Chapter map

1. 1. Observed frequencies
2. 2. Exposure quantities
3. 3. Observed proportions
4. 4. Expected proportions
5. 5. Residual proportions
6. 6. Ordinary chi-square component
7. 7. Spatial connections
8. 8. Diagonal weights
9. 9. Off-diagonal weights
10. 10. Weight matrix W
11. 11. Quadratic form
12. 12. Null moments
13. 13. Local contribution
14. 14. Scale sensitivity
15. 15. Residual diagnostics
16. 16. External validation

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. Observed frequencies

The useful question is not only how to calculate, but what the calculation assumes. Record Ni in every region.

A concrete situation

Counts are 5,3,1,1.

What the mathematics does not license

Changing boundaries changes frequencies. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Version the geography. 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. Exposure quantities

A small example makes the mechanism visible. Store population or another baseline xi.

A concrete situation

Four areas have equal population.

What the mathematics does not license

Crude population may miss risk structure. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Standardize appropriately. 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. Observed proportions

This is where modelling choices begin to affect evidence. Compute ri=Ni/N.

A concrete situation

r=(.5,.3,.1,.1).

What the mathematics does not license

Shares are dependent through N. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Use the multinomial covariance. 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. Expected proportions

The central mathematical idea is concrete. Compute pi=xi/x or Ei/E+.

A concrete situation

p=(.25,.25,.25,.25).

What the mathematics does not license

Estimated baselines add uncertainty. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Refit in simulation. 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. Residual proportions

Here the geometry becomes operational. Subtract p from r.

A concrete situation

d=(.25,.05,−.15,−.15).

What the mathematics does not license

Signs carry spatial meaning. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Map signed residuals. 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. Ordinary chi-square component

This step deserves a careful pause. Compute N sum di²/pi.

A concrete situation

The toy Pearson component is 4.4.

What the mathematics does not license

Small expected counts weaken asymptotics. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Use simulation checks. 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. Spatial connections

The useful question is not only how to calculate, but what the calculation assumes. Define aij from distance or adjacency.

A concrete situation

Areas 1 and 2 are close.

What the mathematics does not license

One matrix may be arbitrary. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Justify the scale. 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. Diagonal weights

A small example makes the mechanism visible. Include self-terms so mismatch remains visible.

A concrete situation

Each region contributes its squared residual.

What the mathematics does not license

Different normalizations change magnitude. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Match the published form. 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. Off-diagonal weights

This is where modelling choices begin to affect evidence. Let nearby residuals interact.

A concrete situation

Positive d1 and d2 reinforce each other.

What the mathematics does not license

Opposite signs can reduce R. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Inspect pair 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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10. Weight matrix W

The central mathematical idea is concrete. Combine connection and expected-share scaling.

A concrete situation

W is symmetric in the toy analysis.

What the mathematics does not license

Asymmetric weights change theory. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Validate matrix properties. 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. Quadratic form

Here the geometry becomes operational. Calculate R=d′Wd.

A concrete situation

Matrix multiplication sums all weighted pairs.

What the mathematics does not license

Software orientation mistakes are easy. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Reproduce by hand. 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. Null moments

This step deserves a careful pause. Use the derived or simulated expectation and variance.

A concrete situation

Observed R is standardized against its null.

What the mathematics does not license

Approximation may fail for rare counts. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Prefer Monte Carlo validation. 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. Local contribution

The useful question is not only how to calculate, but what the calculation assumes. Decompose R by region or neighbourhood.

A concrete situation

Region one has the largest contribution.

What the mathematics does not license

Local values are dependent. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Adjust repeated inspection. 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. Scale sensitivity

A small example makes the mechanism visible. Repeat a small declared weight family.

A concrete situation

20,30,40 km thresholds give different R.

What the mathematics does not license

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

Try it, check it, transfer it

Use global control. 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. Residual diagnostics

This is where modelling choices begin to affect evidence. Compare observed map, Pearson residuals and spatial term.

A concrete situation

One outlier and one neighbour pair are visible.

What the mathematics does not license

A single bad denominator can mimic a cluster. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Audit 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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16. External validation

The central mathematical idea is concrete. Apply fixed W and baseline to held-out data.

A concrete situation

A later period retains the same residual grouping.

