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eduKateSG · Why Mathematics? · Response Surface Methodology: Curvature, Central Composite Designs and Optimisation
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 8–14
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 15–21
Check assumptions
See where design and calibration matter.Route 4 · Chapters 22–28
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 29–42
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why is mathematics important when the best setting lies between the low and high levels already tested? Response surface methodology fits curvature, draws contours and searches a safe design region. Central composite designs add centre and star points so a quadratic model can guide optimisation and validation.
Did You Know? An apparent optimum from a fitted equation is only a prediction. It becomes useful engineering knowledge after the stationary point is inside the design region, the curvature is credible and confirmation runs reproduce the response.
Full chapter index
Chapters 1–7 · Understand the question
Chapters 8–14 · Reproduce the example
Chapters 15–21 · Check assumptions
Chapters 22–28 · Explore applications
Chapters 29–42 · Study and verify
- Final decision
- A spreadsheet workflow
- A coding workflow
- Applications and careers
- A student investigation
- Common misconceptions
- Connection to recent mathematics articles
- Guidance for students and parents
- Practice questions
- Frequently asked questions
- A reporting checklist
- Optimisation follows learning
- A final surface audit
- Useful next reading
After screening identifies important quantitative factors, the next question is where to operate. A two-level model gives direction but cannot locate interior curvature.
Transform physical settings to coded coordinates centred at zero. Coding improves numerical stability and makes design geometry comparable.
Fit a plane with main effects and interactions. If the gradient is strong and curvature absent, move in the direction of steepest ascent or descent.
Compare the mean response at repeated centre runs with the average of factorial corner runs. A meaningful difference signals pure quadratic curvature.
For two factors use intercept, x1, x2, x1x2, x1 squared and x2 squared. These terms allow bowls, peaks, ridges and tilted contours.
Start with a factorial or fractional core, add centre points and star points on each axis. The star distance alpha depends on desired properties and constraints.
NIST explains how CCDs augment a factorial design with centre and star points to estimate curvature. Read the NIST handbook.
Consider predicted y=100+8×1+4×2−6×1 squared−4×2 squared+2x1x2 in coded units.
The derivatives are 8−12×1+2×2 and 4−8×2+2×1. Solving both equations gives x1=0.7826 and x2=0.6957.
Substitution gives approximately 104.52. Because the point lies within the coded square and the Hessian is negative definite, it is a local maximum for the fitted surface.
| Term | Coefficient |
|---|---|
| Intercept | 100 |
| x1 | 8 |
| x2 | 4 |
| x1^2 | -6 |
| x2^2 | -4 |
| x1x2 | 2 |
The second-derivative matrix is [[−12,2],[2,−8]]. Its leading entry is negative and determinant 92 is positive, confirming concavity.
NIST describes quadratic response-surface models, centre-point curvature and the run growth of full three-level factorials. Read the NIST handbook.
| x1 | x2 | Predicted response |
|---|---|---|
| 0.7826 | 0.6957 | 104.5217 |
Contour lines show combinations with equal predicted response. Tight contours indicate rapid change; elongated contours indicate a ridge or weak direction.
An external optimum is extrapolation. Expand the region only when safe and scientifically credible, or optimise on the boundary.
Temperature, cost, fairness or safety can bound feasible settings. Constrained optimisation answers a different question from unconstrained calculus.
When a stationary point is unstable or a saddle, examine the best predicted response at increasing distance from the centre.
These designs fit quadratics without every extreme corner and can be useful when simultaneous factor extremes are unsafe.
A rotatable design gives equal prediction variance at points the same distance from the centre. That geometric property can guide alpha and run choice.
Replicated settings estimate pure error. Compare residual lack of fit with that noise rather than trusting R-squared alone.
Yield, cost and safety may conflict. Desirability functions or Pareto fronts make trade-offs explicit rather than hiding them in one arbitrary score.
A slightly lower mean with flatter curvature may be better in production because small setting errors cause less loss.
Study duration and retrieval spacing may have diminishing returns. A response surface can locate a practical region, but student heterogeneity and ethics constrain settings.
Learning rate and regularisation often interact nonlinearly. Random seeds, compute batches and early stopping must be included in the experimental plan.
Run new observations at the predicted optimum and selected comparison settings. Confirmation tests prediction, not just the fit to old data.
