Comparing Competing Models
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| linear model/ˈlɪnɪə ˈmɒdl/ | mô hình tuyến tính |
| quadratic model/kwɒˈdrætɪk ˈmɒdl/ | mô hình bậc hai |
| exponential model/ˌekspəˈnenʃl ˈmɒdl/ | mô hình mũ |
| residuals/rɪˈsɪdʒuːəlz/ | phần dư |
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?
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.
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.
A curve a straight line cannot match
y = a·bˣ
An exponential bends to hug curved data. Try lowering the base toward 1 to see it flatten toward a line.
A residual is the difference between…
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A residual = observed − predicted. Small residuals mean the model fits the data closely.
To judge how well a model matches data, examine the size of its ____.
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Smaller residuals (and no obvious pattern in them) signal a better-fitting model.
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.
Given enough time, an exponential model eventually grows faster than any polynomial model.
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Exponential growth outpaces $x^2, x^3,$ even $x^{100}$ in the long run — the exponent always wins.
Select all true statements about comparing models.
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The best fit has the smallest residuals, not the biggest. The other three are correct.
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.
Data doubling at a steady rate is best modeled by a…
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Constant-ratio doubling is the hallmark of an exponential model, not a polynomial one.
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.
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.
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.
Write from a residual total of 6
- One model's residuals are 2, negative 1 and 3.
- Another model's residuals are 4, 4 and 4.
- Which total of absolute residuals is smaller?
Model A has residuals 2, negative 1 and 3. Model B has residuals 4, 4 and 4. What is the total of the absolute residuals for model A?
2 plus 1 plus 3 is 6.
What is model A's absolute residual total?
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Absolute residuals add to 6.