Regression, financial models and residuals
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| residual/rɪˈsɪdʒuːəl/ | sai số |
Is the prediction supported by the data?
- A line predicts bus travel time from distance. A small calculation error matters less than using a model outside its evidence.
- This lesson studies residual 残差: The observed value minus the value predicted by a fitted model.
The method
- A linear model y=a+bx has intercept a and slope b with contextual units.
- Inspect residuals and the data range.
- For repeated percentage change use a geometric model; for a loan distinguish principal, rate, repayment and period.
The model t=8+2d fits observed journey times. An observed 21 minutes against a predicted 18 means what?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Residual = observed - predicted = 21-18 = 3. Residuals reveal systematic model weakness; a good fit on average can still hide poor regions.
A worked case
The fitted model t=8+2d predicts 18 minutes at d=5 km. If the observed time is 21, the residual is 3 minutes.
Regression, financial models and residuals
A linear model y=a+bx has intercept a and slope b with contextual units
Use the positive residual to explain how the fitted model misses this observation.
For t=8+2d, find t at d=5.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Substitute distance 5: 8+2×5=18 minutes.
Where it goes wrong
- A good fit does not prove a causal mechanism.
- A correlation coefficient measures linear association, not the gradient.
- Calculator output must be translated into a model, checked and interpreted.
Not true: A high correlation permits reliable extrapolation to any distance. A model fitted to data is reliable only near that data; extrapolating far outside it is unreliable.
Find the residual if observed t=21 and predicted t=18.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Residual=observation-prediction=21-18=3 minutes.
A high correlation permits reliable extrapolation to any distance.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
A model fitted to data is reliable only near that data; extrapolating far outside it is unreliable.
Applying it
- Use the AQA large data set and suitable statistical displays.
- Interpret residuals, correlation and model limitations; association alone does not establish causation.
For t=8+2d, find the predicted time when d=7.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Substitute d=7: t=8+2×7=22 minutes.
Match each modelling quantity to its meaning.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Model parameters carry contextual units, residuals measure per-point error, and repeated percentage change compounds multiplicatively.
In your course
- AQA A-level Mathematics (7357), specification section L.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Residual: The observed value minus the value predicted by a fitted model.
- A linear model y=a+bx has intercept a and slope b with contextual units.
- Watch for: A good fit does not prove a causal mechanism.