Chapter map

1. 1. Aligned values
2. 2. Global mean
3. 3. Global scale
4. 4. Neighbour graph
5. 5. Gi versus Gi star
6. 6. Local weighted sum
7. 7. Weight sum
8. 8. Squared-weight sum
9. 9. Expected local sum
10. 10. Standardizing denominator
11. 11. Gi star z score
12. 12. Conditional permutation
13. 13. Pseudo p-value
14. 14. Multiple testing
15. 15. Scale sensitivity
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.

---

1. Aligned values

The central mathematical idea is concrete. One comparable number is attached to every location.

A concrete situation

Four areas have values 2, 3, 10 and 11.

What the mathematics does not license

Counts can reflect unequal population. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Use rates or model exposure when required. 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.

---

2. Global mean

Here the geometry becomes operational. The overall average anchors the local expectation.

A concrete situation

The mean is 6.5.

What the mathematics does not license

Outliers move the centre. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Inspect distributions and robust diagnostics. 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.

---

3. Global scale

This step deserves a careful pause. The population-form standard deviation enters the denominator.

A concrete situation

sqrt(234/4−6.5²) is about 4.031.

What the mathematics does not license

Zero variance makes the statistic undefined. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Check spread before mapping. 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.

---

4. Neighbour graph

The useful question is not only how to calculate, but what the calculation assumes. Weights encode local reach.

A concrete situation

The focal third area connects to itself and area four.

What the mathematics does not license

Post-hoc bandwidth choice inflates evidence. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Predeclare plausible scales. 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.

---

5. Gi versus Gi star

A small example makes the mechanism visible. The star convention determines whether self-weight enters.

A concrete situation

Gi* uses weights on areas three and four.

What the mathematics does not license

Software defaults differ. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Record the diagonal explicitly. 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.

---

6. Local weighted sum

This is where modelling choices begin to affect evidence. Multiply every value by its focal-row weight and add.

A concrete situation

10+11 gives 21.

What the mathematics does not license

Row standardization changes the sum. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Audit raw and transformed weights. 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.

---

7. Weight sum

The central mathematical idea is concrete. The first moment of the row is sum wij.

A concrete situation

Two binary weights sum to 2.

What the mathematics does not license

Islands have no ordinary neighbourhood. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Define island handling. 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.

---

8. Squared-weight sum

Here the geometry becomes operational. The second moment is sum wij squared.

A concrete situation

Two unit weights give 2.

What the mathematics does not license

Unequal kernels change effective neighbours. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Report both sums. 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.

---

9. Expected local sum

This step deserves a careful pause. The global mean times weight sum gives 13.

A concrete situation

Observed 21 exceeds 13.

What the mathematics does not license

High raw values alone are insufficient. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Compare neighbourhood support. 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.

---

10. Standardizing denominator

The useful question is not only how to calculate, but what the calculation assumes. Scale uses s sqrt((n sumw²−sumw²)/(n−1)).

A concrete situation

Here it is about 4.655.

What the mathematics does not license

Formula conventions depend on weights and null. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Match implementation documentation. 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.

---

11. Gi star z score

A small example makes the mechanism visible. Subtract expectation and divide by scale.

A concrete situation

(21−13)/4.655 is about 1.72.

What the mathematics does not license

A z score is not automatically a normal-theory p-value. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Check permutation 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.

---

12. Conditional permutation

This is where modelling choices begin to affect evidence. Hold the focal value and shuffle neighbour assignments.

A concrete situation

999 replicates produce a local reference.

What the mathematics does not license

Restrictions change the estimand. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

State the randomization scheme. 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.

---

13. Pseudo p-value

The central mathematical idea is concrete. Rank the observed score among simulations.

A concrete situation

Ten as-or-more-extreme results give about .011.

What the mathematics does not license

Resolution depends on replicates. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Include observed arrangement and seed. 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.

---

14. Multiple testing

Here the geometry becomes operational. Every location produces a hypothesis.

A concrete situation

One hundred areas yield one hundred scores.

What the mathematics does not license

Unadjusted .05 labels overstate discoveries. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use FDR or family-wise 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.

---

15. Scale sensitivity

This step deserves a careful pause. Repeat scientifically plausible distance bands.

A concrete situation

A pocket appears at 1 km but not 5 km.

