Chapter map

1. 1. Study boundary
2. 2. Case points
3. 3. Population surface
4. 4. Candidate grid
5. 5. Radius set
6. 6. Circle membership
7. 7. Expected count
8. 8. Poisson tail
9. 9. Search table
10. 10. Maximum evidence
11. 11. Multiplicity calibration
12. 12. Overlap grouping
13. 13. Grid sensitivity
14. 14. Edge correction
15. 15. Independent confirmation
16. 16. Communication

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 boundary

The useful question is not only how to calculate, but what the calculation assumes. Fix the region and any buffer before searching.

A concrete situation

A city plus a 5 km denominator buffer is declared.

What the mathematics does not license

Edges truncate circles and populations. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Map inclusion 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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2. Case points

A small example makes the mechanism visible. Geocode events with time and eligibility fields.

A concrete situation

Rare events are represented as points.

What the mathematics does not license

Privacy and positional error matter. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Use protected, quality-checked coordinates. 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 surface

This is where modelling choices begin to affect evidence. Represent people at risk on compatible geography.

A concrete situation

A gridded population raster supports circle totals.

What the mathematics does not license

Census mismatch biases expectations. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Align year and 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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4. Candidate grid

The central mathematical idea is concrete. Place centres at a declared spacing.

A concrete situation

Centres are 500 m apart.

What the mathematics does not license

Fine grids multiply tests and cost. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Publish spacing. 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. Radius set

Here the geometry becomes operational. Choose one or more radii before inspecting results.

A concrete situation

Radii are 1,2 and 3 km.

What the mathematics does not license

Scale shopping inflates evidence. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Predeclare or adjust. 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. Circle membership

This step deserves a careful pause. Select case points and population cells inside each radius.

A concrete situation

One 2 km circle contains six cases.

What the mathematics does not license

Boundary inclusion and raster fractions matter. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

State the geometric 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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7. Expected count

The useful question is not only how to calculate, but what the calculation assumes. Multiply population by the baseline rate.

A concrete situation

2,000 people at 1/1,000 gives λ=2.

What the mathematics does not license

Crude rate may ignore risk groups. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Standardize expectations. 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. Poisson tail

A small example makes the mechanism visible. Compute P(X≥observed|λ).

A concrete situation

For X≥6 and λ=2, p≈.0166.

What the mathematics does not license

This is one-window evidence. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Keep it separate from global evidence. 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. Search table

This is where modelling choices begin to affect evidence. Store centre, radius, O, E and marginal tail for every window.

A concrete situation

The run evaluates thousands of rows.

What the mathematics does not license

Keeping only hits hides selection. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Archive the full universe. 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. Maximum evidence

The central mathematical idea is concrete. Define a search-wide statistic such as the largest count contrast.

A concrete situation

Each map contributes one maximum.

What the mathematics does not license

Changing statistics after viewing results is circular. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Freeze the 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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11. Multiplicity calibration

Here the geometry becomes operational. Simulate complete maps and rerun all circles.

A concrete situation

Every null map yields its smallest tail.

What the mathematics does not license

Overlap makes Bonferroni conservative but does not remove selection. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Use search-aware 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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12. Overlap grouping

This step deserves a careful pause. Summarize highly overlapping flagged circles.

A concrete situation

Ten circles share one core neighbourhood.

What the mathematics does not license

Merging rules can alter boundaries. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Publish the grouping threshold. 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. Grid sensitivity

The useful question is not only how to calculate, but what the calculation assumes. Shift the grid origin diagnostically.

A concrete situation

A 250 m offset changes one candidate.

What the mathematics does not license

Post-hoc best-grid selection is biased. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Use stability, not cherry-picking. 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. Edge correction

A small example makes the mechanism visible. Track the fraction of each circle inside the risk surface.

A concrete situation

A coastal circle is half land.

What the mathematics does not license

Naive circles include impossible risk space. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Clip population consistently. 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. Independent confirmation

This is where modelling choices begin to affect evidence. Freeze candidate centres and radii for later data.

A concrete situation

A later period tests the same circles.

What the mathematics does not license

Confirmation still supports association only. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Seek exposure 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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16. Communication

The central mathematical idea is concrete. Show marginal and global evidence separately.

A concrete situation

A map legend labels exploratory candidates.

What the mathematics does not license

Bright circles can overstate certainty. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Include counts, denominators and limits. 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

Openshaw’s Geographical Analysis Machine, or GAM, was designed to search a map without committing to one administrative partition or one prespecified centre. A grid of candidate points is laid over the study region. Around each point, circles of one or more sizes accumulate observed events and population at risk. A Poisson tail or related rare-event calculation flags circles whose counts are unusually high under the declared baseline.

