Chapter map

1. 1. Study locations
2. 2. Opportunity measure
3. 3. Null rate
4. 4. Candidate centre
5. 5. Candidate radius
6. 6. Window population
7. 7. Window cases
8. 8. Expected cases
9. 9. Inside likelihood
10. 10. Outside likelihood
11. 11. Candidate log ratio
12. 12. Maximum statistic
13. 13. Null simulation
14. 14. Monte Carlo p-value
15. 15. Secondary clusters
16. 16. Sensitivity and 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. Study locations

This is where modelling choices begin to affect evidence. Events and opportunity are aligned to geographic units or coordinates.

A concrete situation

Ten zones carry cases and population.

What the mathematics does not license

Centroids lose within-zone geometry. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Record spatial resolution. 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. Opportunity measure

The central mathematical idea is concrete. Population, person-time or trials define expected exposure.

A concrete situation

Twenty cases occur in population 1,000.

What the mathematics does not license

Wrong denominators create false clusters. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Validate opportunity data. 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. Null rate

Here the geometry becomes operational. Total cases divided by total opportunity sets the common baseline.

A concrete situation

20/1000=.02.

What the mathematics does not license

Temporal trend can violate a static rate. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Stratify or model time when needed. 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. Candidate centre

This step deserves a careful pause. Every allowed location may centre a window.

A concrete situation

A zone centroid is one centre.

What the mathematics does not license

Centre rules affect discoverable shapes. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Publish the search set. 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. Candidate radius

The useful question is not only how to calculate, but what the calculation assumes. Window size grows through ordered distances.

A concrete situation

Each added centroid creates a new candidate.

What the mathematics does not license

Continuous-looking scans are finite searches. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Count candidates. 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 population

A small example makes the mechanism visible. Sum exposure inside the candidate.

A concrete situation

A window contains population 200.

What the mathematics does not license

Large windows can dominate interpretation. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Set a justified maximum proportion. 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 cases

This is where modelling choices begin to affect evidence. Sum observed events inside.

A concrete situation

The candidate has 8 cases.

What the mathematics does not license

Duplicate or geocoded events bias counts. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Audit records. 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. Expected cases

The central mathematical idea is concrete. Multiply total cases by exposure share.

A concrete situation

20×200/1000=4.

What the mathematics does not license

Expectation inherits denominator error. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Carry uncertainty where material. 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. Inside likelihood

Here the geometry becomes operational. Compare observed inside cases with expected inside cases.

A concrete situation

8 log(8/4) contributes about 5.545.

What the mathematics does not license

Zero counts require limit conventions. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Use verified implementations. 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. Outside likelihood

This step deserves a careful pause. The complement contributes to the conditional ratio.

A concrete situation

12 log(12/16) is about −3.452.

What the mathematics does not license

Ignoring outside data exaggerates evidence. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Calculate both parts. 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. Candidate log ratio

The useful question is not only how to calculate, but what the calculation assumes. Add inside and outside contributions for the high-rate alternative.

A concrete situation

The score is about 2.09.

What the mathematics does not license

A score is not a calibrated p-value. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Search all candidates first. 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. Maximum statistic

A small example makes the mechanism visible. The highest candidate score defines the most likely cluster.

A concrete situation

One of hundreds of windows wins.

What the mathematics does not license

Winner selection causes look-elsewhere bias. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Preserve the maximum in 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. Null simulation

This is where modelling choices begin to affect evidence. Generate data under the fitted null and rescan completely.

A concrete situation

999 replicates yield 999 maxima.

What the mathematics does not license

Simulating only the winning 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 same search pipeline. 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. Monte Carlo p-value

The central mathematical idea is concrete. Rank the observed maximum among null maxima.

A concrete situation

Four null maxima at least as large gives .005.

What the mathematics does not license

Model misspecification invalidates calibration. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Check residual structure. 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. Secondary clusters

Here the geometry becomes operational. Apply declared non-overlap or hierarchical rules.

A concrete situation

A second window is geographically separate.

What the mathematics does not license

Reporting many overlaps confuses readers. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

State selection rules. 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. Sensitivity and follow-up

This step deserves a careful pause. Vary maximum size, geometry and resolution, then investigate.

A concrete situation

A cluster persists from 25% to 40% population caps.

