Chapter map

1. 1. Region table
2. 2. Distance matrix
3. 3. Population target
4. 4. Window seed
5. 5. Population accumulation
6. 6. Window membership
7. 7. Window case count
8. 8. Maximum statistic
9. 9. Expected allocation
10. 10. Fixed-total simulation
11. 11. Complete rerun
12. 12. Monte Carlo p-value
13. 13. Overlapping windows
14. 14. Target sensitivity
15. 15. Boundary audit
16. 16. External follow-up

Visit the How Mathematics Works hub for the wider series and its connections to education, reasoning, science, computing, design and careers.

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1. Region table

The useful question is not only how to calculate, but what the calculation assumes. Store centroid, population, cases and expected count for every region.

A concrete situation

Five equal regions have 100 people each.

What the mathematics does not license

Misaligned census years bias windows. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Version inputs. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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2. Distance matrix

A small example makes the mechanism visible. Order regions around each centroid.

A concrete situation

B and C are nearest to A.

What the mathematics does not license

Centroid distance can cross barriers. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Compare adjacency or travel distance. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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3. Population target

This is where modelling choices begin to affect evidence. Choose N* before examining maxima.

A concrete situation

N*=300 people.

What the mathematics does not license

Scale determines power. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Predeclare or adjust multiple targets. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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4. Window seed

The central mathematical idea is concrete. Start at each eligible region centre.

A concrete situation

Area A seeds one window.

What the mathematics does not license

Centre eligibility affects the search. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

State all centres. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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5. Population accumulation

Here the geometry becomes operational. Add nearest regions until N* is reached.

A concrete situation

A+B+C reach 300.

What the mathematics does not license

Last-region overshoot varies. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Document the stopping rule. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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6. Window membership

This step deserves a careful pause. Record the area set for each centre.

A concrete situation

Window A contains A,B,C.

What the mathematics does not license

Different centres can yield identical sets. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Deduplicate transparently. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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7. Window case count

The useful question is not only how to calculate, but what the calculation assumes. Sum cases inside each set.

A concrete situation

5+4+0=9.

What the mathematics does not license

One miscoded area can dominate. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Audit contributions. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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8. Maximum statistic

A small example makes the mechanism visible. Take the largest window count.

A concrete situation

Observed λ=9.

What the mathematics does not license

The winning window is selected by data. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Use maximum calibration. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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9. Expected allocation

This is where modelling choices begin to affect evidence. Convert population or Ei to null probabilities.

A concrete situation

Equal populations give .2 each.

What the mathematics does not license

Baseline risk may differ by covariates. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Use standardized probabilities. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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10. Fixed-total simulation

The central mathematical idea is concrete. Redistribute O+ cases multinomially.

A concrete situation

Ten cases are allocated with probabilities .2.

What the mathematics does not license

Poisson simulation answers a different null. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

State conditioning. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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11. Complete rerun

Here the geometry becomes operational. Rebuild every window count for each simulated map.

A concrete situation

Each replicate produces one λ*.

What the mathematics does not license

Testing one observed window is invalid. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Repeat the whole search. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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12. Monte Carlo p-value

This step deserves a careful pause. Rank observed λ among simulated maxima.

A concrete situation

Four of 999 exceed nine gives about .005.

What the mathematics does not license

Resolution depends on replicates. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Use enough simulations. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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13. Overlapping windows

The useful question is not only how to calculate, but what the calculation assumes. Report shared regions among top windows.

A concrete situation

Three significant candidates share A and B.

What the mathematics does not license

They may be one signal. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Summarize overlap. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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14. Target sensitivity

A small example makes the mechanism visible. Compare only a declared N* set.

A concrete situation

200,300,500 probe different scales.

What the mathematics does not license

Selecting the best target multiplies tests. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Use global adjustment. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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15. Boundary audit

This is where modelling choices begin to affect evidence. Inspect windows at map edges.

A concrete situation

An edge centre expands inward only.

What the mathematics does not license

Study borders truncate risk populations. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Add buffers or discuss. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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16. External follow-up

The central mathematical idea is concrete. Apply fixed settings to later data.

A concrete situation

A later period repeats the compact excess.

What the mathematics does not license

Replication is not cause. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Investigate mechanisms. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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A seven-day practice lab

1. Day 1: define the decision, observation unit, X variables and Y or reference condition.
2. Day 2: reproduce the smallest worked example and check every dimension, unit and sign.
3. Day 3: build a transparent baseline and write what it assumes.
4. Day 4: fit the main method inside a training-only pipeline; record every parameter.
5. Day 5: challenge one assumption with a deliberately difficult dataset or operating regime.
6. Day 6: compare validation metrics with the practical workload created by errors.
7. Day 7: present one supported conclusion, one unresolved uncertainty and one safe next experiment.

