Chapter map

1. 1. Observation window
2. 2. Coordinates
3. 3. Point count
4. 4. Intensity
5. 5. Distance metric
6. 6. Nearest neighbour
7. 7. Tie handling
8. 8. Observed mean
9. 9. CSR benchmark
10. 10. R calculation
11. 11. Classical standard error
12. 12. Boundary bias
13. 13. Donnelly correction
14. 14. Monte Carlo test
15. 15. Intensity inhomogeneity
16. 16. Multiscale transfer

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. Observation window

The central mathematical idea is concrete. The region where every event could have been observed is explicit.

A concrete situation

A 10×10 square has area 100.

What the mathematics does not license

Convenient boundaries may omit habitat. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Justify the window. 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. Coordinates

Here the geometry becomes operational. Each event has location and units.

A concrete situation

Four points are measured in metres.

What the mathematics does not license

Rounding creates artificial ties. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Audit precision and duplicates. 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. Point count

This step deserves a careful pause. n is the number of distinct observed events.

A concrete situation

Here n=4.

What the mathematics does not license

Repeated records inflate density. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Deduplicate by design 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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4. Intensity

The useful question is not only how to calculate, but what the calculation assumes. lambda equals n divided by area.

A concrete situation

4/100=.04 per square metre.

What the mathematics does not license

Homogeneity may be false. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Map first-order intensity. 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. Distance metric

A small example makes the mechanism visible. The geometry defines separation.

A concrete situation

Euclidean distance is used on a flat plot.

What the mathematics does not license

Roads, barriers or spheres need other metrics. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Choose domain geometry. 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. Nearest neighbour

This is where modelling choices begin to affect evidence. For each event find the smallest positive distance.

A concrete situation

One point’s nearest partner is 2 m away.

What the mathematics does not license

Self-distance must be excluded. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Check indices. 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. Tie handling

The central mathematical idea is concrete. Equal nearest distances require a stable convention.

A concrete situation

Two points share the same nearest pair.

What the mathematics does not license

Ties affect neighbour identities but not distance mean. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Record multiplicities. 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. Observed mean

Here the geometry becomes operational. Average all nearest-neighbour distances.

A concrete situation

Distances 2,2,4,4 average 3.

What the mathematics does not license

A few isolated points pull the mean upward. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Plot the distribution. 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. CSR benchmark

This step deserves a careful pause. Expected mean is 1/(2√lambda).

A concrete situation

At .04 it equals 2.5.

What the mathematics does not license

The formula assumes homogeneous planar Poisson events. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

State the null. 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. R calculation

The useful question is not only how to calculate, but what the calculation assumes. Divide observed mean by expected mean.

A concrete situation

3/2.5=1.2.

What the mathematics does not license

R alone has no uncertainty statement. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Add a calibrated test or interval. 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. Classical standard error

A small example makes the mechanism visible. Asymptotic formulas scale with area and n.

A concrete situation

Larger samples narrow uncertainty.

What the mathematics does not license

Small irregular windows violate approximations. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Prefer simulation when uncertain. 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. Boundary bias

This is where modelling choices begin to affect evidence. Outside neighbours are unobserved for edge points.

A concrete situation

A corner point appears more isolated.

What the mathematics does not license

Naive R can suggest regularity. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use correction or inner guard zones. 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. Donnelly correction

The central mathematical idea is concrete. Finite-window expectations adjust mean and variance using boundary length.

A concrete situation

A rectangle uses area and perimeter.

What the mathematics does not license

Approximations may fail for holes or complex edges. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

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

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14. Monte Carlo test

Here the geometry becomes operational. Generate CSR patterns in the identical window and recompute R.

A concrete situation

999 patterns form a reference distribution.

What the mathematics does not license

Conditioning on n versus Poisson n differs. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Match the intended null. 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. Intensity inhomogeneity

This step deserves a careful pause. A gradient can create short distances without interaction.

A concrete situation

Events concentrate where habitat is suitable.

What the mathematics does not license

Homogeneous CSR confounds trend and interaction. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use inhomogeneous models. 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. Multiscale transfer

The useful question is not only how to calculate, but what the calculation assumes. Compare R with K, pair correlation and held-out samples.

