Chapter map

1. 1. Observation window
2. 2. Point coordinates
3. 3. Intensity estimate
4. 4. Pairwise distance
5. 5. Radius indicator
6. 6. Ordered-pair count
7. 7. Basic K estimator
8. 8. CSR expectation
9. 9. L transformation
10. 10. L minus r
11. 11. Isotropic correction
12. 12. Translation correction
13. 13. Variable-width correction
14. 14. Simulation envelope
15. 15. Inhomogeneous K
16. 16. Marked and multitype extension

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

Here the geometry becomes operational. A bounded region defines where events could have been observed.

A concrete situation

A 10 by 10 square has area 100.

What the mathematics does not license

A convenient rectangle may exclude real habitat. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Map and 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. Point coordinates

This step deserves a careful pause. Every event has a location in stated units.

A concrete situation

Four cells have x-y centroids in micrometres.

What the mathematics does not license

Localization and duplicates affect short radii. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Audit precision and ties. 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. Intensity estimate

The useful question is not only how to calculate, but what the calculation assumes. n divided by area estimates homogeneous event density.

A concrete situation

Four points in area 100 give 0.04.

What the mathematics does not license

Intensity may vary across the window. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Map first-order trends. 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. Pairwise distance

A small example makes the mechanism visible. Euclidean separation is computed for distinct pairs.

A concrete situation

Points (0,0) and (3,4) are distance 5.

What the mathematics does not license

Barriers make Euclidean closeness misleading. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Choose a domain-relevant metric. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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5. Radius indicator

This is where modelling choices begin to affect evidence. A pair contributes one when dij≤r.

A concrete situation

At r=3, three unordered pairs qualify.

What the mathematics does not license

Threshold ties need a consistent rule. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Document ≤ versus <. 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. Ordered-pair count

The central mathematical idea is concrete. K sums i not equal j, so both directions enter.

A concrete situation

Three unordered pairs become six ordered pairs.

What the mathematics does not license

Forgetting direction changes the scale. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Check a hand example. 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. Basic K estimator

Here the geometry becomes operational. Area divided by n(n−1) scales the count.

A concrete situation

100/12 times 6 equals K=50.

What the mathematics does not license

No edge correction biases bounded windows. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Label this estimate uncorrected. 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. CSR expectation

This step deserves a careful pause. A planar homogeneous Poisson process gives πr².

A concrete situation

At r=3, CSR is about 28.27.

What the mathematics does not license

CSR assumes homogeneous independent events. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Test intensity before interaction. 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. L transformation

The useful question is not only how to calculate, but what the calculation assumes. sqrt(K/π) converts area to distance.

A concrete situation

K=50 gives L about 3.99.

What the mathematics does not license

Transformation does not remove bias. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Keep corrections and units visible. 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. L minus r

A small example makes the mechanism visible. Subtracting radius yields a zero CSR reference.

A concrete situation

3.99−3 is about 0.99.

What the mathematics does not license

A positive value is descriptive, not yet significant. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

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

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11. Isotropic correction

This is where modelling choices begin to affect evidence. Pair weights reflect visible circumference around a point.

A concrete situation

A boundary point receives a larger correction.

What the mathematics does not license

Complex windows complicate angles. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Validate 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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12. Translation correction

The central mathematical idea is concrete. Overlap of a window with its translated copy supplies weights.

A concrete situation

A pair displacement reduces overlap area.

What the mathematics does not license

It assumes a correctly specified window. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Compare corrections. 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. Variable-width correction

Here the geometry becomes operational. Only centres with enough boundary clearance contribute at r.

A concrete situation

A point closer than r to the edge may be excluded.

What the mathematics does not license

Effective sample size falls with r. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Plot contributing counts. 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. Simulation envelope

This step deserves a careful pause. Null patterns generate a collection of K or L curves.

A concrete situation

999 CSR patterns define pointwise quantiles.

What the mathematics does not license

Pointwise bands do not control all radii jointly. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Use a global envelope when claimed. 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. Inhomogeneous K

The useful question is not only how to calculate, but what the calculation assumes. Pair contributions are weighted by estimated local intensities.

A concrete situation

Dense habitat regions no longer automatically imply interaction.

What the mathematics does not license

Intensity estimation can overfit. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Cross-validate bandwidth. 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. Marked and multitype extension

A small example makes the mechanism visible. Labels support cross-K and mark-specific questions.

A concrete situation

KAB asks whether type B surrounds type A.

What the mathematics does not license

Abundance and label uncertainty confound. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Use label permutations and biological 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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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

Ripley’s K function asks how many other events lie within distance r of a typical event, after scaling by event intensity. Unlike a single nearest-neighbour distance, K traces spatial structure across many radii. For a homogeneous Poisson process under complete spatial randomness in the plane, the theoretical benchmark is pi times r squared. Values above that benchmark suggest more close pairs than CSR at that scale; values below suggest inhibition or regular spacing.