Questions for a student, parent or mentor

• Which part of the model is a mathematical definition, and which part is a choice?
• What small calculation would catch a sign, scale or dimension error?
• What evidence comes from training, validation and the untouched test?
• What benign condition could look like a fault or prediction failure?
• How would the answer change for another response, asset, school project or time period?
• What should a human review before acting on the score?

A method-specific audit before deployment

The Turnbull cluster evaluation permutation procedure builds search windows that each contain approximately the same population at risk. From a candidate centre, neighbouring regions are added in distance order until the window reaches a predeclared population target N*. Equal-population windows make case counts more comparable across dense and sparse parts of the map.

For every centre, the procedure sums cases inside its fixed-population window. The test statistic is the maximum window count across the complete search. Under the null, the observed total number of cases is redistributed according to population or expected-count probabilities, and the entire maximum search is repeated. Comparing maxima automatically accounts for looking at many overlapping windows in that declared search.

Suppose five equal-population areas have case counts 5,4,0,1,0 and each contains 100 people. With N*=300, every window expands to roughly three areas. If one centre’s three-area window contains the first three counts, it holds nine of the ten cases. The observed maximum of nine is compared with the maximum from each simulated map, not with the distribution of one prespecified window.

N* sets the target scale, while centroids, ties, overshoot and irregular geography affect membership. Trying many N* values without adjustment recreates a multiplicity problem. CEPP detects a compact high-count window under its population model; it does not say the circle is a natural boundary or that every area in it shares one exposure. Follow-up needs local records, covariates and independent periods.

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Frequently asked questions

What is CEPP?

CEPP is the cluster evaluation permutation procedure, a search using approximately fixed-population windows and a maximum count statistic.

Why fix population instead of radius?

Equal risk population makes case counts more comparable between dense and sparse regions.

What is N*?

N* is the target population at risk accumulated into every search window.

What is the global statistic?

It is the largest case count found among all declared windows.

How does simulation control multiplicity?

Every simulated map repeats the same full maximum search, so the reference includes selection over centres.

Does a high window reveal its cause?

No. It identifies a compact excess under the model and requires denominator, exposure and independent follow-up.

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Primary source and fact-check note

Bruce W. Turnbull, Eric J. Iwano, William S. Burnett, Holly L. Howe and Larry C. Clark, “Monitoring for Clusters of Disease: Application to Leukemia Incidence in Upstate New York,” American Journal of Epidemiology 132(Supplement 1), S136–S143 (1990), DOI 10.1093/oxfordjournals.aje.a115775. Definitions and core claims in this article were checked against this primary publication. Calculations shown here were independently recomputed. Examples are educational illustrations; they are not reported industrial performance results.

Useful next reading

• Kulldorff spatial scan statistics
• Besag–Newell cluster detection
• How Mathematics Works

Keep the options open. Mathematics education supports clearer questions, transferable problem-solving skills and informed choices across science, engineering, computing, AI, statistics, design and many other careers. It does not guarantee intelligence, income, admission or employment; those outcomes depend on far more than one subject or model.

Extended laboratory: topic-specific validation and responsible use

Data audit

Verify case definitions, denominators, coordinates and observation periods.

Geometry audit

Publish foci, centroids, distances, windows and weight matrices.

Calculation audit

Reproduce one complete toy statistic by hand.

Null audit

State conditioning, expected probabilities and simulation design.

Multiplicity audit

Control centres, scales, sources and local diagnostics.

Sensitivity audit

Repeat only scientifically plausible geometries and baselines.

Transfer audit

Reserve later periods or separate regions.

Final claim

Report spatial evidence without automatic causal attribution.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Region table

Begin with this mechanism: Store centroid, population, cases and expected count for every region. Recreate the concrete case: Five equal regions have 100 people each. Write region IDs, centroids, populations, cases, distance ranks, population targets, window memberships, accumulated populations, window counts, maximum counts, null allocations and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Misaligned census years bias windows. Complete the transfer step: Version inputs. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 2: Distance matrix

Begin with this mechanism: Order regions around each centroid. Recreate the concrete case: B and C are nearest to A. Write region IDs, centroids, populations, cases, distance ranks, population targets, window memberships, accumulated populations, window counts, maximum counts, null allocations and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Centroid distance can cross barriers. Complete the transfer step: Compare adjacency or travel distance. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 3: Population target

Begin with this mechanism: Choose N* before examining maxima. Recreate the concrete case: N*=300 people. Write region IDs, centroids, populations, cases, distance ranks, population targets, window memberships, accumulated populations, window counts, maximum counts, null allocations and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Scale determines power. Complete the transfer step: Predeclare or adjust multiple targets. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 4: Window seed