What the mathematics does not license

Persistence does not identify cause. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Use field evidence and privacy safeguards. 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

Kulldorff’s spatial scan statistic moves windows of different locations and sizes across a map, compares observed and expected events inside each window, and records the maximum likelihood ratio. The method searches while testing, so ordinary unadjusted p-values for the winning circle would be invalid. Monte Carlo replication repeats the entire search under the null.

For a Poisson model, expected cases are proportional to population or another exposure. A candidate window with more cases than expected is compared with its complement through a conditional likelihood ratio. The winning window maximizes evidence over the allowed search set; secondary clusters must obey declared overlap rules. Bernoulli, space–time and other models use different likelihoods.

Suppose the whole map has 20 cases among population 1,000. A window has eight cases among population 200, so four are expected. The log likelihood ratio for a high-rate alternative is about 8 log(8/4)+12 log(12/16)=2.09. That is a candidate score, not its calibrated significance, because many centres and radii were searched.

Spatial scan statistics support surveillance and cluster detection, but circles are models rather than natural truths. Centroid aggregation, population mismeasurement, maximum cluster size, multiple time scales and repeated routine monitoring affect the answer. Responsible analysis states the likelihood, opportunity measure, search geometry and simulations, checks sensitivity, protects privacy and follows statistical clusters with domain investigation.

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Frequently asked questions

What is a spatial scan statistic?

It searches a family of spatial windows and uses the maximum likelihood ratio to test for a cluster not explained by the baseline.

Why are Monte Carlo tests needed?

The winner is selected after many overlapping windows are searched, so the entire maximization must be repeated under the null.

What does the Poisson model compare?

It compares cases inside and outside a window conditional on total cases, with expected cases proportional to population or exposure.

Must the window be circular?

No, but circles are common; ellipses, networks and flexible shapes answer different questions and have different search burdens.

What is a secondary cluster?

It is another high-scoring window reported under declared overlap and ranking rules after the most likely cluster.

Does a detected cluster prove a cause?

No. It identifies excess events relative to a model and opportunity measure; data quality and investigation determine explanation.

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Primary source and fact-check note

Martin Kulldorff, “A Spatial Scan Statistic,” Communications in Statistics—Theory and Methods 26(6), 1481–1496 (1997), DOI 10.1080/03610929708831995. 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

• Ripley’s K point patterns
• Getis–Ord Gi* hotspot analysis
• 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 definitions, alignment, missingness and units.

Structure audit

Publish the graph, window, weights or metric.

Calculation audit

Reproduce a hand-worked example and software output.

Null audit

State exactly what is randomized or simulated.

Multiplicity audit

Account for local tests or searched windows.

Sensitivity audit

Repeat plausible scales, thresholds and boundaries.

Transfer audit

Reserve independent regions, periods or samples.

Final claim

Report association under the model without claiming automatic cause.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Study locations

Begin with this mechanism: Events and opportunity are aligned to geographic units or coordinates. Recreate the concrete case: Ten zones carry cases and population. Write cases, opportunity, candidate centres, radii, window totals, expected counts, likelihood contributions, maxima, simulated maxima and cluster 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: Centroids lose within-zone geometry. Complete the transfer step: Record spatial resolution. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 2: Opportunity measure

Begin with this mechanism: Population, person-time or trials define expected exposure. Recreate the concrete case: Twenty cases occur in population 1,000. Write cases, opportunity, candidate centres, radii, window totals, expected counts, likelihood contributions, maxima, simulated maxima and cluster 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: Wrong denominators create false clusters. Complete the transfer step: Validate opportunity data. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 3: Null rate

Begin with this mechanism: Total cases divided by total opportunity sets the common baseline. Recreate the concrete case: 20/1000=.02. Write cases, opportunity, candidate centres, radii, window totals, expected counts, likelihood contributions, maxima, simulated maxima and cluster 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: Temporal trend can violate a static rate. Complete the transfer step: Stratify or model time when needed. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 4: Candidate centre

Begin with this mechanism: Every allowed location may centre a window. Recreate the concrete case: A zone centroid is one centre. Write cases, opportunity, candidate centres, radii, window totals, expected counts, likelihood contributions, maxima, simulated maxima and cluster 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 rules affect discoverable shapes. Complete the transfer step: Publish the search set. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 5: Candidate radius

