、抛物线或指数曲线来拟合。
- 每一条都会从数据附近经过,但只有一条抓住了它真正的形状。
- 选错了族,预测就会严重偏离。
- 那么,我们如何判断哪个模型真正值得信赖?
Try each family
- Fit a linear model 线性模型, a quadratic model 二次模型, and an exponential model 指数模型 to the same points.
- Each one produces a formula and a curve through the cloud of data.
- Look at where each curve sits relative to the points.
- A line cannot bend; a parabola bends once; an exponential curves ever upward.
试遍每个族
- 用线性模型(linear model)、二次模型(quadratic model)和指数模型(exponential model)拟合同一批点。
- 每一个都产生一个公式和一条穿过数据云的曲线。
- 看每条曲线相对于这些点的位置。
- 直线不能弯;抛物线弯一次;指数曲线不断向上弯。
Residuals judge the fit
- A residual 残差 is the gap between a data value and the model's prediction there.
- Small residuals, scattered with no pattern, mean a good fit.
- Large residuals, or residuals that curve, mean the wrong family.
- Comparing residuals is how you rank competing models with evidence.
残差判断拟合
- 残差(residuals)是一个数据值与模型在那里的预测值之间的差距。
- 残差小、且散布没有规律,意味着拟合得好。
- 残差大,或残差呈弯曲趋势,意味着选错了族。
- 比较残差,就是用证据给相互竞争的模型排名的方法。
A curve a straight line cannot match · 一条直线无法匹配的曲线
y = a·bˣ
An exponential bends to hug curved data. Try lowering the base toward 1 to see it flatten toward a line. · 指数函数会弯曲去贴合弯曲的数据。试着把底数降到接近 1,看它趋平成一条直线。
A residual is the difference between… · 残差是……之间的差。
A residual = observed − predicted. Small residuals mean the model fits the data closely. · 残差 = 观测值 − 预测值。残差小意味着模型与数据拟合得紧密。
To judge how well a model matches data, examine the size of its ____. · 要判断一个模型与数据匹配得多好,检查它的____的大小。
Smaller residuals (and no obvious pattern in them) signal a better-fitting model. · 更小的残差(且其中没有明显的规律)标志着一个拟合更好的模型。
Exponential eventually wins
- Over a long enough range, an exponential outgrows every polynomial.
- Even $x^{100}$ is left behind once $2^x$ gets going.
- So for data that keeps multiplying, the exponential is the natural long-run model.
- Polynomials curve, but they cannot keep up with steady doubling.
指数最终获胜
- 在足够长的范围上,指数函数会超过每一个多项式。
- 一旦 $2^x$ 起势,连 $x^{100}$ 都被甩在后面。
- 所以对于不断相乘的数据,指数函数是自然的长期模型。
- 多项式会弯曲,但跟不上稳定的翻倍。
Given enough time, an exponential model eventually grows faster than any polynomial model. · 只要时间足够,指数模型最终会比任何多项式模型增长得更快。
Exponential growth outpaces $x^2, x^3,$ even $x^{100}$ in the long run — the exponent always wins. · 从长远看,指数增长会超过 $x^2, x^3$,甚至 $x^{100}$——指数总是获胜。
Select all · 所有 true statements about comparing models. · 选出关于比较模型的所有正确说法。
The best fit has the smallest residuals, not the biggest. The other three are correct. · 最佳拟合的残差最小,而不是最大。其余三条正确。
Choose with context and evidence
- Use the residuals, but also use what you know about the situation.
- A savings account earns interest → exponential; a falling object → quadratic.
- Prefer the simplest model that fits well and makes physical sense.
- Justify your choice with both the data and the story behind it.
用情境和证据来选择
- 使用残差,但也要用你对情境的了解。
- 储蓄账户赚利息 → 指数;下落的物体 → 二次。
- 优先选择拟合得好且在物理上说得通的最简单模型。
- 用数据和它背后的故事一起来支撑你的选择。
Data doubling at a steady rate is best modeled by a… · 以稳定速率翻倍的数据,最适合用……建模。
Constant-ratio doubling is the hallmark of an exponential model, not a polynomial one. · 按恒定比例翻倍是指数模型的标志,而不是多项式模型。
A wiggly high-degree polynomial can pass through every point and still be a terrible model — it swings wildly between the data and predicts nonsense outside it. A close fit alone is not enough; the family must match the underlying behavior.
一个扭动的高次多项式可以穿过每一个点,却仍是一个糟糕的模型——它在数据之间剧烈摆动,在数据之外预测出胡话。仅仅拟合得近是不够的;族必须与背后的行为相符。
Sales over four months: $100, 150, 225, 340$.
- Ratios: $150/100 = 1.5$, $225/150 = 1.5$, $340/225 \approx 1.5$ — nearly constant.
- A constant ratio points to an exponential model, not a line.
- Model: $S \approx 100 \cdot 1.5^{\,m}$, matching the roughly $50\%$ monthly growth.
四个月的销量:$100, 150, 225, 340$。
- 比值:$150/100 = 1.5$,$225/150 = 1.5$,$340/225 \approx 1.5$——几乎恒定。
- 恒定的比值指向指数模型,而不是直线。
- 模型:$S \approx 100 \cdot 1.5^{\,m}$,符合每月约 $50\%$ 的增长。
To compare a linear model, quadratic model, and exponential model on the same data, examine the residuals — smaller is better. Remember an exponential eventually outgrows any polynomial, and let both the evidence and the context guide your final choice.
要在同一批数据上比较线性模型、二次模型和指数模型,检查残差——越小越好。记住指数函数最终会超过任何多项式,并让证据和情境共同指导你最终的选择。