The machine is mathematically interesting because it combines geometry, computation and hypothesis generation. It can reveal patterns hidden by arbitrary district boundaries, but it also tests many overlapping circles. A circle that looks rare in isolation may be unsurprising after thousands of centres and radii have been examined. The search is exploratory unless the full multiplicity and selection process is calibrated.

Suppose a circle contains 2,000 people and the baseline event rate is 1 per 1,000, giving a Poisson expectation λ=2. If six events are observed, P(X≥6|λ=2)≈0.0166. That sounds unusual for one fixed circle. Across 1,000 independent circles, about 16.6 such exceedances would be expected on average at the same marginal threshold; real GAM circles overlap, so dependence changes the exact calculation but not the need for search-wide control.

GAM output is a hypothesis map, not proof that each flagged circle is a natural cluster or shares one exposure. Grid spacing, radius steps, edge effects, denominator quality and case geocoding all shape the result. A careful workflow stores every tested circle, calibrates the maximum or false-discovery process, merges overlapping candidates transparently and validates a frozen candidate set in independent data.

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Frequently asked questions

What is a Geographical Analysis Machine?

It is an automated search that moves windows over a map to generate and evaluate candidate spatial patterns.

Why use a grid of centres?

The grid searches locations without restricting hypotheses to administrative centroids.

What is tested inside a circle?

Observed events are compared with the count expected from the population at risk, commonly through a Poisson rare-event calculation.

Why are marginal p-values insufficient?

The machine examines many overlapping centres and radii, so selection makes the smallest isolated p-value optimistic.

How should overlapping circles be reported?

Store the full search, group candidates transparently and preserve the distinction between tested windows and inferred phenomena.

Does GAM prove a disease cause?

No. It generates geographic evidence and hypotheses that need independent confirmation and exposure investigation.

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Primary source and fact-check note

Stan Openshaw, Alan W. Craft, Martin Charlton and J. M. Birch, “Investigation of leukaemia clusters by use of a Geographical Analysis Machine,” The Lancet 331(8580), 272–273 (1988), DOI 10.1016/S0140-6736(88)90352-2. 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
• Turnbull CEPP 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

Universe audit

Archive every grid centre, radius and tested window.

Denominator audit

Align cases, population surface and baseline rate.

Geometry audit

Publish spacing, radii, edge and membership rules.

Tail audit

Reproduce one Poisson calculation by hand.

Multiplicity audit

Calibrate the complete overlapping search.

Stability audit

Shift grids and radii only as declared diagnostics.

Transfer audit

Confirm frozen candidates in independent data.

Final claim

Label output as geographic hypothesis evidence, not causal proof.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Study boundary

Begin with this mechanism: Fix the region and any buffer before searching. Recreate the concrete case: A city plus a 5 km denominator buffer is declared. Write grid coordinates, radii, case memberships, population totals, baseline rates, expected counts, Poisson tails, search maxima, overlap fractions, edge fractions and simulation 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: Edges truncate circles and populations. Complete the transfer step: Map inclusion rules. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 2: Case points

Begin with this mechanism: Geocode events with time and eligibility fields. Recreate the concrete case: Rare events are represented as points. Write grid coordinates, radii, case memberships, population totals, baseline rates, expected counts, Poisson tails, search maxima, overlap fractions, edge fractions and simulation 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: Privacy and positional error matter. Complete the transfer step: Use protected, quality-checked coordinates. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior 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 surface

Begin with this mechanism: Represent people at risk on compatible geography. Recreate the concrete case: A gridded population raster supports circle totals. Write grid coordinates, radii, case memberships, population totals, baseline rates, expected counts, Poisson tails, search maxima, overlap fractions, edge fractions and simulation 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: Census mismatch biases expectations. Complete the transfer step: Align year and resolution. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 4: Candidate grid

Begin with this mechanism: Place centres at a declared spacing. Recreate the concrete case: Centres are 500 m apart. Write grid coordinates, radii, case memberships, population totals, baseline rates, expected counts, Poisson tails, search maxima, overlap fractions, edge fractions and simulation 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: Fine grids multiply tests and cost. Complete the transfer step: Publish spacing. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 5: Radius set

Begin with this mechanism: Choose one or more radii before inspecting results. Recreate the concrete case: Radii are 1,2 and 3 km. Write grid coordinates, radii, case memberships, population totals, baseline rates, expected counts, Poisson tails, search maxima, overlap fractions, edge fractions and simulation 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 shopping inflates evidence. Complete the transfer step: Predeclare or adjust. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 6: Circle membership