What the mathematics does not license

Selecting the best map is cherry-picking. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Report the full sensitivity grid. 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.

---

16. External validation

The useful question is not only how to calculate, but what the calculation assumes. Hold out places or dates from tuning.

A concrete situation

The same pocket reappears next year.

What the mathematics does not license

Replication still is not causation. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

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

---

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

Getis–Ord local G statistics ask whether unusually high or low values gather around a location under an explicit spatial weights rule. Gi excludes the focal value; Gi* includes it through the diagonal. That small-looking distinction changes the numerator, expectation and interpretation, so a hotspot map is only meaningful when the weights and star convention are published.

The standardized Gi* score compares a local weighted sum with the amount expected from the whole data set. Its denominator uses the global standard deviation and the first two sums of the location's weights. A large positive z score describes a high-value pocket, while a large negative score describes a low-value pocket. It does not merely identify a large observation: neighbours must support the local concentration.

Permutation inference recomputes the statistic under a declared rearrangement scheme. Conditional permutations can hold the focal value fixed while shuffling its neighbours; total randomization answers a different question. Pseudo p-values depend on the alternative, replicate count and random seed, and local analysis creates a multiple-testing problem across locations.

Gi* supports hotspot questions in public health, ecology, retail, transport and spatial omics, but it cannot prove a mechanism. Population opportunity, unequal measurement effort, spatial trend, boundary choice and post-hoc distance selection can create impressive colours. Responsible analysis checks rates and denominators, compares plausible weights, controls multiplicity and validates the pattern in new regions or periods.

---

Frequently asked questions

What is Getis–Ord Gi*?

It is a standardized local statistic comparing a location’s weighted sum, including the focal value, with the global distribution.

What is a hotspot?

A hotspot is a location with a high positive standardized score supported by neighbouring high values under the stated weights and null.

How does Gi differ from Gi*?

Gi conventionally excludes the focal observation; Gi* includes it through a nonzero diagonal or equivalent star weight.

Why are weights important?

Distance bands, contiguity, kernels and row transformations change which values contribute and how strongly.

Why adjust for multiple comparisons?

One local test is performed at every location, so unadjusted significance labels create too many false discoveries.

Does Gi* prove a cause?

No. It detects local concentration under a selected spatial structure; exposure, sampling and trend remain alternative explanations.

---

Primary source and fact-check note

Arthur Getis and J. K. Ord, “The Analysis of Spatial Association by Use of Distance Statistics,” Geographical Analysis 24(3), 189–206 (1992), DOI 10.1111/j.1538-4632.1992.tb00261.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
• Geary’s C local contrasts
• 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: Aligned values

Begin with this mechanism: One comparable number is attached to every location. Recreate the concrete case: Four areas have values 2, 3, 10 and 11. Write values, means, standard deviations, weights, row sums, squared weights, weighted sums, expectations, z scores, permutation ranks and adjusted p-values 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: Counts can reflect unequal population. Complete the transfer step: Use rates or model exposure when required. 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: Global mean

Begin with this mechanism: The overall average anchors the local expectation. Recreate the concrete case: The mean is 6.5. Write values, means, standard deviations, weights, row sums, squared weights, weighted sums, expectations, z scores, permutation ranks and adjusted p-values 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: Outliers move the centre. Complete the transfer step: Inspect distributions and robust diagnostics. 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: Global scale

Begin with this mechanism: The population-form standard deviation enters the denominator. Recreate the concrete case: sqrt(234/4−6.5²) is about 4.031. Write values, means, standard deviations, weights, row sums, squared weights, weighted sums, expectations, z scores, permutation ranks and adjusted p-values 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 variance makes the statistic undefined. Complete the transfer step: Check spread before mapping. 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: Neighbour graph

Begin with this mechanism: Weights encode local reach. Recreate the concrete case: The focal third area connects to itself and area four. Write values, means, standard deviations, weights, row sums, squared weights, weighted sums, expectations, z scores, permutation ranks and adjusted p-values 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 bandwidth choice inflates evidence. Complete the transfer step: Predeclare plausible scales. 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: Gi versus Gi star

