Regression, financial models and residuals
مُقدَّم| English | المعنى |
|---|---|
| residual/rɪˈsɪdʒuːəl/ | الباقية |
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?
الصعوبة 1/5 · 100 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
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. A ¥1000 deposit at 5% compound annual interest becomes 1000×1.05³=1157.625 after 3 years.
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.
الصعوبة 2/5 · 140 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
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.
الصعوبة 2/5 · 140 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
Residual=observation-prediction=21-18=3 minutes.
A high correlation permits reliable extrapolation to any distance.
الصعوبة 1/5 · 100 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
A model fitted to data is reliable only near that data; extrapolating far outside it is unreliable.
Applying it
- In IB AI, record the data source, domain, assumptions and calculator method.
- Compare an alternative model and judge predictions against residuals.
- A financial answer must state payment timing and whether interest is compounded.
Find the value of 1000 after 3 years at 5% compound annual interest. Give it to 3 significant figures.
الصعوبة 2/5 · 140 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
Compound annually: 1000×1.05³=1157.625.
Match each modelling quantity to its meaning.
الصعوبة 2/5 · 140 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
Model parameters carry contextual units, residuals measure per-point error, and repeated percentage change compounds multiplicatively.
In your course
- Pearson Edexcel International A Level Mathematics, unit S1.
- 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.
Write from a prediction of 16
- The model is t equals 4 plus 3d.
- Predict t when d equals 4.
The model t equals 4 plus 3d predicts the time. What is the prediction when d equals 4?
4 plus 12 is 16.
What does the model predict?
الصعوبة 1/5 · 100 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
4 plus 3 times 4 is 16. Other options — 18 (the lesson prediction); 4 (the starting value of the model).