Begin with this mechanism: Select case points and population cells inside each radius. Recreate the concrete case: One 2 km circle contains six cases. Write grid coordinates, radii, case memberships, population totals, baseline rates, expected counts, Poisson tails, search maxima, overlap fractions, edge fractions and simulation 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: Boundary inclusion and raster fractions matter. Complete the transfer step: State the geometric rule. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 7: Expected count

Begin with this mechanism: Multiply population by the baseline rate. Recreate the concrete case: 2,000 people at 1/1,000 gives λ=2. Write grid coordinates, radii, case memberships, population totals, baseline rates, expected counts, Poisson tails, search maxima, overlap fractions, edge fractions and simulation 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: Crude rate may ignore risk groups. Complete the transfer step: Standardize expectations. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 8: Poisson tail

Begin with this mechanism: Compute P(X≥observed|λ). Recreate the concrete case: For X≥6 and λ=2, p≈.0166. Write grid coordinates, radii, case memberships, population totals, baseline rates, expected counts, Poisson tails, search maxima, overlap fractions, edge fractions and simulation 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: This is one-window evidence. Complete the transfer step: Keep it separate from global evidence. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 9: Search table

Begin with this mechanism: Store centre, radius, O, E and marginal tail for every window. Recreate the concrete case: The run evaluates thousands of rows. Write grid coordinates, radii, case memberships, population totals, baseline rates, expected counts, Poisson tails, search maxima, overlap fractions, edge fractions and simulation 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: Keeping only hits hides selection. Complete the transfer step: Archive the full universe. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 10: Maximum evidence

Begin with this mechanism: Define a search-wide statistic such as the largest count contrast. Recreate the concrete case: Each map contributes one maximum. Write grid coordinates, radii, case memberships, population totals, baseline rates, expected counts, Poisson tails, search maxima, overlap fractions, edge fractions and simulation 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: Changing statistics after viewing results is circular. Complete the transfer step: Freeze the rule. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 11: Multiplicity calibration

Begin with this mechanism: Simulate complete maps and rerun all circles. Recreate the concrete case: Every null map yields its smallest tail. Write grid coordinates, radii, case memberships, population totals, baseline rates, expected counts, Poisson tails, search maxima, overlap fractions, edge fractions and simulation 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: Overlap makes Bonferroni conservative but does not remove selection. Complete the transfer step: Use search-aware calibration. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 12: Overlap grouping

Begin with this mechanism: Summarize highly overlapping flagged circles. Recreate the concrete case: Ten circles share one core neighbourhood. Write grid coordinates, radii, case memberships, population totals, baseline rates, expected counts, Poisson tails, search maxima, overlap fractions, edge fractions and simulation 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: Merging rules can alter boundaries. Complete the transfer step: Publish the grouping threshold. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 13: Grid sensitivity

Begin with this mechanism: Shift the grid origin diagnostically. Recreate the concrete case: A 250 m offset changes one candidate. Write grid coordinates, radii, case memberships, population totals, baseline rates, expected counts, Poisson tails, search maxima, overlap fractions, edge fractions and simulation 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: Post-hoc best-grid selection is biased. Complete the transfer step: Use stability, not cherry-picking. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 14: Edge correction

Begin with this mechanism: Track the fraction of each circle inside the risk surface. Recreate the concrete case: A coastal circle is half land. Write grid coordinates, radii, case memberships, population totals, baseline rates, expected counts, Poisson tails, search maxima, overlap fractions, edge fractions and simulation 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: Naive circles include impossible risk space. Complete the transfer step: Clip population consistently. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 15: Independent confirmation

Begin with this mechanism: Freeze candidate centres and radii for later data. Recreate the concrete case: A later period tests the same circles. Write grid coordinates, radii, case memberships, population totals, baseline rates, expected counts, Poisson tails, search maxima, overlap fractions, edge fractions and simulation 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: Confirmation still supports association only. Complete the transfer step: Seek exposure records. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.

Workshop 16: Communication

Begin with this mechanism: Show marginal and global evidence separately. Recreate the concrete case: A map legend labels exploratory candidates. Write grid coordinates, radii, case memberships, population totals, baseline rates, expected counts, Poisson tails, search maxima, overlap fractions, edge fractions and simulation 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: Bright circles can overstate certainty. Complete the transfer step: Include counts, denominators and limits. Change exactly one case count, population denominator, expected count, risk stratum, coordinate, distance metric, pair membership, grid spacing, radius, circle edge rule, adjacency edge, matrix weight, diagonal choice, dispersion parameter, null allocation, replicate count, prior precision, mixing proportion, observation period or validation region. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, geometry, statistic or likelihood, null or prior model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods or cohorts outside tuning when testing transfer.