A concrete situation

R suggests regularity and K confirms it only at short radii.

What the mathematics does not license

One-scale agreement still is descriptive. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Validate with process 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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A seven-day practice lab

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

Questions for a student, parent or mentor

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

A method-specific audit before deployment

The Clark–Evans nearest-neighbour index compresses a point pattern into the mean distance from each event to its closest other event, then compares that mean with the expectation under complete spatial randomness. Values below one indicate shorter-than-CSR nearest-neighbour distances; values above one indicate longer distances and more regular spacing.

For a homogeneous planar Poisson process with intensity lambda, the expected nearest-neighbour distance is 1 divided by twice the square root of lambda. With four events in area 100, lambda is 0.04 and the benchmark is 2.5 distance units. If observed nearest-neighbour distances average 3, R equals 1.2.

The elegant formula assumes an effectively unbounded or appropriately corrected window. Points near a boundary have potential neighbours outside the observed region, so naive mean distances can be too large. Donnelly-style corrections, guard zones, toroidal corrections or simulation inside the exact window address different designs; obstacles and irregular habitat still need explicit geometry.

The index is useful in ecology, facility placement, cell patterns and materials science, yet it describes only the nearest scale. Two patterns can share R while differing strongly at larger radii, intensity gradients or subgroup relationships. A careful study audits duplicates and units, corrects boundaries, compares simulations, complements R with K or pair-correlation curves and validates any mechanistic interpretation.

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Frequently asked questions

What is the Clark–Evans index?

R is observed mean nearest-neighbour distance divided by the complete-spatial-randomness expectation at the observed intensity.

How is R interpreted?

R below one suggests clustering at the nearest-neighbour scale; R above one suggests inhibition or regular spacing.

What is the CSR expectation?

In an unbounded homogeneous planar Poisson process it is 1/(2√λ), where λ is events per unit area.

Why do edges matter?

Boundary points may have unseen neighbours outside the window, making observed nearest distances too large.

Is R a multiscale statistic?

No. It summarizes only each point’s closest neighbour; K, L or pair-correlation functions examine broader distances.

Does R reveal mechanism?

No. Intensity trend, observation effort, obstacles and point uncertainty can produce the same ratio.

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Primary source and fact-check note

Philip J. Clark and Francis C. Evans, “Distance to Nearest Neighbor as a Measure of Spatial Relationships in Populations,” Ecology 35(4), 445–453 (1954), DOI 10.2307/1931034. Definitions and core claims in this article were checked against this primary publication. Calculations shown here were independently recomputed. Examples are educational illustrations; they are not reported industrial performance results.

Useful next reading

• Ripley’s K point patterns
• Persistent homology for data shape
• 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: Observation window

Begin with this mechanism: The region where every event could have been observed is explicit. Recreate the concrete case: A 10×10 square has area 100. Write window area, perimeter, coordinates, density, nearest distances, mean distance, CSR expectation, R values, edge corrections 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: Convenient boundaries may omit habitat. Complete the transfer step: Justify the window. 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: Coordinates

Begin with this mechanism: Each event has location and units. Recreate the concrete case: Four points are measured in metres. Write window area, perimeter, coordinates, density, nearest distances, mean distance, CSR expectation, R values, edge corrections 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: Rounding creates artificial ties. Complete the transfer step: Audit precision and duplicates. 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: Point count

Begin with this mechanism: n is the number of distinct observed events. Recreate the concrete case: Here n=4. Write window area, perimeter, coordinates, density, nearest distances, mean distance, CSR expectation, R values, edge corrections 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: Repeated records inflate density. Complete the transfer step: Deduplicate by design rules. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 4: Intensity

Begin with this mechanism: lambda equals n divided by area. Recreate the concrete case: 4/100=.04 per square metre. Write window area, perimeter, coordinates, density, nearest distances, mean distance, CSR expectation, R values, edge corrections 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: Homogeneity may be false. Complete the transfer step: Map first-order intensity. 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: Distance metric

Begin with this mechanism: The geometry defines separation. Recreate the concrete case: Euclidean distance is used on a flat plot. Write window area, perimeter, coordinates, density, nearest distances, mean distance, CSR expectation, R values, edge corrections 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: Roads, barriers or spheres need other metrics. Complete the transfer step: Choose domain geometry. 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: Nearest neighbour