What the mathematics does not license

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

Try it, check it, transfer it

Investigate causes separately. 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

Rogerson’s spatial chi-square statistic starts with the same observed-versus-expected question as an ordinary goodness-of-fit test, then adds information about where deviations occur. Observed case proportions form r and expected population or risk proportions form p. Their difference is passed through a spatial weight matrix W, producing the quadratic form R=(r−p)′W(r−p).

The diagonal contribution measures within-region mismatch much like Pearson’s chi-square. Off-diagonal terms reward or penalize neighbouring residuals according to the chosen connection matrix, allowing several adjacent high deviations to matter differently from the same deviations scattered across the map. Distance-decay or adjacency weights, their normalization and the diagonal convention are therefore scientific choices.

With ten cases across four equal-expected regions, observed counts 5,3,1,1 give r=(.5,.3,.1,.1) and p=(.25,.25,.25,.25). The ordinary Pearson component is 10 times the sum of (ri−pi)²/pi, which equals 4.4. Rogerson’s spatial term then changes the result according to whether the positive residuals in the first two regions are close and how W is scaled.

A large R can arise from one severe mismatch, spatially coherent deviations, or both. The local version helps identify contributions, but local inspection introduces repeated testing. A responsible analysis maps residuals, publishes W, compares plausible distance ranges, checks denominator quality, calibrates the global and local statistics correctly, and describes a spatial pattern without converting it into an unsupported mechanism.

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Frequently asked questions

What is Rogerson’s R?

It is a quadratic form in observed-minus-expected regional proportions using a matrix that includes within-region and spatial connection terms.

How does it extend chi-square?

Diagonal terms capture goodness-of-fit while off-diagonal terms capture spatial alignment among deviations.

What are r and p?

ri is the observed share of all cases and pi is the expected share based on population or another baseline.

Why does W matter?

W determines which regions interact, at what strength and with what normalization.

What does a large R mean?

There is substantial regional mismatch, spatially coherent residuals, or both under the stated null.

Does the local version prove a hot spot?

No. It identifies contribution under the model and must be adjusted for repeated local inspection.

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Primary source and fact-check note

Peter A. Rogerson, “The Detection of Clusters Using a Spatial Version of the Chi-Square Goodness-of-Fit Statistic,” Geographical Analysis 31(2), 130–147 (1999), DOI 10.1111/j.1538-4632.1999.tb00973.x. 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

• Moran’s I spatial autocorrelation
• Tango’s C clustering statistic
• 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: Observed frequencies

Begin with this mechanism: Record Ni in every region. Recreate the concrete case: Counts are 5,3,1,1. Write regional counts, exposure quantities, observed proportions, expected proportions, residual vectors, ordinary chi-square terms, connection weights, spatial matrices, quadratic products, local contributions, null moments and tail 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 boundaries changes frequencies. Complete the transfer step: Version the geography. 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: Exposure quantities

Begin with this mechanism: Store population or another baseline xi. Recreate the concrete case: Four areas have equal population. Write regional counts, exposure quantities, observed proportions, expected proportions, residual vectors, ordinary chi-square terms, connection weights, spatial matrices, quadratic products, local contributions, null moments and tail 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 population may miss risk structure. Complete the transfer step: Standardize appropriately. 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: Observed proportions

Begin with this mechanism: Compute ri=Ni/N. Recreate the concrete case: r=(.5,.3,.1,.1). Write regional counts, exposure quantities, observed proportions, expected proportions, residual vectors, ordinary chi-square terms, connection weights, spatial matrices, quadratic products, local contributions, null moments and tail 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: Shares are dependent through N. Complete the transfer step: Use the multinomial covariance. 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: Expected proportions

Begin with this mechanism: Compute pi=xi/x or Ei/E+. Recreate the concrete case: p=(.25,.25,.25,.25). Write regional counts, exposure quantities, observed proportions, expected proportions, residual vectors, ordinary chi-square terms, connection weights, spatial matrices, quadratic products, local contributions, null moments and tail 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: Estimated baselines add uncertainty. Complete the transfer step: Refit in simulation. 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: Residual proportions

Begin with this mechanism: Subtract p from r. Recreate the concrete case: d=(.25,.05,−.15,−.15). Write regional counts, exposure quantities, observed proportions, expected proportions, residual vectors, ordinary chi-square terms, connection weights, spatial matrices, quadratic products, local contributions, null moments and tail 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: Signs carry spatial meaning. Complete the transfer step: Map signed residuals. 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: Ordinary chi-square component