Begin with this mechanism: The star convention determines whether self-weight enters. Recreate the concrete case: Gi* uses weights on areas three and four. Write values, means, standard deviations, weights, row sums, squared weights, weighted sums, expectations, z scores, permutation ranks and adjusted p-values 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 defaults differ. Complete the transfer step: Record the diagonal explicitly. 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: Local weighted sum

Begin with this mechanism: Multiply every value by its focal-row weight and add. Recreate the concrete case: 10+11 gives 21. Write values, means, standard deviations, weights, row sums, squared weights, weighted sums, expectations, z scores, permutation ranks and adjusted p-values 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: Row standardization changes the sum. Complete the transfer step: Audit raw and transformed weights. 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: Weight sum

Begin with this mechanism: The first moment of the row is sum wij. Recreate the concrete case: Two binary weights sum to 2. Write values, means, standard deviations, weights, row sums, squared weights, weighted sums, expectations, z scores, permutation ranks and adjusted p-values 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: Islands have no ordinary neighbourhood. Complete the transfer step: Define island handling. 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: Squared-weight sum

Begin with this mechanism: The second moment is sum wij squared. Recreate the concrete case: Two unit weights give 2. Write values, means, standard deviations, weights, row sums, squared weights, weighted sums, expectations, z scores, permutation ranks and adjusted p-values 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: Unequal kernels change effective neighbours. Complete the transfer step: Report both sums. 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: Expected local sum

Begin with this mechanism: The global mean times weight sum gives 13. Recreate the concrete case: Observed 21 exceeds 13. Write values, means, standard deviations, weights, row sums, squared weights, weighted sums, expectations, z scores, permutation ranks and adjusted p-values 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: High raw values alone are insufficient. Complete the transfer step: Compare neighbourhood support. 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: Standardizing denominator

Begin with this mechanism: Scale uses s sqrt((n sumw²−sumw²)/(n−1)). Recreate the concrete case: Here it is about 4.655. Write values, means, standard deviations, weights, row sums, squared weights, weighted sums, expectations, z scores, permutation ranks and adjusted p-values 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: Formula conventions depend on weights and null. Complete the transfer step: Match implementation documentation. 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: Gi star z score

Begin with this mechanism: Subtract expectation and divide by scale. Recreate the concrete case: (21−13)/4.655 is about 1.72. Write values, means, standard deviations, weights, row sums, squared weights, weighted sums, expectations, z scores, permutation ranks and adjusted p-values 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 z score is not automatically a normal-theory p-value. Complete the transfer step: Check permutation evidence. 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: Conditional permutation

Begin with this mechanism: Hold the focal value and shuffle neighbour assignments. Recreate the concrete case: 999 replicates produce a local reference. Write values, means, standard deviations, weights, row sums, squared weights, weighted sums, expectations, z scores, permutation ranks and adjusted p-values 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: Restrictions change the estimand. Complete the transfer step: State the randomization scheme. 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: Pseudo p-value

Begin with this mechanism: Rank the observed score among simulations. Recreate the concrete case: Ten as-or-more-extreme results give about .011. Write values, means, standard deviations, weights, row sums, squared weights, weighted sums, expectations, z scores, permutation ranks and adjusted p-values with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Resolution depends on replicates. Complete the transfer step: Include observed arrangement and seed. 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: Multiple testing

Begin with this mechanism: Every location produces a hypothesis. Recreate the concrete case: One hundred areas yield one hundred scores. Write values, means, standard deviations, weights, row sums, squared weights, weighted sums, expectations, z scores, permutation ranks and adjusted p-values 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: Unadjusted .05 labels overstate discoveries. Complete the transfer step: Use FDR or family-wise control. 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: Scale sensitivity

Begin with this mechanism: Repeat scientifically plausible distance bands. Recreate the concrete case: A pocket appears at 1 km but not 5 km. Write values, means, standard deviations, weights, row sums, squared weights, weighted sums, expectations, z scores, permutation ranks and adjusted p-values with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Selecting the best map is cherry-picking. Complete the transfer step: Report the full sensitivity grid. 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: External validation

Begin with this mechanism: Hold out places or dates from tuning. Recreate the concrete case: The same pocket reappears next year. Write values, means, standard deviations, weights, row sums, squared weights, weighted sums, expectations, z scores, permutation ranks and adjusted p-values with units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Replication still is not causation. Complete the transfer step: Investigate exposures and mechanisms. 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.