Begin with this mechanism: Start at each eligible region centre. Recreate the concrete case: Area A seeds one window. Write region IDs, centroids, populations, cases, distance ranks, population targets, window memberships, accumulated populations, window counts, maximum counts, null allocations and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Centre eligibility affects the search. Complete the transfer step: State all centres. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 5: Population accumulation

Begin with this mechanism: Add nearest regions until N* is reached. Recreate the concrete case: A+B+C reach 300. Write region IDs, centroids, populations, cases, distance ranks, population targets, window memberships, accumulated populations, window counts, maximum counts, null allocations and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Last-region overshoot varies. Complete the transfer step: Document the stopping rule. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 6: Window membership

Begin with this mechanism: Record the area set for each centre. Recreate the concrete case: Window A contains A,B,C. Write region IDs, centroids, populations, cases, distance ranks, population targets, window memberships, accumulated populations, window counts, maximum counts, null allocations and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Different centres can yield identical sets. Complete the transfer step: Deduplicate transparently. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 7: Window case count

Begin with this mechanism: Sum cases inside each set. Recreate the concrete case: 5+4+0=9. Write region IDs, centroids, populations, cases, distance ranks, population targets, window memberships, accumulated populations, window counts, maximum counts, null allocations and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: One miscoded area can dominate. Complete the transfer step: Audit contributions. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 8: Maximum statistic

Begin with this mechanism: Take the largest window count. Recreate the concrete case: Observed λ=9. Write region IDs, centroids, populations, cases, distance ranks, population targets, window memberships, accumulated populations, window counts, maximum counts, null allocations and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: The winning window is selected by data. Complete the transfer step: Use maximum calibration. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 9: Expected allocation

Begin with this mechanism: Convert population or Ei to null probabilities. Recreate the concrete case: Equal populations give .2 each. Write region IDs, centroids, populations, cases, distance ranks, population targets, window memberships, accumulated populations, window counts, maximum counts, null allocations and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Baseline risk may differ by covariates. Complete the transfer step: Use standardized probabilities. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 10: Fixed-total simulation

Begin with this mechanism: Redistribute O+ cases multinomially. Recreate the concrete case: Ten cases are allocated with probabilities .2. Write region IDs, centroids, populations, cases, distance ranks, population targets, window memberships, accumulated populations, window counts, maximum counts, null allocations and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Poisson simulation answers a different null. Complete the transfer step: State conditioning. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 11: Complete rerun

Begin with this mechanism: Rebuild every window count for each simulated map. Recreate the concrete case: Each replicate produces one λ*. Write region IDs, centroids, populations, cases, distance ranks, population targets, window memberships, accumulated populations, window counts, maximum counts, null allocations and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Testing one observed window is invalid. Complete the transfer step: Repeat the whole search. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 12: Monte Carlo p-value

Begin with this mechanism: Rank observed λ among simulated maxima. Recreate the concrete case: Four of 999 exceed nine gives about .005. Write region IDs, centroids, populations, cases, distance ranks, population targets, window memberships, accumulated populations, window counts, maximum counts, null allocations and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Resolution depends on replicates. Complete the transfer step: Use enough simulations. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 13: Overlapping windows

Begin with this mechanism: Report shared regions among top windows. Recreate the concrete case: Three significant candidates share A and B. Write region IDs, centroids, populations, cases, distance ranks, population targets, window memberships, accumulated populations, window counts, maximum counts, null allocations and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: They may be one signal. Complete the transfer step: Summarize overlap. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 14: Target sensitivity

Begin with this mechanism: Compare only a declared N* set. Recreate the concrete case: 200,300,500 probe different scales. Write region IDs, centroids, populations, cases, distance ranks, population targets, window memberships, accumulated populations, window counts, maximum counts, null allocations and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Selecting the best target multiplies tests. Complete the transfer step: Use global adjustment. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 15: Boundary audit

Begin with this mechanism: Inspect windows at map edges. Recreate the concrete case: An edge centre expands inward only. Write region IDs, centroids, populations, cases, distance ranks, population targets, window memberships, accumulated populations, window counts, maximum counts, null allocations and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Study borders truncate risk populations. Complete the transfer step: Add buffers or discuss. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 16: External follow-up

Begin with this mechanism: Apply fixed settings to later data. Recreate the concrete case: A later period repeats the compact excess. Write region IDs, centroids, populations, cases, distance ranks, population targets, window memberships, accumulated populations, window counts, maximum counts, null allocations and Monte Carlo ranks with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Replication is not cause. Complete the transfer step: Investigate mechanisms. Change exactly one case, expected count, risk denominator, focus, coordinate, centroid, distance metric, direction, scale parameter, population target, area order, stopping rule, matrix weight, diagonal convention, simulation model, replicate count, random seed, local threshold, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.