Begin with this mechanism: Compute N sum di²/pi. Recreate the concrete case: The toy Pearson component is 4.4. Write regional counts, exposure quantities, observed proportions, expected proportions, residual vectors, ordinary chi-square terms, connection weights, spatial matrices, quadratic products, local contributions, null moments and tail 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: Small expected counts weaken asymptotics. Complete the transfer step: Use simulation checks. 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: Spatial connections

Begin with this mechanism: Define aij from distance or adjacency. Recreate the concrete case: Areas 1 and 2 are close. Write regional counts, exposure quantities, observed proportions, expected proportions, residual vectors, ordinary chi-square terms, connection weights, spatial matrices, quadratic products, local contributions, null moments and tail 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 matrix may be arbitrary. Complete the transfer step: Justify the scale. 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: Diagonal weights

Begin with this mechanism: Include self-terms so mismatch remains visible. Recreate the concrete case: Each region contributes its squared residual. Write regional counts, exposure quantities, observed proportions, expected proportions, residual vectors, ordinary chi-square terms, connection weights, spatial matrices, quadratic products, local contributions, null moments and tail 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 normalizations change magnitude. Complete the transfer step: Match the published form. 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: Off-diagonal weights

Begin with this mechanism: Let nearby residuals interact. Recreate the concrete case: Positive d1 and d2 reinforce each other. Write regional counts, exposure quantities, observed proportions, expected proportions, residual vectors, ordinary chi-square terms, connection weights, spatial matrices, quadratic products, local contributions, null moments and tail 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: Opposite signs can reduce R. Complete the transfer step: Inspect pair 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 10: Weight matrix W

Begin with this mechanism: Combine connection and expected-share scaling. Recreate the concrete case: W is symmetric in the toy analysis. Write regional counts, exposure quantities, observed proportions, expected proportions, residual vectors, ordinary chi-square terms, connection weights, spatial matrices, quadratic products, local contributions, null moments and tail 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: Asymmetric weights change theory. Complete the transfer step: Validate matrix properties. 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: Quadratic form

Begin with this mechanism: Calculate R=d′Wd. Recreate the concrete case: Matrix multiplication sums all weighted pairs. Write regional counts, exposure quantities, observed proportions, expected proportions, residual vectors, ordinary chi-square terms, connection weights, spatial matrices, quadratic products, local contributions, null moments and tail 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: Software orientation mistakes are easy. Complete the transfer step: Reproduce by hand. 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: Null moments

Begin with this mechanism: Use the derived or simulated expectation and variance. Recreate the concrete case: Observed R is standardized against its null. Write regional counts, exposure quantities, observed proportions, expected proportions, residual vectors, ordinary chi-square terms, connection weights, spatial matrices, quadratic products, local contributions, null moments and tail 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: Approximation may fail for rare counts. Complete the transfer step: Prefer Monte Carlo validation. 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: Local contribution

Begin with this mechanism: Decompose R by region or neighbourhood. Recreate the concrete case: Region one has the largest contribution. Write regional counts, exposure quantities, observed proportions, expected proportions, residual vectors, ordinary chi-square terms, connection weights, spatial matrices, quadratic products, local contributions, null moments and tail 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: Local values are dependent. Complete the transfer step: Adjust repeated inspection. 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: Scale sensitivity

Begin with this mechanism: Repeat a small declared weight family. Recreate the concrete case: 20,30,40 km thresholds give different R. Write regional counts, exposure quantities, observed proportions, expected proportions, residual vectors, ordinary chi-square terms, connection weights, spatial matrices, quadratic products, local contributions, null moments and tail 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: Weight shopping inflates evidence. Complete the transfer step: Use global control. 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: Residual diagnostics

Begin with this mechanism: Compare observed map, Pearson residuals and spatial term. Recreate the concrete case: One outlier and one neighbour pair are visible. Write regional counts, exposure quantities, observed proportions, expected proportions, residual vectors, ordinary chi-square terms, connection weights, spatial matrices, quadratic products, local contributions, null moments and tail 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 single bad denominator can mimic a cluster. Complete the transfer step: Audit 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 16: External validation

Begin with this mechanism: Apply fixed W and baseline to held-out data. Recreate the concrete case: A later period retains the same residual grouping. Write regional counts, exposure quantities, observed proportions, expected proportions, residual vectors, ordinary chi-square terms, connection weights, spatial matrices, quadratic products, local contributions, null moments and tail 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: Stability is not mechanism. Complete the transfer step: Investigate causes separately. 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.