Begin with this mechanism: For each event find the smallest positive distance. Recreate the concrete case: One point’s nearest partner is 2 m away. Write window area, perimeter, coordinates, density, nearest distances, mean distance, CSR expectation, R values, edge corrections 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: Self-distance must be excluded. Complete the transfer step: Check indices. 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: Tie handling

Begin with this mechanism: Equal nearest distances require a stable convention. Recreate the concrete case: Two points share the same nearest pair. Write window area, perimeter, coordinates, density, nearest distances, mean distance, CSR expectation, R values, edge corrections 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: Ties affect neighbour identities but not distance mean. Complete the transfer step: Record multiplicities. 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: Observed mean

Begin with this mechanism: Average all nearest-neighbour distances. Recreate the concrete case: Distances 2,2,4,4 average 3. Write window area, perimeter, coordinates, density, nearest distances, mean distance, CSR expectation, R values, edge corrections 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: A few isolated points pull the mean upward. Complete the transfer step: Plot the distribution. 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: CSR benchmark

Begin with this mechanism: Expected mean is 1/(2√lambda). Recreate the concrete case: At .04 it equals 2.5. Write window area, perimeter, coordinates, density, nearest distances, mean distance, CSR expectation, R values, edge corrections 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: The formula assumes homogeneous planar Poisson events. Complete the transfer step: State the null. 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: R calculation

Begin with this mechanism: Divide observed mean by expected mean. Recreate the concrete case: 3/2.5=1.2. Write window area, perimeter, coordinates, density, nearest distances, mean distance, CSR expectation, R values, edge corrections 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: R alone has no uncertainty statement. Complete the transfer step: Add a calibrated test or interval. 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: Classical standard error

Begin with this mechanism: Asymptotic formulas scale with area and n. Recreate the concrete case: Larger samples narrow uncertainty. Write window area, perimeter, coordinates, density, nearest distances, mean distance, CSR expectation, R values, edge corrections 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: Small irregular windows violate approximations. Complete the transfer step: Prefer simulation when uncertain. 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: Boundary bias

Begin with this mechanism: Outside neighbours are unobserved for edge points. Recreate the concrete case: A corner point appears more isolated. Write window area, perimeter, coordinates, density, nearest distances, mean distance, CSR expectation, R values, edge corrections 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 R can suggest regularity. Complete the transfer step: Use correction or inner guard zones. 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: Donnelly correction

Begin with this mechanism: Finite-window expectations adjust mean and variance using boundary length. Recreate the concrete case: A rectangle uses area and perimeter. Write window area, perimeter, coordinates, density, nearest distances, mean distance, CSR expectation, R values, edge corrections 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: Approximations may fail for holes or complex edges. Complete the transfer step: Simulate exact geometry. Change exactly one value, class label, threshold, edge, weight, distance band, self-weight, graph transformation, window boundary, opportunity denominator, candidate radius, maximum cluster size, simulation count, coordinate unit, metric, edge correction, alternative hypothesis or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, periods, specimens or patients outside tuning when testing transfer.

Workshop 14: Monte Carlo test

Begin with this mechanism: Generate CSR patterns in the identical window and recompute R. Recreate the concrete case: 999 patterns form a reference distribution. Write window area, perimeter, coordinates, density, nearest distances, mean distance, CSR expectation, R values, edge corrections 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: Conditioning on n versus Poisson n differs. Complete the transfer step: Match the intended null. 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: Intensity inhomogeneity

Begin with this mechanism: A gradient can create short distances without interaction. Recreate the concrete case: Events concentrate where habitat is suitable. Write window area, perimeter, coordinates, density, nearest distances, mean distance, CSR expectation, R values, edge corrections 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: Homogeneous CSR confounds trend and interaction. Complete the transfer step: Use inhomogeneous models. 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: Multiscale transfer

Begin with this mechanism: Compare R with K, pair correlation and held-out samples. Recreate the concrete case: R suggests regularity and K confirms it only at short radii. Write window area, perimeter, coordinates, density, nearest distances, mean distance, CSR expectation, R values, edge corrections 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: One-scale agreement still is descriptive. Complete the transfer step: Validate with process 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.