Begin with this mechanism: Window size grows through ordered distances. Recreate the concrete case: Each added centroid creates a new candidate. Write cases, opportunity, candidate centres, radii, window totals, expected counts, likelihood contributions, maxima, simulated maxima and cluster 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: Continuous-looking scans are finite searches. Complete the transfer step: Count candidates. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 6: Window population

Begin with this mechanism: Sum exposure inside the candidate. Recreate the concrete case: A window contains population 200. Write cases, opportunity, candidate centres, radii, window totals, expected counts, likelihood contributions, maxima, simulated maxima and cluster 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: Large windows can dominate interpretation. Complete the transfer step: Set a justified maximum proportion. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 7: Window cases

Begin with this mechanism: Sum observed events inside. Recreate the concrete case: The candidate has 8 cases. Write cases, opportunity, candidate centres, radii, window totals, expected counts, likelihood contributions, maxima, simulated maxima and cluster 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: Duplicate or geocoded events bias counts. Complete the transfer step: Audit records. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 8: Expected cases

Begin with this mechanism: Multiply total cases by exposure share. Recreate the concrete case: 20×200/1000=4. Write cases, opportunity, candidate centres, radii, window totals, expected counts, likelihood contributions, maxima, simulated maxima and cluster 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: Expectation inherits denominator error. Complete the transfer step: Carry uncertainty where material. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 9: Inside likelihood

Begin with this mechanism: Compare observed inside cases with expected inside cases. Recreate the concrete case: 8 log(8/4) contributes about 5.545. Write cases, opportunity, candidate centres, radii, window totals, expected counts, likelihood contributions, maxima, simulated maxima and cluster 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: Zero counts require limit conventions. Complete the transfer step: Use verified implementations. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 10: Outside likelihood

Begin with this mechanism: The complement contributes to the conditional ratio. Recreate the concrete case: 12 log(12/16) is about −3.452. Write cases, opportunity, candidate centres, radii, window totals, expected counts, likelihood contributions, maxima, simulated maxima and cluster 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: Ignoring outside data exaggerates evidence. Complete the transfer step: Calculate both parts. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 11: Candidate log ratio

Begin with this mechanism: Add inside and outside contributions for the high-rate alternative. Recreate the concrete case: The score is about 2.09. Write cases, opportunity, candidate centres, radii, window totals, expected counts, likelihood contributions, maxima, simulated maxima and cluster 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: A score is not a calibrated p-value. Complete the transfer step: Search all candidates first. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 12: Maximum statistic

Begin with this mechanism: The highest candidate score defines the most likely cluster. Recreate the concrete case: One of hundreds of windows wins. Write cases, opportunity, candidate centres, radii, window totals, expected counts, likelihood contributions, maxima, simulated maxima and cluster 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: Winner selection causes look-elsewhere bias. Complete the transfer step: Preserve the maximum in simulations. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 13: Null simulation

Begin with this mechanism: Generate data under the fitted null and rescan completely. Recreate the concrete case: 999 replicates yield 999 maxima. Write cases, opportunity, candidate centres, radii, window totals, expected counts, likelihood contributions, maxima, simulated maxima and cluster 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: Simulating only the winning window is invalid. Complete the transfer step: Repeat the same search pipeline. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 14: Monte Carlo p-value

Begin with this mechanism: Rank the observed maximum among null maxima. Recreate the concrete case: Four null maxima at least as large gives .005. Write cases, opportunity, candidate centres, radii, window totals, expected counts, likelihood contributions, maxima, simulated maxima and cluster 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: Model misspecification invalidates calibration. Complete the transfer step: Check residual structure. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 15: Secondary clusters

Begin with this mechanism: Apply declared non-overlap or hierarchical rules. Recreate the concrete case: A second window is geographically separate. Write cases, opportunity, candidate centres, radii, window totals, expected counts, likelihood contributions, maxima, simulated maxima and cluster 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: Reporting many overlaps confuses readers. Complete the transfer step: State selection rules. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 16: Sensitivity and follow-up

Begin with this mechanism: Vary maximum size, geometry and resolution, then investigate. Recreate the concrete case: A cluster persists from 25% to 40% population caps. Write cases, opportunity, candidate centres, radii, window totals, expected counts, likelihood contributions, maxima, simulated maxima and cluster 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: Persistence does not identify cause. Complete the transfer step: Use field evidence and privacy safeguards. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.