An estimator counts ordered pairs whose separation is no greater than r and multiplies by study-window area divided by n times n minus one. Boundary points lose potential neighbours outside the observed window, so uncorrected counts are biased downward. Isotropic, translation, variable-width and set-covariance corrections solve different versions of that missing-neighbour problem. The window and correction are part of the model, not display settings.

The transformation L(r)=sqrt(K(r)/pi) puts distance back into the original units. Plotting L(r)-r gives a zero benchmark under planar CSR. Simulation envelopes are constructed by generating patterns under a fully stated null model, recomputing the curve and comparing the observed function with the simulated collection. Pointwise envelopes do not automatically provide a global family-wise test across all radii.

K can support ecology, astronomy, epidemiology, materials analysis and cell-pattern research, yet inhomogeneous intensity can mimic clustering. Repeated points, anisotropy, obstacles, uncertain locations and the choice of maximum radius all matter. A careful study separates first-order intensity from second-order interaction, uses an inhomogeneous or marked extension when needed, repeats edge corrections and validates conclusions at scientifically meaningful scales.

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Frequently asked questions

What is Ripley’s K?

Ripley’s K is a cumulative second-order point-pattern statistic: after intensity scaling, it counts other events within radius r of a typical event.

What is the CSR benchmark?

For a homogeneous Poisson process in the plane, K(r)=πr².

Why is K a curve?

Computing it over many radii reveals whether apparent clustering or inhibition changes with scale.

Why correct edges?

Points near the observation boundary have unobserved neighbourhood area outside the window, which biases uncorrected pair counts.

What does L(r)−r show?

L=sqrt(K/π) returns distance units, so L−r has a zero CSR benchmark in two dimensions.

Does K above CSR prove interaction?

No. Inhomogeneous intensity, sampling design, shared habitat and localization error can all produce excess close pairs.

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Primary source and fact-check note

B. D. Ripley, “The Second-Order Analysis of Stationary Point Processes,” Journal of Applied Probability 13(2), 255–266 (1976), DOI 10.1017/S0021900200094328. 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

• Trendsceek marked point processes
• scNiche cellular microenvironments
• MILWRM regional domains
• 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

Window audit

Justify area, holes, barriers and observation effort.

Coordinate audit

Check units, duplicates and localization error.

Intensity audit

Separate first-order trend from interaction.

Radius audit

Predeclare meaningful range and effective counts.

Edge audit

Compare isotropic, translation and variable-width correction.

Envelope audit

Distinguish pointwise from global inference.

Extension audit

Use inhomogeneous or marked K when required.

Final claim

Excess close pairs do not alone prove interaction.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Observation window

Begin with this mechanism: A bounded region defines where events could have been observed. Recreate the concrete case: A 10 by 10 square has area 100. Write window area, coordinates, distances, radii, ordered-pair indicators, edge weights, intensities, K values, CSR expectations, L values, contributing counts and envelope ranks with shapes, 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 convenient rectangle may exclude real habitat. Complete the transfer step: Map and justify the window. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 2: Point coordinates

Begin with this mechanism: Every event has a location in stated units. Recreate the concrete case: Four cells have x-y centroids in micrometres. Write window area, coordinates, distances, radii, ordered-pair indicators, edge weights, intensities, K values, CSR expectations, L values, contributing counts and envelope ranks with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Localization and duplicates affect short radii. Complete the transfer step: Audit precision and ties. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 3: Intensity estimate

Begin with this mechanism: n divided by area estimates homogeneous event density. Recreate the concrete case: Four points in area 100 give 0.04. Write window area, coordinates, distances, radii, ordered-pair indicators, edge weights, intensities, K values, CSR expectations, L values, contributing counts and envelope ranks with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Intensity may vary across the window. Complete the transfer step: Map first-order trends. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 4: Pairwise distance

Begin with this mechanism: Euclidean separation is computed for distinct pairs. Recreate the concrete case: Points (0,0) and (3,4) are distance 5. Write window area, coordinates, distances, radii, ordered-pair indicators, edge weights, intensities, K values, CSR expectations, L values, contributing counts and envelope ranks with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Barriers make Euclidean closeness misleading. Complete the transfer step: Choose a domain-relevant metric. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 5: Radius indicator

Begin with this mechanism: A pair contributes one when dij≤r. Recreate the concrete case: At r=3, three unordered pairs qualify. Write window area, coordinates, distances, radii, ordered-pair indicators, edge weights, intensities, K values, CSR expectations, L values, contributing counts and envelope ranks with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Threshold ties need a consistent rule. Complete the transfer step: Document ≤ versus <. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 6: Ordered-pair count

Begin with this mechanism: K sums i not equal j, so both directions enter. Recreate the concrete case: Three unordered pairs become six ordered pairs. Write window area, coordinates, distances, radii, ordered-pair indicators, edge weights, intensities, K values, CSR expectations, L values, contributing counts and envelope ranks with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Forgetting direction changes the scale. Complete the transfer step: Check a hand example. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 7: Basic K estimator