Inspect residuals, leverage, contour uncertainty, centre replication, variance surface and sensitivity to influential runs.
Solve the two gradient equations, verify y=104.52, classify the Hessian and compare confirmation observations with prediction intervals.
A local quadratic is not a universal law. It can fail outside the region, under process drift or when important factors were omitted.
Report physical and coded units, design points, replication, fitted equation, stationary point, Hessian, constraints, prediction interval and confirmation.
Optimise only after the surface is adequately explored. Prefer a stable, safe operating region over a fragile mathematical peak.
Store physical settings, coded values, blocks, run order, response, replicate flag, prediction and residual.
Generate the CCD matrix, fit the full quadratic, solve the gradient, compute Hessian eigenvalues and validate with held-out runs.
Chemical engineers, food scientists, manufacturers, educators and machine-learning teams use response surfaces to tune quantitative systems.
Optimise a paper-airplane design over wing angle and nose weight. Add centre points, fit curvature and confirm the predicted best setting.
A stationary point is not automatically a maximum, R-squared is not validation, and extrapolated peaks are not evidence.
The factorial guide explains main effects and interactions. The fractional guide explains screening before optimisation.
Students should sketch contours and check whether the proposed optimum sits inside tested space. Parents can ask whether a learning “optimal dose” was confirmed with new students.
- Fit a quadratic.
- Solve the gradient.
- Classify the Hessian.
- Add a constraint.
- Plan confirmation runs.
### Why centre points? To estimate pure error and detect curvature.
### What are star points? Axis points that help estimate quadratic terms.
### Is the stationary point the optimum? Only after classification and constraints.
### Must it be confirmed? Yes.
Report factors and ranges, coding, design type, alpha, replication, blocks, model, coefficients, lack of fit, stationary point, constraints, prediction uncertainty and confirmation.
The fastest route to a peak often starts with screening, then local modelling, then confirmation.
Confirm the optimum is inside justified space, curvature is supported by data and new runs validate both response and uncertainty.
Continue through the Mathematics Learning Hub. Recalculate the central example, identify the assumptions and explain which claim the result can and cannot support.
Extended practice laboratory
Practice case 1: Choose factors
Setup: Start from the article’s response surface mechanism for case 1. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 2: Set safe region
Setup: Start from the article’s response surface mechanism for case 2. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 3: Code variables
Setup: Start from the article’s response surface mechanism for case 3. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 4: Fit first order
Setup: Start from the article’s response surface mechanism for case 4. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 5: Test curvature
Setup: Start from the article’s response surface mechanism for case 5. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 6: Add centre points
Setup: Start from the article’s response surface mechanism for case 6. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 7: Add star points
Setup: Start from the article’s response surface mechanism for case 7. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 8: Build CCD
Setup: Start from the article’s response surface mechanism for case 8. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 9: Fit quadratic
Setup: Start from the article’s response surface mechanism for case 9. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 10: Calculate gradient
Setup: Start from the article’s response surface mechanism for case 10. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 11: Solve stationary point
Setup: Start from the article’s response surface mechanism for case 11. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 12: Classify Hessian
Setup: Start from the article’s response surface mechanism for case 12. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 13: Plot contours
Setup: Start from the article’s response surface mechanism for case 13. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 14: Apply constraints
Setup: Start from the article’s response surface mechanism for case 14. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 15: Use ridge path
Setup: Start from the article’s response surface mechanism for case 15. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 16: Add replication
Setup: Start from the article’s response surface mechanism for case 16. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 17: Estimate pure error
Setup: Start from the article’s response surface mechanism for case 17. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 18: Check residuals
Setup: Start from the article’s response surface mechanism for case 18. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 19: Check lack of fit
Setup: Start from the article’s response surface mechanism for case 19. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 20: Validate optimum
Setup: Start from the article’s response surface mechanism for case 20. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 21: Add multiple responses
Setup: Start from the article’s response surface mechanism for case 21. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 22: Use desirability
Setup: Start from the article’s response surface mechanism for case 22. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 23: Audit extrapolation
Setup: Start from the article’s response surface mechanism for case 23. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 24: Name limits
Setup: Start from the article’s response surface mechanism for case 24. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 25: Write conclusion
Setup: Start from the article’s response surface mechanism for case 25. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the estimand and design, verify the statistical identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.