Begin with this mechanism: Area divided by n(n−1) scales the count. Recreate the concrete case: 100/12 times 6 equals K=50. Write window area, coordinates, distances, radii, ordered-pair indicators, edge weights, intensities, K values, CSR expectations, L values, contributing counts and envelope ranks with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: No edge correction biases bounded windows. Complete the transfer step: Label this estimate uncorrected. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 8: CSR expectation

Begin with this mechanism: A planar homogeneous Poisson process gives πr². Recreate the concrete case: At r=3, CSR is about 28.27. Write window area, coordinates, distances, radii, ordered-pair indicators, edge weights, intensities, K values, CSR expectations, L values, contributing counts and envelope ranks with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: CSR assumes homogeneous independent events. Complete the transfer step: Test intensity before interaction. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 9: L transformation

Begin with this mechanism: sqrt(K/π) converts area to distance. Recreate the concrete case: K=50 gives L about 3.99. Write window area, coordinates, distances, radii, ordered-pair indicators, edge weights, intensities, K values, CSR expectations, L values, contributing counts and envelope ranks with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Transformation does not remove bias. Complete the transfer step: Keep corrections and units visible. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 10: L minus r

Begin with this mechanism: Subtracting radius yields a zero CSR reference. Recreate the concrete case: 3.99−3 is about 0.99. Write window area, coordinates, distances, radii, ordered-pair indicators, edge weights, intensities, K values, CSR expectations, L values, contributing counts and envelope ranks with shapes, 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 positive value is descriptive, not yet significant. Complete the transfer step: Use simulations. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 11: Isotropic correction

Begin with this mechanism: Pair weights reflect visible circumference around a point. Recreate the concrete case: A boundary point receives a larger correction. Write window area, coordinates, distances, radii, ordered-pair indicators, edge weights, intensities, K values, CSR expectations, L values, contributing counts and envelope ranks with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Complex windows complicate angles. Complete the transfer step: Validate geometry. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 12: Translation correction

Begin with this mechanism: Overlap of a window with its translated copy supplies weights. Recreate the concrete case: A pair displacement reduces overlap area. Write window area, coordinates, distances, radii, ordered-pair indicators, edge weights, intensities, K values, CSR expectations, L values, contributing counts and envelope ranks with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: It assumes a correctly specified window. Complete the transfer step: Compare corrections. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 13: Variable-width correction

Begin with this mechanism: Only centres with enough boundary clearance contribute at r. Recreate the concrete case: A point closer than r to the edge may be excluded. Write window area, coordinates, distances, radii, ordered-pair indicators, edge weights, intensities, K values, CSR expectations, L values, contributing counts and envelope ranks with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Effective sample size falls with r. Complete the transfer step: Plot contributing counts. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 14: Simulation envelope

Begin with this mechanism: Null patterns generate a collection of K or L curves. Recreate the concrete case: 999 CSR patterns define pointwise quantiles. Write window area, coordinates, distances, radii, ordered-pair indicators, edge weights, intensities, K values, CSR expectations, L values, contributing counts and envelope ranks with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Pointwise bands do not control all radii jointly. Complete the transfer step: Use a global envelope when claimed. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 15: Inhomogeneous K

Begin with this mechanism: Pair contributions are weighted by estimated local intensities. Recreate the concrete case: Dense habitat regions no longer automatically imply interaction. Write window area, coordinates, distances, radii, ordered-pair indicators, edge weights, intensities, K values, CSR expectations, L values, contributing counts and envelope ranks with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Intensity estimation can overfit. Complete the transfer step: Cross-validate bandwidth. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.

Workshop 16: Marked and multitype extension

Begin with this mechanism: Labels support cross-K and mark-specific questions. Recreate the concrete case: KAB asks whether type B surrounds type A. Write window area, coordinates, distances, radii, ordered-pair indicators, edge weights, intensities, K values, CSR expectations, L values, contributing counts and envelope ranks with shapes, units, denominators and assumptions. Calculate the smallest non-trivial example by hand and reproduce it in a spreadsheet or short program. Challenge this limitation: Abundance and label uncertainty confound. Complete the transfer step: Use label permutations and biological validation. Change exactly one value definition, coordinate unit, weights rule, radius, neighbour count, row transformation, self-weight, mean, outlier, permutation restriction, alternative hypothesis, window boundary, edge correction, intensity model, radius grid, simulation count, metric, feature scaling, filtration, coefficient field, simplex dimension, tie order, resampling unit or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep observation, preprocessing, spatial or metric structure, statistic, null model, computation, uncertainty, interpretation and independent validation on separate notebook lines. Reserve complete regions, time periods, specimens or patients outside tuning when testing transfer.